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Complex Systems - Jean-Philippe Bouchaud - Lecture 6: Hawkes processes. Networks. — Transcript

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0:00this conference will now be recorded
0:06press the record button button
0:09and like last time sorry for that
0:13okay good so
0:15last time I told you about branching
0:17processes and Galton Watson model
0:20today I want to talk to you about horse
0:22processors
0:24I'll give you an introduction to network
0:26Theory speaking about yeah there's many
0:29networks which in a sense are very close
0:32to the Galton Watson process that I
0:35talked about and I'll expand on that and
0:38also the very famous by now but people
0:41call scale free Networks
0:44and then I'll talk about the giant
0:47components in these Networks and their
0:50resilience and this has a lot to do with
0:52uh
0:53uh efficiency of vaccination campaigns
0:56and all these uh
0:58interesting topics
1:00Okay so
1:02let me talk about hoax processes and
1:05I'll start by reminding you what
1:08question processes are
1:14so a question process is a sequence of
1:17events okay and this is the time axis
1:20and personal process are events that
1:23occur at random times so what does it
1:26really mean it means that
1:30um if you introduce
1:33accounts of the number of events that
1:36happen
1:37up to time T then each time something
1:41happens this n of P
1:43increased by one okay
1:46so you just count the number of events
1:48that occurred and you have something
1:50like this where each jump occurs as
1:54where one of these events takes place
1:57and so the randomness is defined by the
2:00fact that
2:01the probability that DN is equal to one
2:06is equal to at time T is equal to Lambda
2:11DT
2:12where Lambda is called the rate
2:16of the person process
2:19okay
2:21so this is very simple I mean things
2:23happen at random and on every little
2:26time interval DT there's a probability
2:28Lambda DT for something to happen and of
2:31course the probability that nothing
2:32happens
2:33is
2:361 minus Lambda DT
2:40okay
2:42so it's a very well known object and I'm
2:45sure many of you have seen it uh on many
2:48occasions so let me remind you a few
2:51basic properties of this question
2:54process
2:55so for example you can ask well I'm
2:58giving myself a time interval
3:01capital T
3:03okay
3:04and the question I'm asking is what is
3:07the probability P of n knowing T that I
3:12observed exactly n events in this
3:15interval of length capital T okay so in
3:19this particular case I observe six
3:21events
3:22but more generally I can ask the
3:24question and it's a little exercise for
3:27you if you want to derive it but it's
3:30very easy to show that this is given by
3:33what's called a poisson distribution
3:36so it's Lambda and Lambda t to the n
3:41exponential of minus Lambda T capital T
3:44divided by
3:46factorial n okay
3:49so this is just a consequence of the
3:52definition of the question process
3:54there's no correlation whatsoever
3:57between events and therefore you get you
4:00get this so for example
4:03and I've said that already several times
4:06before the average value of n is Lambda
4:10capital T this is pretty intuitive if
4:13Lambda is the rate of events per unit
4:15time so if you want this is the units is
4:18second minus one
4:20then in an interval of time of of length
4:24capital p uh they are on average Lambda
4:27T events
4:29and another property of personal
4:31processes is that the variance is equal
4:34to the mean
4:36so average value of N squared minus the
4:39average value of n
4:41squared is also Lambda T not squared
4:44Lambda t
4:47and so this leads to introduced to the
4:51introduction of a quantity that's that's
4:54quite interesting to consider in
4:57particular for practical application
4:59which is what I'm going to call the
5:01clustering ratio
5:10row and I'm going to define the
5:13clustering ratio by the the ratio of the
5:16variance
5:19over the mean
5:24and of course I'm going to go
5:27too far so let me move my camera already
5:31foreign
5:40process this is equal to one
5:45okay
5:47another interesting quantity that one
5:50can compute uh for personal processes
5:54is
5:55the insert time distribution so if I
5:58call S
5:59the difference between two consecutive
6:01events
6:03then it's easy again to show that P of s
6:07the probability to observe a certain
6:09interval between two consecutive events
6:12is
6:13a LaPlace distribution Lambda
6:16exponential minus Lambda s
6:21okay so these are my basic observations
6:25about
6:26personal processes
6:29shouldn't go too high
6:35and I'm going to tell you about various
6:39objects that are not personal processes
6:42and that need something different to be
6:46uh modeled with
6:57sorry I'm trying to find them right
7:00height
7:02okay
7:06a little bit
7:16so
7:18let me give an example of something that
7:22Ashley has a clustering ratio less than
7:26one
7:27and
7:28um and examples of things that have a
7:31clustering ratio larger than one
7:33so intuitively
7:35clustering ratio
7:40less than one
7:42means that the fluctuations
7:46are smaller than you would expect for a
7:48personal process okay so the way these
7:52events occur tend to be more regular
7:54than for a question process imagine for
7:58example that you have a perfectly
8:00regular set of events so these events
8:03occur not randomly but you know like
8:05every second exactly then
8:09for a very large T very large capital T
8:11there would be no fluctuations at all in
8:13the number of events and so the
8:16clustering ratio would be less than one
8:20so clustering less than one means that
8:23there's some kind of repulsion
8:27between events
8:29foreign
8:37so if something happens now
8:42it's less likely that something is going
8:44to happen just after that
8:47and a very well-known example of a set
8:50of points that follows
8:52um a repulsion type of process
8:57is the the set of eigenvalues of a
9:01random Matrix so if you take a large
9:03random Matrix and you diagonalize this
9:05random Matrix you can plot
9:08the eigenvalue
9:10like in quantum mechanics and if I plot
9:14the eigenvalues of a random Matrix
9:18I will find something that's much more
9:20regular than the personal process
9:22and this is called uh
9:26level repulsion
9:33and there's a long story about this in
9:36the
9:37Nuclear Physics literature fast this was
9:41introduced by wigner
9:43and and of course all
9:46the massive developments of random
9:48Matrix theory in the last 50 years a lot
9:51of that is about the precise description
9:54of the set of points so I won't go much
9:58into details but that's an example of
10:00clustering ratio lesson one
10:03and of course
10:05you know
10:07taking the the opposite of what I just
10:10said if you have a clustering ratio
10:12greater than one
10:14it means that uh fluctuations are larger
10:18than for a question process and this
10:21means that you have what's called
10:22clustering
10:29so on the contrary there are correlation
10:31between events if something happens it's
10:34more likely than for a personal process
10:37that something else happens very soon
10:38after and because of that you will have
10:42bursts of activity
10:44and then activity that's lower
10:48in such a way that the fluctuations of
10:50this of the number of events in the time
10:52interval is greater than one okay
10:57and so if you do this very simple test
11:00for several time series you realize that
11:03a lot of Time series that we are
11:06interested in in this uh in these
11:08lectures
11:09actually
11:10show some clustering properties so for
11:14example I told you about earthquakes
11:22and if you look at the time series
11:24of earthquakes then there's clear
11:27clustering I mean it's what people call
11:29aftershocks for example so we know that
11:32once an earthquake has happened somehow
11:35the crust the the crust of the earth
11:39it's rationalized and this leads to a
11:42higher probability to observe more
11:44earthquakes and in in what I'm going to
11:48talk to you about today hoax processors
11:50actually they were introduced by Mr
11:53Hawks in 72
11:56and uh it was in the context of
12:00uh earthquakes Dynamics
12:06so
12:07it is uh
12:10fair to associate as an example of
12:13clustering time series earthquakes as
12:15the primary ones but I told you about
12:18financial markets financial markets is
12:21also a prime example
12:24of clustering
12:27of activity so for example the activity
12:30in the financial markets is every time
12:32the price of a stock changes for example
12:35you can put a mark when the price
12:40changes
12:41and of course this um the the the means
12:45value of the activity rate the Lambda
12:48has increased over the years because as
12:51you know now uh people trade at the
12:54higher frequency so Lambda itself has
12:57increased but apart from this increase
13:00of activity when see something that has
13:02always been the case
13:04before uh high frequency trading before
13:08the electronization of markets and since
13:11then that hasn't changed is the fact
13:13that when the price when the price
13:15changes it's actually more probable that
13:18the price is going to change again very
13:20soon after
13:21and so you have this very very strong
13:23clustering I'm going to go back to that
13:26later
13:28in social sciences there's a very
13:31interesting story as well and hoax
13:33processors are used to model earthquake
13:37financial markets and some social
13:39sciences phenomena
13:43so for example a a case that's been
13:47studied empirically is is the case of
13:49riots
13:54so when you see
13:56um the way the Dynamics of Rise takes
13:58place it is clear that there is also uh
14:02in as in earthquakes in a sense the fact
14:04that when a riot has occurred then in
14:08the nearby City another ride tends to
14:10happen uh very close after and this was
14:13actually studied in detail in the case
14:16of what happened in France in 2005 there
14:19was a wave of riots uh some of you may
14:23remember in the suburbs of Paris and
14:26there is a very strong signal of
14:29clustering of these events
14:33okay so a lot of example of clustering
14:37starting from earthquakes and this has
14:40led Hawks to
14:43um
14:44provide a model
14:46a simple model for these clustering
14:48events
14:53that are named own
14:56these models as a box processors
15:15foreign
15:30processes are also called subordinated
15:35in a way that I'm going to define or
15:38donated
15:41question processes
15:44so what does that mean well it simply
15:46means that
15:48the personal process that I've
15:50considered up tuna had a fixed rate of
15:54events Lambda
15:56now I'm going to
15:57imagine that this person process has uh
16:01time varying
16:03rate of of events so
16:06the probability that DNT is equal to one
16:08is equal to Lambda T DT okay
16:14where Lambda T is now time dependent and
16:17the way it's time dependent is going to
16:20depend on what happened in the past
16:22so what hoax postulates
16:26is that Lambda t
16:28is equal to some
16:30base rate value mu
16:34which is going to be constant
16:36Plus
16:38the sum over all past events
16:42so I'm going to write TJ is the time at
16:45which the JS Event happened and so it's
16:49the sum Over All J's such that TJ is
16:52less than T of a sudden kernel k
16:56of T minus t j
17:01and K
17:03is the Hulk's kernel as a function of
17:06the lag Tau between
17:08the event that happened in the past and
17:10now
17:11this kernel K of tar
17:14is a certain decaying function
17:18for example
17:21exponential
17:25so for example I'm going to take a
17:28just after that
17:31a specific example where K of tar is
17:34exponential of minus
17:36Alpha Tau
17:38but it could be something else
17:40so what it means is that
17:43you're very affected by events that just
17:46happened
17:47which increase the probability of having
17:50something happening again
17:52but far away events so when T minus DJ
17:55becomes
17:57in this case much larger than one over
17:59Alpha then you tend to forget
18:02these past events okay
18:06so this is if you want
18:08what people it's called an influence
18:11kernel as well
18:15which means to insist on the fact that
18:18it's the influence of past events on the
18:21current level of activity
18:23and mu is a Baseline
18:27level
18:30okay
18:33so there's another way to rewrite this
18:36equation which in some cases is the
18:39useful so let me rewrite this as an
18:42identity as Mu the same Plus instead of
18:46having a discrete sum I'm going to write
18:49this as an integral it's minus integral
18:52from minus infinity to T
18:54of GNT Prime
18:58k
18:59of T minus t Prime
19:04okay and you see it's the same because
19:07BNT Prime this is equal to uh one if
19:12something happened
19:16if if there's an event at C Prime and
19:19zero if not okay
19:25so let me give you a few uh properties
19:28of this
19:29um
19:31of this model
19:33the first property is what happens for
19:36the average rate of events in this model
19:39so you see that Lambda now is time
19:42dependent
19:44so
19:46if I if I
19:47make a little drawing
19:50of Lambda t as a function of t
19:54then Lambda T is going to look look like
19:57this a little bit
20:01okay
20:02and so you already see uh clustering by
20:06the eye because there are moments where
20:08Lambda is going to be larger so many
20:11things will happen around these times
20:13but I can Define the average value of
20:16Lambda
20:18across time
20:21so that would be what I call Lambda bar
20:23okay this is the average level of
20:26activity
20:27when I average across time
20:30so one can show that in certain
20:33condition this hoax process is ergotic
20:36which means that I can think of Lambda
20:39bar either as a Time average or as a as
20:43an ensemble average in the sense that
20:45you see that things are all random here
20:48things happen at random so Lambda T is a
20:52random quantity
20:58it's random because it depends on random
21:00events and these random events are the
21:04underlying question process so you see
21:06why we one why this is called the
21:09subordinated platform process is because
21:11conditional to Lambda T is a personal
21:14process but it is subordinated on
21:18another process which in this case self
21:21consistently depends on the process
21:22itself okay
21:26so
21:27uh
21:29what happens if I try to compute uh lamb
21:32that the average value of Lambda well
21:35let me take the average of this equation
21:37here
21:40and what I get
21:42on the left hand side is Lambda bar by
21:44definition
21:46and on the right hand side I have mu
21:48plus and here I'm going to use this
21:52um expression
21:54and in this expression I'm going to note
21:56that the average value of BT DNC Prime
22:00this is equal
22:02to Lambda bar DG Prime
22:06well TT
22:10okay
22:11so
22:13what I get by take make taking the
22:15average of both sides of this equation
22:17is
22:20an equation where I get Lambda bar in
22:22both sides of the equation
22:30this is what I get
22:32okay
22:34and so I'm going to give a name
22:37to this integral over T Prime
22:41so you see first that I I can always
22:43change variables I can always call T
22:45minus t Prime Tau
22:48like I did in my drawing
22:50and this integral
22:52over T Prime from minus infinity T to T
22:55of K of T minus P Prime this is also
22:58equal to Lambda bar
23:01integral from 0 to Infinity D Tau KF
23:06Style
23:08okay
23:11so let me rewrite this equation
23:18here
23:38so what I get is Lambda bar
23:41equals mu
23:43plus Lambda Bar times what I'm going to
23:47call r0
23:48and of course I'm not using this
23:51notation uh randomly with r0
23:56which is defined as the integral from 0
23:58to Infinity
23:59beta of K of Tau
24:04and therefore you see that Lambda bar
24:07is equal to Mu over 1 minus r0
24:14okay well if you see this expression you
24:17see that it can only make sense if r0 is
24:20less than one
24:27because otherwise you find the Lambda
24:29bar which is negative which doesn't make
24:31sense and in this case when r0 is
24:34greater than one it means that the
24:36feedback is so strong that actually the
24:39process diverges
24:41there's no stationary State here I've
24:44assumed I got this t as I've mentioned
24:46before and this assumes that the process
24:49enters some kind of stationary state but
24:52if r0 is larger than one then each event
24:56affects so strongly the future rate of
24:59events that the process kind of gets
25:03self-excited so much that it runs the
25:06way to Infinity
25:08and actually I've used the term that's
25:10often used in this context which is
25:13self-excited process
25:17okay
25:23so K of tau is the self-excitation
25:26feedback which makes the the system uh
25:31interesting but potentially unstable
25:34so if r0 is greater than one
25:39the ox process is unstable
25:55so there's a lot of things that one can
25:57do with these uh polks processes they're
26:00they're actually very very nice from a
26:02mathematical point of view you can
26:03compute all sorts of things exactly
26:06for example you can compute correlation
26:09functions here I've computed the average
26:11but I can also compute correlation
26:13functions and correlation functions are
26:16going to be needed if you want to
26:18measure the variance because a variance
26:21is directly related to the second the
26:24two point correlation function of the
26:25process
26:27so one can do that one can actually very
26:30easily calibrate this process on data
26:32and that's why it's very successful in
26:35the literature there's a long list now
26:37of Time series on which people have
26:40tried to fit
26:42uh hoax processes
26:45and in particular what's really
26:46interesting about
26:49um
26:50the the two point function is the
26:53clustering ratio
26:55which is a summary of the two point
26:57function if you want and in this case
27:01it's actually uh well not completely
27:05trivial but one can show and I I'm not
27:08showing here
27:09so let me insist that it's not proven
27:11here
27:14and those who are interested uh I can
27:17give you some literature but it's not
27:19completely trivial actually
27:21is that for hoax processes row is given
27:24by one over one minus r0
27:28squared
27:34so what's nice about this result
27:38is that as you see rho this clustering
27:41ratio is independent
27:46of mu
27:48and
27:50of KF Tau
27:53except from the norm of K of Tau so what
27:57I'm saying is that
27:58you don't care about the detailed shape
28:01of K of Tau the only thing you care
28:03about
28:04except
28:06r0 of course
28:08so once you know the the norm of K
28:12then you don't need to know anything
28:15else you don't need to know mu either
28:17so you don't need to know the the
28:19underlying
28:21um uh based level Baseline level of
28:25activity you can derive the value of r0
28:29from the measurement of row only okay
28:32because for example imagine that you say
28:35but here I maybe that's a way to measure
28:38r0 well it's not because that priority
28:41you don't know what mu is okay
28:43so what's nice about the
28:46clustering ratio is that all reference
28:49to the details of the hoax process that
28:52is Mu or the detail shape of K is
28:56actually disappeared so you can have an
28:59idea of the value of r0 just but by
29:02looking at the clustering ratio
29:05so let me give you two limits
29:08one is the limit where r0 goes to zero
29:13so what happens if kl0 goes to zero well
29:15you see that if r0 goes to zero it means
29:18that somehow
29:20there's no feedback and what we recover
29:23is the poisson result row tensor on
29:28but what's more interesting is the case
29:31when r0 a goes to one
29:35by the way you know I haven't insisted
29:38on that but clearly this formula shows
29:41that it's always greater or equal to one
29:44right
29:45so there's always clustering in the Box
29:47processes
29:49which is what we wanted we wanted to
29:52describe
29:53um uh a clustering a model with
29:58clustering and let me come back to
30:00something I haven't said I should have
30:02said before in a second which is I'm
30:05assuming that K is is strictly positive
30:07here
30:08and if K is negative then the model is
30:12not very well defined so here I should
30:15have said that
30:17k
30:19is
30:20is positive
30:22so it's really something that describes
30:25self-excitation and not self-inhibition
30:30um
30:31so when all zero goes to 1 then you see
30:33from the formula that rows goes to
30:35Infinity
30:39and so what's surprising
30:42is that if you calibrate the hoax
30:44process to financial markets you find
30:50that
30:52r0 is indeed very close to one
31:02and this is interesting because it means
31:04that somehow financial markets are
31:07extremely fragile you see that our zero
31:10equal one again is the uh is the
31:13boundary between the stable process and
31:16an unstable process if the process would
31:19be slightly more
31:21self-exciting then it would be unstable
31:23and for some reason which are not which
31:27is not completely clear even at this
31:29stage
31:30financial markets if you calibrate them
31:33with a question with a hoax process
31:36systematically gives gives you values of
31:39r0 which are close to one suggesting
31:43um you know
31:45close to instability that
31:48financial markets are
31:50toys at the verge of instability
31:58so of course you could say well maybe
32:00it's because the Hawks process is not a
32:01good model for financial markets and
32:03that by calibrating your hoax model
32:05which is not the right model you get of
32:08zero close to one and that may well be
32:10but I'm just pointing that out to you
32:14which is a very interesting I think
32:16observation is that when calibrated to
32:21financial markets
32:23one finds a a kind of incipient
32:27instability
32:28from the model
32:31which goes actually hand in hand with
32:34many things I told you about financial
32:35markets already it seems that you know
32:38there are parallel distributions
32:40crashes happen all the time so maybe all
32:43this is related to some kind of
32:45intrinsic instability
32:48um
32:51of financial markets at least this is a
32:54this is something that researchers are
32:56investigating
32:58now is trying to understand whether or
33:02not there are good reasons for financial
33:04markets to be close to a critical point
33:10okay so let me finish this chapter on
33:14Hawke's processes by
33:16uh giving you
33:19the example of
33:21KF Tau
33:25exponential so if I write K of pi equal
33:28r0
33:29uh Alpha exponential of minus Alpha Tau
33:35then I've normalized it correctly in
33:38such a way that
33:40um it starts at r0 Alpha here
33:44and the area is is given by r0
33:50then let me give again
33:52the the equation Lambda T is Mu plus
33:56integral up to T
33:58of um
34:00DT DNT Prime
34:04K of T minus D Prime
34:11and the exponential uh distribution the
34:14exponential shape of K of tau is an
34:16interesting property which is always you
34:19know the things that work well with
34:21exponential is that if I take the
34:24derivative of this equation with respect
34:27to time
34:28then
34:31e Lambda T DT
34:34then I'm not going to you know detail
34:37the calculation it's it's uh easy enough
34:39to do what you get is
34:43minus Alpha
34:46Lambda T minus mu
34:50this comes from the the the derivative
34:54of K with respect to T and using the
34:58fact that K is an exponential which
35:00actually reproduces Lambda T minus mu
35:04and then Plus
35:07r0
35:12Alpha
35:15GNT DT
35:21and so what we know is that DNT
35:25is a is is a conditional question
35:28process
35:29so GNT
35:33can be written as a deterministic part
35:37which is related to the average value of
35:40VNT which by definition is Lambda T DT
35:44plus the fluctuating part
35:47and the fluctuating part so if you if
35:51you think of this process in a kind of
35:52coarse grain manner you see that the
35:55average value of the NT conditions to
35:57Lambda T is Lambda T DT but there are
36:00fluctuations
36:01and the fluctuations of a personal
36:03process is proportional out proportional
36:06to the rate of event so the the
36:09fluctuations can be written as
36:12PSI some noise
36:14square root of Lambda T DT
36:18so here I'm cheating a little bit
36:19because
36:20um
36:21this this is not very rigorous but I'm
36:23writing I mean DNT is either zero or one
36:26so in order to go from here to there I'm
36:30kind of course grading the system and
36:32applying a kind of central limit theorem
36:34to DNT so you see what I'm going and
36:37what I'm trying to give you here is not
36:38a mathematical proof although everything
36:41that I'm going to say can be shown to be
36:44rigorous in some limits but it's just a
36:48an intuition about what's going on in
36:50this model so what I'm saying is that
36:52this dntdp is going to be is going to
36:55have an average value Lambda T and
36:58fluctuation
37:00and so if I follow this thing at least
37:03naively
37:04what you get is that there's a piece
37:08here of this guy which is going to add
37:12to this one
37:14and a piece that's going to remain as a
37:17noise
37:18so again
37:20I'm not dictating the calculation that
37:22what you get by regrouping terms
37:26is minus alpha 1 minus r0
37:30Lambda T minus Lambda bar
37:35where Lambda bar is the Lambda bar that
37:37I defined before it's it's it's mu
37:40divided by 1 minus r0
37:45so actually here Lambda Bar times 1
37:48minus r0 is just the MU guy that you had
37:51here
37:52and the Lambda t one minus r0 comes from
37:56this sum plus the one coming from there
38:01Plus
38:03are zero
38:05Alpha
38:07square root of Lambda t
38:10times the noise PSI
38:15a large band noise let me call it ETA
38:18this is a larger noise
38:24you remember larger noise are
38:25ill-defined there of all the one over
38:27square root of DT and this is what you
38:30would get from here dividing by DT gives
38:33you a noise that's one over square root
38:35of DT so it's a large amount of noise
38:39okay so that's where I wanted to uh
38:43to arrive at because what you see is
38:47that you have a differential equation
38:49which is such that
38:52there's an average
38:54so what does this stochastic
38:56differential equation mean it means that
38:58Lambda T is fluctuating around Lambda
39:02bar you see this is a mean reversion
39:04term this is a kind of harmonic
39:05calculate oscillator term which
39:08a pool's Lambda T back to its average
39:11value so we already know that we already
39:13know that Lambda T fluctuates around an
39:17average value Lambda T but the strength
39:20of the reversion
39:22is decreased as r0 approaches one
39:28so if you want to see it simply you see
39:32that alpha 1 minus r0 is the time scale
39:35of the of the mean reversion
39:37you see dimensionally one alpha one
39:40minus r0 is the frequency
39:42and so this is the relaxation time
39:46or the inverse relaxation time
39:52and the in-house relaxation time is 1
39:54over Alpha One minus r0
40:00so this is you know if if I write here
40:03in a corner something that
40:07if I write a harmonic oscillator D
40:10Lambda T equals minus Omega
40:12Lambda T minus Lambda bar
40:15plus noise
40:18this is also called the Ornstein
40:21ullenbeck process then one knows that
40:24one over Omega is the characteristic
40:26time scale of the relaxation of such a
40:29process you can show it very easily so
40:32what plays the role of Omega here is
40:35Alpha One minus r0 so I get the
40:37relaxation time that is 1 over alpha 1
40:40minus r0
40:41so what it means is that when r0 goes to
40:43one
40:48you have two things happening at the
40:50same time you have that Lambda bar
40:53diverges
40:57so the process becomes more and more
40:58intense
40:59but at the same time
41:01the relaxation time of the process
41:04diverges as well
41:07so relaxation time
41:11also
41:15so this is this is a process this is a
41:17phenomenon that's very usual in
41:20statistical mechanics model where the
41:23approach to a critical point because r0
41:25equal one is a critical point it's a
41:27point Beyond which the model is unstable
41:31then at the same time there's something
41:34that diverges which here is the average
41:38rate of events so in a physical system
41:41it's often for example the
41:42susceptibility of the system that
41:44diverges and at the same time the
41:47relaxation time the speed of at which
41:50the system relaxes to equilibrium also
41:53diverges and that's what you see uh
41:56simply coming out
41:58of this equation and that was not clear
42:01from the
42:03quantities I've told you about earlier
42:06here one has on top of these results on
42:11Lambda bar and rho one has a result on
42:13the relaxation time
42:15so how this folks processes
42:19evolve with time really and we see that
42:22close to criticality they actually
42:23becomes very slow
42:27so the other thing I wanted to tell you
42:31is that there's a very close analogy
42:33between hoax processes
42:35and uh
42:37the
42:38Gelson Watson branching process that
42:40I've told you about one way to think
42:44about it is just looking at this
42:46equation here and this equation in a
42:50very precise mathematical way can be
42:53shown to mean that there are ancestors
42:57arriving in the system at right mu
42:59and then these ancestors they give rise
43:02to children
43:03and they give rise to Children exactly
43:06like in the Calvin Watson process and so
43:10what you get what you gain compared to
43:12the Galton Watson process here is that
43:15you have some time Dimension that allows
43:18you to to know exactly when these
43:21children get born if you want but in
43:24terms of the structure of the families
43:27um it's everything that happens
43:29underlying this model is the same as
43:31what happens in the gelton Watson
43:33process
43:35and one way to realize that this is the
43:37case
43:38is looking at this equation again
43:42and if you remember I told you that the
43:45Galton Watson process in the continuing
43:47continuous time limit had a term a
43:50growth term proportional to
43:52r0 minus one Lambda which is what I
43:56wrote last time I wrote something like
43:57oh 0 minus one
43:59n so that was the vndt
44:04and then a fluctuation term which is
44:07proportional to square root of n
44:11so I told you last time this is the
44:13continuous time limit of the Gauss and
44:14Watson process and you see that it's uh
44:17very very similar here
44:19so it's just a hint that these two
44:22models are very closely related
44:25I realized that there's something I
44:26didn't tell you uh to justify my
44:30my Mumble here about taking a a cause
44:35graining I mean in making this bold
44:39interpretation of the NDT as a noise
44:43actually it becomes more and more
44:45Justified to do this as the process
44:47becomes slower and slower so when r0
44:50goes to 1 as I just said the relaxation
44:53time goes to Infinity the system becomes
44:55slower and slower and so I I it it is
44:59possible to make a change of scale
45:02to read to cosgrain the system if you
45:04want and to actually think of this ENT
45:07as a gaussian process in the limit
45:09because the process is very slowly
45:12evolving so on the time scale of the
45:15evolution of the process I can aggregate
45:17many of these DNT which are only zero or
45:21one and by aggregating them I get what I
45:25wrote here which is a The Continuous
45:28time gaussian process
45:30so for those of you who are worried by
45:32my Cavalier way of handling this term
45:35it's actually everything is is well
45:37defined and justified in the limit when
45:39r0 goes to one and in that limit the
45:42process is equivalent to a critical
45:45branching process
45:51okay so let me leave hooks processes and
45:55go to networks now
45:57um the the only thing I wanted to tell
46:00you before leaving them is that hooks
46:03processes can be generalized in many
46:05ways so for example you can have a
46:09multi-dimensional hoax process you can
46:11have you can add indices you can have
46:15n
46:17processes happening simultaneously so
46:19for example imagine the activity of n
46:22different stocks in the stock market
46:24and then you can add
46:28vectorial
46:35and a matrix for generalization of these
46:38talks processes saying that what happens
46:40on I is affected by what happened on J
46:43mediated by an influence kernel that now
46:46is becomes a matrix
46:48in the stock space
46:50and has some time dependence so the
46:54whole artillery can be generalized to
46:57many cases and this is one of the case
47:01that's been considered in the literature
47:04okay
47:26so hoax processes are important because
47:28they can be calibrated to data and
47:30they're important because they are very
47:32closely related to
47:34branching processes
47:39so now let me
47:43leave or apparently leave what I told
47:47you about branching processes but you'll
47:49see that will soon recover them in
47:52another uh
47:54Incarnation and go to
47:58Network Theory
48:09so
48:10the reason people are very interested in
48:14networks is that their networks
48:17everywhere around us and they're
48:19extremely relevant in many uh
48:22situations for example electric grids or
48:26networks of cables and nodes which are
48:30the the places where electricity is is
48:33produced
48:34there's a physical example that you all
48:36know well which which is polymer gels
48:40for example uh the white of an egg is a
48:44liquid when the polymers are independent
48:47but then when you heat uh white the
48:51white of an egg then these polymers
48:54um
48:55attached to each other
48:57they create a kind of network
49:00and once the network is completed the
49:03object is is solid it's it's not not a
49:07liquid anymore and this is related to
49:09the phenomenon of calculation which I'm
49:12going to talk about so in the in the
49:14case of uh the why the the the white of
49:17an egg it's really interesting because
49:18one sees a transition that will comment
49:22later between a case where the network
49:25is made of disconnected clusters
49:28and it's in the case of Jag it's a
49:30liquid and the case when there is a
49:33giant component
49:35where the the network includes a very
49:39large number of different molecules and
49:43in the physical sense it's a it's a
49:45solid
49:47so social networks of course no need to
49:50speak about them social networks whether
49:53they are virtual like Facebook or other
49:56type of social network or actual
49:59physical Networks
50:00networks created by contacts between
50:03people and you know your social network
50:06in the sense of who you meet every day
50:09is of course extremely important in in
50:12in view of uh of pandemic transmission
50:15and so on
50:17but there are also economic networks
50:20like banking networks pool lens to whom
50:23some networks who produces what so
50:27that's called the input output Network
50:29some firms produce Goods that are useful
50:32for other firms who themselves produce
50:35uh things so there's a network of
50:39interaction between firms there are
50:42ecological Networks
50:44so some species eat other species so
50:48there's a predator for prey relation
50:49between species and all these networks
50:53are crucial to understand emerging
50:56phenomena
50:57because depending on the on on the
51:00property of the underlying networks you
51:01can have in the case of kovitz for
51:04example you can have either propagation
51:07across the whole population or only
51:10limited clusters of infection and in the
51:13case of the wife of an egg as I said you
51:16can either be in a liquid phase or in a
51:20in a solid state there are many many
51:22other examples like
51:24and if you put conductors at random in
51:29space you only have little islands of
51:31conductors that overall do not conduct
51:33electricity or is there a giant
51:36component which allows the whole thing
51:38to become carrying electricity from one
51:42side to the other
51:43of the samples
51:45so there's a huge
51:47um
51:49incentive to understand the properties
51:52of networks in general
51:55so a network is made of nodes
52:01okay so I have nodes
52:04and I have links or edges
52:07okay
52:10so for example this is a network that
52:14contains two clusters
52:20so I'm going to use the terms edges or
52:23links
52:24to mean the same thing
52:28and so you see here that
52:30an ensemble of nodes that are
52:33such that I can go from anyone to any
52:37other one following edges is called the
52:40cluster
52:45and the question is do I have in my
52:48network a lot of small independent
52:51clusters or do I have on top of small
52:54clusters do I have a giant cluster which
52:57in this case would be something like
53:00this which involves
53:02maybe not all the nodes but at least the
53:05finite fraction of all the nodes
53:08so I will speak more about giant
53:10components later but you have an
53:12intuitive feeding already at this level
53:14is to know whether there is a cluster
53:18that covers a finite fraction of the
53:21number of nodes in the limit Square this
53:23number goes to infinity or if I only
53:25have you know either isolated guys and
53:29of course I can have a mixture of both
53:30things for example in this case I would
53:33have a big cluster and an isolated
53:35cluster but of course this is hand
53:37waving and one has to make more precise
53:40statement later on okay
53:44so I'm going to give you two ways to
53:46construct these networks one
53:50is the simplest and the oldest random
53:54construction of network which is called
53:57the the random range the Erdos Raini
54:00Networks
54:07and then
54:08I'll give you another type of
54:10construction which leads to in a sense
54:12more interesting graphs which are scale
54:15free Networks
54:18okay so another training Network how's
54:20how does that work
54:22well so imagine that you have
54:27end node
54:35I pick at random a pair of nodes for
54:38example this one and this one i j
54:42and for each of these possible pair I
54:45decide that there is a link
54:47with probability p over n
54:50and there is no link with probability 1
54:53minus t over n
54:55okay
54:58so this is my Construction
55:01it's only independent every link is
55:03independent
55:04I mean every potential link is
55:06independent
55:07and it becomes filled with probability p
55:10over n
55:12and it's left empty with probability 1
55:14minus p over n
55:16so you might ask why do I put a p over n
55:19here
55:21well it's because I want the model to
55:23remain well defined in the limit when n
55:25goes to Infinity
55:27and if you think for two seconds you
55:29realize that for each node there are n
55:33minus one possible Neighbors
55:36and since each
55:38Edge is present with product DP over n
55:42the average degree
55:45so the average number of Neighbors
55:56is given by P over n times n minus 1
56:00. so for large n
56:03this is equal to p
56:07so in this model p is the average
56:09connectivity of the graph or the average
56:12number of Neighbors
56:15okay
56:16yes
56:21no no p is the p is of all the one I'm
56:23going to speak about this so the
56:25question was if p is smaller than one so
56:27here p
56:30is the older one it's no other one
56:33number but as we're going to see it can
56:35be either smaller or larger than one and
56:38as you've anticipated is going to be
56:41important
56:52so in the rest of these lectures I'm
56:55going to call K
56:58the degree
57:00of a node
57:04okay
57:05which is the number of Neighbors
57:08and so what I just shown is that the
57:11average value of K is equal to p
57:17but you can be a little more greedy and
57:21ask what is the full distribution of of
57:24K
57:25which I'm going to call P of K
57:28this is a distribution degree
57:31distribution
57:38foreign
57:41well
57:43if you want to know how many neighbors a
57:46certain node has
57:49it's going to be
57:51the number of ways to choose K among n
57:54minus one
57:56times t t divided by n to the K 1 minus
58:00P divided by n to the N minus 1 minus K
58:04so it's a binomial distribution
58:07but I'm not redoing the calculation it's
58:09always the same calculation and the
58:11limit when n goes to Infinity this
58:12binomial distribution becomes a personal
58:15distribution
58:16and so what I get is that in this case
58:19in the other shrenic case this is the
58:22question distribution that we've seen
58:24already today
58:26which is p to the K exponential of minus
58:29p
58:30divided by factorial k okay
58:34so this is a very simple graph
58:36and it has a very simple distribution
58:39which is a personal distribution
58:42and note that this distribution decays
58:45extremely quickly when K increases
58:49so you know if you want a numerical
58:52example
58:53if I take P equal 1
58:57so the average degree is one
58:59then the probability to find
59:03a node with 10 Neighbors which is not
59:05that big it's just 10 times the average
59:08this is already like 10 to the minus 7.
59:14so the personal distribution is a little
59:17bit like the gaussian distribution that
59:19I talked about is it's extremely thin
59:21distribution thin tail distribution that
59:23decays incredibly quickly as K increases
59:27and therefore it's a graph where
59:30there's no hub
59:32there's no site that has a very large
59:36degree that has a very large number of
59:39of of of
59:41of neighbors or friends if you think
59:44about
59:45a a social network
59:48and we know from empirical data that
59:51many graphs are actually uh scale free
59:54in the sense that they have a parallel
59:56distribution of degrees some nodes have
1:00:00an incredibly large number of of
1:00:02neighbors While others have a small
1:00:04number of Neighbors
1:00:06so I'll go back to that in in the next
1:00:10paragraph on scale free networks but let
1:00:13me tell you a little more about
1:00:17um what happens in the other ready
1:00:20Network
1:00:21well
1:00:23although everything looks simple
1:00:26there's still something non-trivial that
1:00:28happens
1:00:29in the regime where pay p is of all the
1:00:32one
1:00:33and what happens was is related to the
1:00:36question that someone asked in the room
1:00:37we have two
1:00:39students today in the room
1:00:42and pleases some of you want to come and
1:00:45we can organize the
1:00:48um that others take that turn so what
1:00:51happens is that there's a phase
1:00:53transition in Elder training networks
1:00:55which is that when p is less than one
1:00:58there are only
1:01:03finite5 clusters
1:01:11so all this is of course in the limits
1:01:13where n goes to Infinity
1:01:17so in when p is less than one I only
1:01:20have you know things like this
1:01:22isolated nodes or nodes that are
1:01:25connected to a near
1:01:26a few other nodes in the in the same
1:01:30cluster but if p is greater than one
1:01:34there exists a giant component
1:01:38so GC
1:01:39is going to be giant component
1:01:45which means that
1:01:48if I pick a node at random
1:01:51there is a probability P Infinity
1:01:54which is greater than zero that this
1:01:57node belongs to an infinite cluster okay
1:02:03so it doesn't mean that
1:02:06this property is equal to one I may
1:02:08still find nodes that are in isolated
1:02:11clusters in in finite5 clusters even
1:02:14nodes that are completely alone okay by
1:02:18the way the probability
1:02:19for K to be zero
1:02:22is is always non-zero it's the it's
1:02:25exponential of minus p
1:02:27so there are always isolated class
1:02:30isolated nodes we have zero is
1:02:33exponential of minus B so
1:02:36clearly even when T is greater than one
1:02:38there are isolated nodes but there is
1:02:41also an infinite size cluster
1:02:45which is called a giant component or
1:02:48it's also called in physics a
1:02:50percolation cluster
1:02:58so you'll see these two names depending
1:03:00on the literature giant component or
1:03:02percolation cluster but it means the
1:03:04same and in my egg white analogy it's
1:03:09when p is greater than one then when you
1:03:12pull on a node you know imagine that
1:03:14you're actually taking a an optical
1:03:17tweezer and grabbing a polymer and try
1:03:20to pull on it well if you're in the P
1:03:23less than one phase you're going to only
1:03:25pour finite clusters
1:03:27so it's going to be easy and the thing
1:03:29is a liquid so it is the reason why it's
1:03:32it's easy
1:03:33but if you're in the
1:03:36um calculating phase then by taking a
1:03:39node at random you might have to pull
1:03:41the entire system
1:03:43which means that it's a solid okay so
1:03:47this is not a trivial
1:03:52um effect that that happens in these
1:03:55others ready Networks
1:03:57and
1:03:59I'm going to speak more about
1:04:02the existence of giant components why is
1:04:04it P equal 1 that makes the difference
1:04:07here and the stability of these giant
1:04:10components a little later
1:04:21so this was my first construction of a
1:04:24random Network
1:04:26the other shreni which is simple enough
1:04:28but as I said it's disappointing quote
1:04:31unquote because there are a lot of
1:04:34networks for which this personal
1:04:36distribution of degree falls short of
1:04:39explaining what's going on I mean as I
1:04:41said as I said the probability of having
1:04:43a node that's highly connected is so
1:04:46small that you're never going to be able
1:04:48to describe social networks for example
1:04:52foreign
1:04:54so let's look at another Construction
1:05:19skill free
1:05:22Networks
1:05:27so these construction these
1:05:30constructions are meant to uh generate
1:05:33much broader distribution of degrees and
1:05:37the most famous of them
1:05:39is due to
1:05:42um
1:05:43who physicists
1:05:48and all but
1:05:54and that's a model from 2000
1:05:59and it's an incredibly popular model
1:06:02with this this paper of biology Albert
1:06:05has a amazing number of citations I I
1:06:08don't know it always already has like
1:06:10several tens of thousands citations it's
1:06:13cited in physics in computer science in
1:06:16biology and in many different fields
1:06:19uh by the way I told you at the
1:06:21beginning of these lectures that if you
1:06:22look at the distribution of
1:06:24uh the citations a paper have it's a
1:06:28very broad parallel distribution and
1:06:30this clearly is one of the paper
1:06:32contributing to the tale of uh the
1:06:36distribution of citations
1:06:39and by the way citations can be thought
1:06:41of as
1:06:43the network you know if you put a link
1:06:45between a paper and another paper when
1:06:49the the second paper is citing the first
1:06:51then you construct a network and that's
1:06:54a little bit the type of models that
1:06:56we're going to construct here
1:06:59so
1:07:01what I'm going to
1:07:02tell you about is a model that
1:07:05is grown dynamically so there's some
1:07:09kind of Dynamics
1:07:13it's a dynamical model
1:07:18and it works as follows
1:07:21so I'm going to start at T equals 0
1:07:25with a single node
1:07:28then at t equal 1
1:07:31I'm adding a node
1:07:33and with this node
1:07:35I'm adding a link so each new node adds
1:07:39one Link in the system
1:07:42and of course at this stage there's no
1:07:44choice the link has to go to the
1:07:47previously present node
1:07:50then at t equal to
1:07:53I already have these two guys
1:07:56and I'm adding one more node and this
1:07:59node has to connect to one of them
1:08:02and in this case they have the same
1:08:04degree each node already present node
1:08:08has degree one
1:08:09and so in this case it's going to be
1:08:12with probability one half one half I'm
1:08:15connecting to one of these two nodes so
1:08:18at this level nothing much happens but
1:08:20then at T equals three
1:08:25when I'm adding
1:08:26the fourth node
1:08:30then the rule will be that I'm attaching
1:08:33the new Edge preferentially to nodes
1:08:36with high degrees
1:08:38and the rule will be that I'm attaching
1:08:41with a rule with a property that's
1:08:43proportional to the already existing
1:08:46number of of uh neighbors so to the
1:08:49already existing degree
1:08:52so here you see that I have three
1:08:55choices but the probability to connect
1:08:57to this thing will be twice the
1:08:59probability to connect to these uh other
1:09:03two guys so I have a probably two-thirds
1:09:05to be connected to this one and one
1:09:08third to be connected to one of these
1:09:11two
1:09:12and so you see what what's going to
1:09:14happen so sometimes I'm going to connect
1:09:16to this one but sometimes I'm going to
1:09:18connect to that one
1:09:19and because I'm connecting to that one
1:09:21with higher probability is again going
1:09:25to increase its degree and the next time
1:09:28step is going to attract even more uh
1:09:31links so it's a kind of Rich get richer
1:09:34effect
1:09:46and if you are following these lectures
1:09:48since the beginning
1:09:50uh you have already
1:09:53um
1:09:55recognized
1:09:57proportional growth models because in a
1:10:01sense the more you have grown
1:10:04the more likely it's going it is going
1:10:06to be that you grow again
1:10:08and we know already that these types of
1:10:10models tend to generate power laws
1:10:15Okay so
1:10:17at time t
1:10:22I have a bunch of nodes
1:10:24and the first node has
1:10:28um
1:10:29K1 Neighbors
1:10:31the second node is K2 Neighbors
1:10:34and so on
1:10:36so
1:10:39I have these uh
1:10:43these degrees
1:10:45and what I know is that each time and I
1:10:48add a node a node
1:10:51I also add a link and this link connects
1:10:54to two nodes
1:10:55and therefore the total
1:10:58degree of the network
1:11:00is twice the number of links I've added
1:11:04you see here for example I have K1 equal
1:11:081 K2 equal 1. so the sum over I of k i
1:11:14is equal in general to 2 times t equal
1:11:18to 2 here it's 4 here and so on okay
1:11:24and the rule of the game will be that I
1:11:27attach
1:11:28to the ice side
1:11:31probability to attach
1:11:36to I
1:11:38is equal to KI
1:11:40divided by the sum of the kis which is
1:11:432T
1:11:46and so this is what's called
1:11:48preferential attachment
1:11:57and precisely it's preferential in a
1:12:00linear way the more degree you have the
1:12:03more likely it is to um
1:12:07become the target of the new incoming
1:12:10node
1:12:12so
1:12:13you can ask why do I choose the strict
1:12:16linear uh relation here
1:12:20well I'm going to give you a
1:12:23generalizations of this result later on
1:12:26at least discuss them quickly but it
1:12:29seems reasonable that in many cases
1:12:32your popularity is is going to be linear
1:12:36for example if you think about papers
1:12:39the more a paper is cited the more
1:12:42likely it is that you're going to see it
1:12:44cited in a paper you read and the more
1:12:47likely it is that you're actually going
1:12:49to cite your paper the same paper as
1:12:51well in your own paper
1:12:54so of course this has to be decided on
1:12:56an empirical basis but it turns out that
1:12:59in many cases this is not a bad
1:13:01approximation
1:13:06Okay so
1:13:08now how does it work
1:13:12thank you
1:13:16[Music]
1:13:25so the first thing I want to emphasize
1:13:28is that this dynamical process is
1:13:31stochastic it's not a deterministic
1:13:33process
1:13:35you see each time step there's a Droid
1:13:38I said here I've chosen to connect to
1:13:41the most likely side to side but of
1:13:44course with probability uh I'm sorry
1:13:46I've said something wrong here you
1:13:48should have stopped me
1:13:50this doesn't sound too
1:13:52one
1:13:55that's the problem when
1:14:00so this is once half one fourth
1:14:04one fourth
1:14:08surprised nobody shouted but anyway
1:14:11so what I'm saying is that there's a
1:14:14higher probability to connect to that
1:14:15one but it could have been that in the
1:14:17in another history in another world I
1:14:20would have connected to this one
1:14:22okay and then this one and this one
1:14:25would have had the same degree at the
1:14:28name next time step so I'm generating an
1:14:32ensemble of graph that are not all the
1:14:34same
1:14:36and so I can speak about averaging over
1:14:40histories in this model okay
1:14:44and that's what I'm going to introduce
1:14:46now I'm going to call n of k and t
1:14:51this is the average number
1:14:56a side of nodes
1:15:01with degree k
1:15:05at time t
1:15:10and it's the average over what
1:15:13it's the average over all possible
1:15:15histories of the construction
1:15:17okay
1:15:20and so having understood that it's an
1:15:23average of a construction it's very easy
1:15:25to
1:15:27come up with a recursion relation for
1:15:30nfk and T so let me write it
1:15:33and then
1:15:35I will comment
1:15:38and justify
1:15:40what I'm claiming is that n of k and t
1:15:44plus one is n of k and t
1:15:47plus K minus 1
1:15:50divided by 2T
1:15:52n of K minus 1
1:15:55and T
1:15:58minus K Over 2T
1:16:01n of k and t
1:16:05plus Delta of K and 1.
1:16:11so this is the fundamental equation the
1:16:14master equation if you want
1:16:18that
1:16:20um I want you to understand so what does
1:16:23it mean it means that
1:16:25in order to have K degree k at time t
1:16:29plus one
1:16:30then either you uh then you had the
1:16:34number of sites is
1:16:36at time T is n of k and t but you can
1:16:39add to this number of sides that have
1:16:41degree K by connecting the new incoming
1:16:45node to a side that had degree K minus
1:16:49one
1:16:49and this happens with probability K
1:16:52minus 1 divided by 2T remember this is
1:16:56this is this
1:16:57so on average is going to be the
1:16:59probability to connect to a site to a
1:17:02node of the green K minus 1 times the
1:17:04number the average number of such sites
1:17:08but if you connect to a site
1:17:11to node with degree k then you're taking
1:17:14away some of these nodes from the count
1:17:19so there are less at time C plus one
1:17:22there are less nodes with degree K and
1:17:24this occurs before DK over to T so
1:17:27there's a minus sign here and then every
1:17:29time you add a node you add a degree
1:17:34which is equal to one because the
1:17:36incoming node always has a degree called
1:17:38one see this guy here it has a only one
1:17:41Edge going out
1:17:43and um so this explains the Delta of K
1:17:47and one here
1:17:52so what can we say about this equation
1:17:54well
1:17:56it's very natural to think that
1:17:59the number of nodes
1:18:01of degree k
1:18:04at large times
1:18:09well first let me write it this way it's
1:18:11it's natural to write it like that
1:18:14T times P of k and t
1:18:19where because the num the total number
1:18:21of uh of sites is
1:18:27sorry I'm using sites and nodes to mean
1:18:31the same thing so the number of nodes is
1:18:33t plus one
1:18:35so maybe I should write t plus one here
1:18:39and doing this I'm defining not the
1:18:42number but the the probability this
1:18:44thing is the probability
1:18:46that
1:18:48a node
1:18:50has degree k
1:18:57so that's a definition if you want but
1:18:59then my assumption will be that when T
1:19:02goes to Infinity
1:19:05assumption which is very natural in a
1:19:08way
1:19:09is that when t
1:19:12goes to Infinity
1:19:15P of k and t
1:19:17converges
1:19:19through a stationary distribution which
1:19:21I'm calling Peak star of K
1:19:24okay
1:19:25so I'm repeating this process over and
1:19:27over again
1:19:28so the number of sides the number of
1:19:31nodes is increasing
1:19:33so I take care of that by introducing
1:19:36this Factor t or t plus 1 here
1:19:39and then at very long times I'm
1:19:42expecting this distribution to reach a
1:19:44stationary state
1:19:46so with these two items
1:19:50this one and this assumption I can plug
1:19:53into
1:19:55um
1:19:56the the master equation above and find
1:19:59an equation for the stationary State
1:20:01itself
1:20:02and so what you find
1:20:04again I'm not dictating the calculations
1:20:07but it it's really easy to do that just
1:20:10take this plug it in here
1:20:13and then assume this to be true and then
1:20:16what you find is that P star of K obeys
1:20:18the following discrete equation one half
1:20:23of K minus 1
1:20:26P star of K minus 1.
1:20:31minus k
1:20:33T star of K
1:20:36plus Delta of K 1.
1:20:41so the one half here comes from the one
1:20:43half that's here
1:20:45K minus 1 P star of K minus one this is
1:20:48this guy
1:20:49KP star k this is this guy and Delta of
1:20:51k n one remains unchanged okay
1:20:55so this is the equation you should solve
1:20:58to get the stationary state of this
1:21:00Albert barabati model
1:21:03and it's fortunate but it's it's it
1:21:06happens that this equation can be
1:21:08exactly solved
1:21:11and what you find
1:21:28foreign
1:21:43if you want to check it yourself but
1:21:46what you find is that P star of K
1:21:51equals
1:21:52k 4 over k
1:21:55K plus 1
1:21:57K plus 2
1:22:00is an exact solution
1:22:12okay so you know just have to put it in
1:22:15there and check that it works and this
1:22:17thing happens to be normalized to one so
1:22:20it's the correct probability
1:22:21distribution
1:22:23and what you can notice is that when K
1:22:27becomes large
1:22:30this decays it's 4 over K to the 1 plus
1:22:34mu
1:22:36with mu equal to
1:22:43so this mechanism this Rich get richer
1:22:45mechanism
1:22:46this proportional growth model leads to
1:22:51um
1:22:52an exponent with a parallel pair with an
1:22:54exponent which is two
1:22:56so the average degree exists but the
1:23:00variance of the degree distribution
1:23:02is infinite
1:23:05okay
1:23:08so that's one possible model there are
1:23:12many variations around this model
1:23:15that you can consider for example you
1:23:18could consider an attachment rule
1:23:22or some variations
1:23:27so you could have an attachment rule
1:23:29that's proportional to K
1:23:33which becomes
1:23:35proportional to K plus
1:23:37some intercept
1:23:40okay
1:23:45personal
1:23:47so it's still linear at large K but
1:23:49there's a
1:23:51there's an intercept m
1:23:53and in this case you can also show that
1:23:56the the solution
1:23:58is a parallel
1:24:04with an exponent
1:24:06view that depends on M
1:24:12okay so that's that's the natural uh
1:24:15change but maybe you want to make us a
1:24:18more drastic change what happens if
1:24:21your attachment rule is proportional to
1:24:24K to the beta
1:24:26so
1:24:28beta equal one
1:24:30this is linear
1:24:32attachments which leads to parallels
1:24:39when beta is less than 1
1:24:43but greater than zero
1:24:45then I'm not doing the calculation but
1:24:48what you get is that TFK
1:24:51is not a parallel anymore but it decays
1:24:54as exponential of minus
1:24:57k
1:24:58to the 1 minus beta
1:25:03so this is called the stretch
1:25:04exponential
1:25:10you see it's when beta is positive it's
1:25:14slower than any exponential
1:25:16but it's faster than any power
1:25:19so it's an intermediate regime
1:25:23and what happens if beta is greater than
1:25:25one
1:25:28well when beta is greater than one
1:25:30you're in a situation that in a sense
1:25:33is similar to the situation of the hoax
1:25:35processes when r0 is greater than one in
1:25:38the sense that this assumption here
1:25:42is no longer true there's no stationary
1:25:45State because you see that the
1:25:48attachment rule
1:25:49grows so quickly with k that's that's
1:25:54condensation
1:26:01in the sense that
1:26:03a finite number of nodes becomes become
1:26:06attract everything
1:26:09and have a degree that grows uh with
1:26:13with tea
1:26:14in an unbounded fashion
1:26:17okay so here you have a completely
1:26:20different regime where there's no
1:26:22uh proper
1:26:24stationary distribution for pfk
1:26:27so in a sense if you want to have power
1:26:31laws
1:26:32this is the only case which naturally
1:26:34gives rise to
1:26:36power laws so if you have a parallel
1:26:38degree distribution in your empirical
1:26:41Network
1:26:42then presumably this is because there is
1:26:45a process behind it which is a linear
1:26:50preferential attachment process
1:26:55okay
1:26:57so
1:27:00this is um
1:27:04what I wanted to tell you about
1:27:08Networks
1:27:11I mean
1:27:12about the construction of networks so
1:27:14two very different constructions of
1:27:16course I'm not exhausting all the
1:27:19possible ways to build networks I'm only
1:27:21giving you two extreme cases
1:27:24which leads to personal distribution of
1:27:27degrees and is a the the classical uh
1:27:31Network model and the Barabbas the
1:27:33Albert model which gives the parallel
1:27:36distribution of degrees
1:27:37but you can invent many other ways to
1:27:39build uh networks and they have to be
1:27:43analyzed on the case-by-case basis but
1:27:46what is important from a technical point
1:27:48of view is and I encourage you to think
1:27:51about this more if you haven't followed
1:27:54all the steps but
1:27:56you know behind all these models there
1:28:00is often
1:28:01a master equation an equation an
1:28:03evolution equation like this and so this
1:28:06is this is really from a technical point
1:28:08of view the the main message of these
1:28:09lectures
1:28:12okay
1:28:19so let me now talk about
1:28:23um
1:28:24a giant component
1:28:37foreign
1:28:43of
1:28:46a giant component
1:28:58so I've already insisted on why it's
1:29:00important to know whether in your
1:29:03network there is a giant component or
1:29:05not
1:29:07so what tools do we have to answer that
1:29:10question I have a a given Network
1:29:15how can I know
1:29:16analytically
1:29:18from a theoretical point of view whether
1:29:21or not I have a giant component
1:29:24that is whether or not my white will be
1:29:27solid or liquid or whether or not my
1:29:30epidemic will propagate among a social
1:29:34network
1:29:35and so on or whether a crisis will
1:29:38propagate across a firm Network and
1:29:41invade the whole economy or be localized
1:29:44in some subparts of the economy okay so
1:29:48this is a very important question indeed
1:29:52so what I'm going to give you as a tool
1:29:56to answer that question
1:29:58is a little bit of a hand waving
1:30:00argument which relies on everything
1:30:03we've understood about branching
1:30:05processes
1:30:08um
1:30:09it's it's going to be actually
1:30:12restricted to some
1:30:16some types of graph
1:30:18which are what one called tree light
1:30:21graph
1:30:29so I'm going to explain in a minute what
1:30:32the tree line graph but you can think of
1:30:35it as an approximation for graph or
1:30:39arbitrary graph to treat these graphs as
1:30:42if they were trees but some graphs are
1:30:45really
1:30:46very close to being trees and for which
1:30:49the approximation I'm going to talk
1:30:51about actually is exact
1:30:54and
1:30:55what I'm going to derive for you is the
1:30:58so-called Molloy read Criterion
1:31:02so this is going to be called the Molloy
1:31:06read Criterion
1:31:08for the existence of a giant component
1:31:15but the way I'm going to present this
1:31:16Mallory Criterion is a little bit hand
1:31:19waving
1:31:21if you want to have a more mathematical
1:31:24proof of the Molloy read Criterion
1:31:27I'm going to if I have time give you a
1:31:30hint of that
1:31:31later on otherwise
1:31:34um you will find it in in the notes that
1:31:38I'm currently writing or actually in the
1:31:41acculturally technique
1:31:43um
1:31:45lecture notes
1:31:48but the way I'm going to present it I
1:31:50think is nice because it emphasizes
1:31:52um
1:31:54the relation with what we've seen
1:31:55already branching processes and I think
1:31:59it's it's a very nice way to see things
1:32:01okay so what's a tree
1:32:05uh so a tree
1:32:09is a graph with no loops
1:32:19so for example
1:32:26this is a regular tree
1:32:28and you see I can continue that to
1:32:31Infinity it has no loot
1:32:36sometimes I have loops
1:32:39in My Graph but the loops are are not
1:32:43small Loops they're they're long loose
1:32:45so maybe for example I continue the
1:32:48construction here
1:32:49and at this point
1:32:52these two points merge together
1:32:55so there's a loop in my graph
1:32:58but the loop
1:32:59here it's not that big but you can
1:33:01imagine that if these Loops are big
1:33:04enough
1:33:05you know so it means that in a sense
1:33:07they're rare enough
1:33:09I can view locally the graph as a tree
1:33:14and and so
1:33:16if you look at for example the others
1:33:18ready graph
1:33:20well
1:33:22in the limit where close to the giant
1:33:25the appearance of the design component
1:33:27the very large clusters in a in an
1:33:31address many graph they're in a proper
1:33:34mathematical sense tree-like there are
1:33:37some Loops but they're somehow uh very
1:33:40large compared to to one the the the
1:33:43size of the loop is much larger than uh
1:33:48than one so for example it grows like
1:33:51the log of the number of
1:33:53of sites so in these in some cases you
1:33:57can really justify the approximation
1:33:59what I'm that I'm going to present to
1:34:00you in other cases the graph is really a
1:34:04tree
1:34:04so it's exact
1:34:07and in still other cases it's just an
1:34:10approximation so for example let me give
1:34:12you a graph that's another tree
1:34:15so if you take a euclidean lattice like
1:34:17this
1:34:18then you see that there's no real
1:34:21meaning in saying that it's a tree
1:34:23because even at the shortest length
1:34:25scale there are Loops okay so this is
1:34:28not a tree
1:34:33but even even if it's not a tree you can
1:34:36you know deem it as a tree treat it as a
1:34:38tree and and it what I'm going to tell
1:34:41you about is just an approximation which
1:34:43has no reason to be good but uh you can
1:34:47try it and in some cases it's a very
1:34:49good approximation in other cases It's
1:34:52actually an exact uh
1:34:55treatment an exact argument okay
1:34:59so my argument is about three-digraphs
1:35:10so if I have a
1:35:12cluster that's a tree
1:35:18or a tree like object
1:35:20I can always think of this tree as a
1:35:24genealogical tree okay
1:35:27of course it has nothing to do with the
1:35:29population growth but I can think of the
1:35:31tree
1:35:32as something that has some temporal
1:35:36structure where I choose a node
1:35:39arbitrarily This Is The Answer ancestor
1:35:45and then these guys are the child the
1:35:47children of the first ancestor and these
1:35:50guys are the children of the children
1:35:51and so on okay so I'm just thinking of a
1:35:55tree like a genealogical tree
1:35:58so you see that if I have loops it it
1:36:00doesn't mean anything because if I have
1:36:02loops it means that I can beat a child
1:36:04of my
1:36:06um of my children
1:36:08but if these Loops are rare enough I can
1:36:12forget these
1:36:13um these these anomalies and think of
1:36:17that as a genealogical tree
1:36:22and so what I know is that my general
1:36:24genealogical tree
1:36:28is growing so this tree
1:36:32is
1:36:34a finite size
1:36:41so the family
1:36:43that as this guy as an ancestor as a
1:36:48finite spies if the reproduction rate of
1:36:50zero is less than one
1:36:53and it can be of infinite size
1:37:04if a zero
1:37:06is greater than one
1:37:09okay and r0 equal one is this strange
1:37:12critical thing that is in between
1:37:16so if I want to know whether there's a
1:37:19giant component
1:37:20I want to know whether there can be
1:37:22infinite size clusters in my in my graph
1:37:26right that's the same question
1:37:29so I have to answer the question in
1:37:31terms of population growth
1:37:34is whether the reproduction rate in this
1:37:39fictitious tree in the sixth genological
1:37:43tree is this uh genealogic is this
1:37:46reproduction rate greater than one or
1:37:49less than one okay so this is the
1:37:51question that I'm going to try to
1:37:52address
1:37:53and it's going to lead to the Malloy
1:37:56read criteria
1:38:02so what I'm going to need
1:38:07is
1:38:09to describe my tree
1:38:12in terms of
1:38:14the probability of the degree
1:38:17distribution
1:38:26so the only ingredient I'm going to need
1:38:32is that
1:38:35is this object T of K
1:38:38this is the unconditional
1:38:44degree distribution
1:38:54and I'm insisting on unconditional here
1:38:56which means that I have my my My Graph
1:39:01my network
1:39:03oh maybe it looks like like this okay
1:39:09and what unconditional means is that you
1:39:11know I'm picking at random
1:39:13this node and asking what is the
1:39:16probability that this node has
1:39:18degree k this is pfk
1:39:22and you'll see in a second
1:39:24another degree distribution which is
1:39:26going to be conditioned and which is not
1:39:28going to be the same as P of K and which
1:39:30is going to play a crucial role
1:39:34so I'm assuming that each node has
1:39:37an independent degree and each of these
1:39:41degrees is drawn according to pfk okay
1:39:47nodes
1:39:49have
1:39:52Independence
1:40:01so ID if you want
1:40:05cross nodes
1:40:12so there are no correlation between the
1:40:15degrees it's not because this one has a
1:40:17high degree that its neighbor will have
1:40:19a high degree
1:40:22okay so this is TFK now you know you
1:40:25should pay attention because I'm going
1:40:27to introduce
1:40:29um and unfortunately of course as usual
1:40:31I won't have time to finish today that's
1:40:33too bad
1:40:34but let me at least go until the Malloy
1:40:37read Criterion
1:40:39so
1:40:41what I'm claiming is now that I'm going
1:40:45to do another construction to uh to
1:40:48introduce another degree distribution
1:40:51which is the following
1:40:53I'm going to take
1:40:55a note at random like before this one
1:40:59and then take an edge out which is
1:41:02outgoing from that node
1:41:04and Target another
1:41:07node okay so I've taken this node in at
1:41:10random I've chosen this particular
1:41:13Edge
1:41:15and I'm I'm landing on the second node
1:41:17so this if you want is i0 which is the
1:41:20node I chose at random the degree of
1:41:23that node is distributed according to
1:41:25pfk
1:41:26and now I'm
1:41:29looking at another node which is a
1:41:31neighbor of mine and I'm asking what's
1:41:33the distribution the degree distribution
1:41:35of this guy J okay
1:41:38so think of it in terms of your friends
1:41:41in a social network
1:41:43I'm taking one guy at random on a social
1:41:45network and I'm counting the number of
1:41:47friends this is distributed according to
1:41:50pfk
1:41:51now I'm taking one friend of this guy
1:41:54and I'm asking what is the degree
1:41:57distribution of that guy okay
1:42:00and what I'm going to say is that this
1:42:04degree distribution is what I'm going to
1:42:06call Q of K
1:42:08so it's the degree
1:42:11distribution
1:42:15of a neighbor
1:42:18of a friend
1:42:24and what I'm claiming is that Q of K is
1:42:26not equal to pfk
1:42:28and there's a clear reason already for
1:42:31that to be the case which is that
1:42:34you see that this node here has zero
1:42:38Neighbors
1:42:40so if I choose at random a site in my
1:42:42network I I have a sudden probability to
1:42:45find that P of k equals 0
1:42:47is non-zero I mean I have a probability
1:42:49to find a node with now with without any
1:42:52neighbors so because P of k equals 0 is
1:42:55not zero these are the guys that I'm
1:42:58going to choose but this guy has
1:43:01absolutely no chance of being chosen as
1:43:03a friend because he has no friends
1:43:06and if you think about this
1:43:09this process of choosing a friend is
1:43:12going to select preferentially friends
1:43:14who have a large number of friends
1:43:17because the more friends they have the
1:43:19more likely it is that you are one of
1:43:22his friends and the more likely it is
1:43:24that you're going to choose him in your
1:43:26selection process
1:43:28so you can make this argument precise by
1:43:32using base theorem for example and what
1:43:35you find is that Q of K
1:43:40is K times larger than TFK
1:43:44because of this amplification effect
1:43:46because there are k
1:43:48ways of choosing a neighbor with a large
1:43:52degree is going to be chosen more often
1:43:55through this process okay
1:43:58so
1:44:03if you want to have a normalized
1:44:05distribution
1:44:07I should divide
1:44:09k p of K by the expectation the average
1:44:13value of K in p
1:44:16so I've defined
1:44:19EP of K
1:44:21as the sum
1:44:25from k equals 0 to Infinity of K
1:44:28times P of K
1:44:31such that the sum over o k of Q of K is
1:44:35equal to one okay
1:44:38and now something strange happens which
1:44:41in a sense is contained in my argument
1:44:43is that if I compute
1:44:45the average value of the number of
1:44:48friends of my friends
1:44:49which is EQ of K
1:44:53this is equal to sum over k
1:44:57of K squared P of K
1:45:00over p p of K
1:45:06which is e t of K squared
1:45:10divided by ep
1:45:12of K
1:45:15and I guess that I shouldn't go
1:45:18other than that and this is larger or
1:45:20equal than e key okay
1:45:26so the average number of your friends is
1:45:29larger than your average number of
1:45:31friends so this is called the the
1:45:34Friendship Paradox in Networks
1:45:36which means it's very frustrating but it
1:45:39means that uh
1:45:40that's this uh selection bias the fact
1:45:44that your friends are you know by by
1:45:48making
1:45:50you know mechanistic mechanistical way
1:45:53uh more uh have more friends than you on
1:45:57average means that they uh they're more
1:46:00popular
1:46:02okay
1:46:03so this is going to have a very
1:46:06important consequence for vaccination
1:46:08campaigns but let me come back to what I
1:46:13was interested in which is the existence
1:46:15of a giant component
1:46:21so what I'm claiming here
1:46:23and I'll be done in five minutes and
1:46:27then
1:46:28unfortunately then next week is vacation
1:46:30so
1:46:34there will be a two-week Gap
1:46:36for me to come back
1:46:38on the last little thing that I wanted
1:46:41to talk about but it's I I don't think I
1:46:43should do that today it takes me too
1:46:45long
1:46:46um so what I'm saying is that you know
1:46:50these children
1:46:52by construction they are children of an
1:46:54ancestor and so they are they are in my
1:46:58analogy they are friends of a guy
1:47:01and so if I want to know the
1:47:04distribution of degrees of these
1:47:07children I have to use Q of K and not P
1:47:10of K
1:47:12and so what happens is that r0
1:47:17is equal to the average number of
1:47:22children
1:47:24which is
1:47:26the expectation of K the average degree
1:47:29on the Q
1:47:31because again these are not random
1:47:34people there are people selected as
1:47:37being the child of someone
1:47:39or the friend of someone minus one
1:47:42because one of the links is the one that
1:47:45defines their uh their ancestry okay
1:47:50and so the multi read Criterion
1:47:53is that r0 should be greater than one
1:47:57which translates into
1:48:00uh
1:48:01EP of K squared
1:48:06so EQ of K I said it's e p of K Square
1:48:09divided by etfk so here I have EQ of K
1:48:14must be greater than two so EPF K Square
1:48:16must be greater than two times
1:48:19EP
1:48:21of K
1:48:26so that exists
1:48:29a giant component
1:48:31if and only if
1:48:33this Criterion holds okay
1:48:37so that's that's what I wanted to arrive
1:48:41at so you see that
1:48:43I've I've been very hand waving here
1:48:45I've used this uh General genealogical
1:48:49tree analogy and I've relied on the
1:48:52branching process on the on what we saw
1:48:55last week about uh the Galton Watson
1:48:58model
1:48:59but I assure you that there's a better
1:49:01way to do all this
1:49:03which involves more mathematics and so I
1:49:05didn't want to go into the mathematics
1:49:08but you can certainly have a go in the
1:49:11lecture note to see how you do this a
1:49:14little better
1:49:15anyway
1:49:17um
1:49:19let's let's do the elders ready case
1:49:25ER is Elder shreni
1:49:29um
1:49:30other shreni means that P is a personal
1:49:32distribution
1:49:35so the question distribution is such
1:49:37that EP is K squared
1:49:41is equal to p squared plus p
1:49:46why because the variance of K is p so
1:49:51the variance is the average of K squared
1:49:53minus the average of K the whole thing
1:49:56squared that's what I get t squared plus
1:49:59p and the average
1:50:02of K is p so I have that this must be
1:50:05larger than twice
1:50:07and 2p
1:50:09which means that P must be greater than
1:50:13one
1:50:14and so we recover what was anticipated
1:50:17by
1:50:18some in the room that if p is greater
1:50:20than one in the others raining graph you
1:50:23have a giant component
1:50:27let's see what happens in the barability
1:50:29Albert model
1:50:31you remember the Albert
1:50:35we had that pfk
1:50:38was decaying as one over K Cube
1:50:41which means formally that the average
1:50:44value of K squared is infinite
1:50:47so for Barbara Albert
1:50:50there exists a giant component
1:50:54and the reason is that
1:50:57you know because of these hubs because
1:50:59there are nodes that are so connected to
1:51:01many others
1:51:03then different parts of the subgraph
1:51:06will likely be connected together
1:51:08through these highly connected nodes
1:51:11okay
1:51:13so in again in the epidemic analogy
1:51:17these are super spreaders
1:51:18and because of their existence the the
1:51:22disease is going to spread very quickly
1:51:24in the population but at a formal level
1:51:27we see that the Molloy read Criterion
1:51:30tells you immediately that biology
1:51:32output graphs are have a giant component
1:51:37okay so what I wanted to tell you about
1:51:42is okay there is a giant component
1:51:46catastrophe
1:51:48the epidemic is the propagating
1:51:52what can I do to uh
1:51:55to prevent it to cut
1:51:58the uh the spread of of the disease and
1:52:03this translates into how robust is this
1:52:06giant component if I start
1:52:08removing some links so imagine that
1:52:11through vaccination
1:52:12I'm removing some of the links of the of
1:52:15that Network how many links should I
1:52:17remove
1:52:18to remove to kill the giant component to
1:52:21to make such that the giant component
1:52:25disintegrates and there are only finite
1:52:27clusters
1:52:29so this is a question that we can very
1:52:31easily answer with the
1:52:34formalism I've given you up to now
1:52:37but um I think I have to stop at this
1:52:39point unfortunately
1:52:42so that's that's that's me for today
1:52:46uh I don't know if there are questions
1:52:52I have a question
1:52:53yes uh can you explain again the
1:52:56expression of Q of K uh
1:52:59On The Other Board why I don't really
1:53:02understand why the it's equal to K not
1:53:05care over the expectation
1:53:07okay so you know the proper way is to
1:53:10use the base argument
1:53:13but intuitively it means that
1:53:17as I said if Jay has a large number of
1:53:20Neighbors
1:53:21okay it's highly probable that you will
1:53:25be one of them
1:53:27so the more Jay has neighbors the more
1:53:31probable it is that it's going to be
1:53:33selected through this mean that I
1:53:36explained that is I first choose i0 at
1:53:40random
1:53:41and then I choose J as a friend of i0
1:53:44but if J is a friend of I zero it means
1:53:47that I zero is a friend of J and because
1:53:50Jay had many friends is going to be much
1:53:52more likely to choose J if J as many
1:53:55friends than if J has a few friends and
1:53:58in the extreme you see that if Jay has
1:54:01no friend there's no way to select J
1:54:04using this procedure and that's exactly
1:54:06what you find here when K is 0 Q of Q of
1:54:100 is 0 there's no way to choose a
1:54:13neighbor with zero neighbor with with
1:54:16yeah it's impossible to choose J if J is
1:54:19no friends so this is an extreme case
1:54:22but you can intuitively understand that
1:54:26the you know the more K the larger K the
1:54:30more probable it is to choose the
1:54:31southern J and as I said I didn't do the
1:54:35argument in a in a rigorous way but if
1:54:38you want to do it in a rigorous way you
1:54:40can think of it in in the biased fashion
1:54:44okay okay thanks
1:54:56my theorem
1:55:00so what is the probability that I'm
1:55:02chosen knowing that I have K Neighbors
1:55:05and it's the probability that I have K
1:55:07Neighbors times the probability time
1:55:09chosen but the probability that I am
1:55:11chosen is proportional to k okay
1:55:14okay
1:55:19other questions
1:55:25can you explain again how you obtained
1:55:28the Criterion
1:55:32so what I'm saying is that
1:55:36these guys they're they're threads they
1:55:40are selected as I've explained here so
1:55:42the degree distribution of these people
1:55:45is Q of K
1:55:47and because the degree distribution of Q
1:55:49is Q of K the average number of children
1:55:53is the average degree minus one because
1:55:56there was one link that comes from the
1:55:59ancestor from the parent so the number
1:56:02of outgoing links here minus the one
1:56:05that makes him or her a child is the
1:56:11expected value of K called in queue
1:56:14because I have to use q and not P minus
1:56:171.
1:56:19and the minus one here counts away the
1:56:23ancestor
1:56:25is the new length
1:56:27coming out of the selected child
1:56:36but you know make no mistake I I'm I'm I
1:56:39know that this is a this is a rough
1:56:41argument this is a kind of hand waving
1:56:43argument
1:56:44but I thought that it was uh maybe more
1:56:48interesting to present this in this way
1:56:51for you to understand the Deep analogy
1:56:53between the existence of a giant
1:56:55component and uh the criticality of
1:56:58branching processes which we've seen
1:57:00last time rather than to give you a more
1:57:03formal uh way to think about this this
1:57:06this problem
1:57:09as I said there is there's a way and
1:57:11you'll see them this these calculation
1:57:14in the notes there's a way to be much
1:57:17more precise than this and in particular
1:57:19to compute uh the probability for a node
1:57:23to belong to the infinite cluster so
1:57:26what I call P Infinity
1:57:29you see here I have no way of computing
1:57:32P Infinity it's just it's just an
1:57:34existence Criterion
1:57:36but it doesn't tell me what the
1:57:38probability to belong to the giant
1:57:40cluster is
1:57:41whereas the more technical approach to
1:57:45this problem gives you an answer for p
1:57:47infinity and of course it gives you that
1:57:49P Infinity is not zero when
1:57:53e of Q is greater than 2.
1:57:58thank you
1:58:05okay
1:58:09well have a good vacation week and um
1:58:12we'll see each other
1:58:14quotes and quotes
1:58:16first week of March
1:58:24thank you
1:58:27thanks bye this conference will now be
1:58:31recorded
1:58:32okay great so let's start with this uh
1:58:36last day before of uh the holidays so as
1:58:40you see we uh kind of catched up with a
1:58:43lecture because this today is about hoax
1:58:45processes that have been discussed
1:58:48during the lecture today
1:58:49but it is a little bit long so what
1:58:52we're gonna do is to split it into two
1:58:55so today we will do only the first part
1:58:57which is a little bit connected to what
1:59:02was discussed in the lectures in
1:59:03particular it focuses on linear hoax
1:59:06processes and then the second part which
1:59:09is about non-linear hoax processes and
1:59:11meta stability this will be postponed to
1:59:14the 10th of March at this point
1:59:18and inside what we can do in the
1:59:21remaining time today for those who are
1:59:23interested is to go back to uh what we
1:59:26left behind from the three which was the
1:59:28derivation of soccer plan for a
1:59:30multiplicative noise and I just want to
1:59:32sketch how you derive it both in detail
1:59:35and in the set on which prescription and
1:59:37just to tell you how to compute
1:59:39stationary states which is uh pretty
1:59:42easy
1:59:43uh okay a comment so unfortunately I
1:59:46realized that the notation of the today
1:59:48and one of the lectures are totally
1:59:50messed up so what I will do is this
1:59:54afternoon to somehow modify that a day
1:59:56in such a way that we stick to the
1:59:58notation of the lectures so I will
2:00:00upload a version with the solutions
2:00:02which is compatible with the lecture but
2:00:05uh in here since I gave you the text
2:00:07already I think it's better to speak to
2:00:11the notation that we have in the text so
2:00:13I wanted to be a little bit elastic so
2:00:15what I did is to call the kernel that in
2:00:18the lecture was K and now I call it high
2:00:21then what was called row 0 as you will
2:00:23see and becomes alcine here the average
2:00:27uh let's say rate that was Lambda bar I
2:00:32will call it g infinity infinity because
2:00:34because of this ergodicity you can also
2:00:37relate it to uh how the stationary value
2:00:40of
2:00:42of Lambda and then we will be working
2:00:45today with exponential clearness so in
2:00:47the lecture the uh let's say
2:00:51Decay time was one over Alpha here since
2:00:53we use Alpha instead of r0 I will
2:00:57introduce beta so forgive me for this
2:01:00annotational issue but keep in mind that
2:01:03somehow we are talking about the same
2:01:04things
2:01:06okay so let's go to
2:01:08then exercise one so let me recall the
2:01:12expression for this conditional
2:01:14intensity of the Oaks process that I
2:01:17call Lambda tip so this is given by some
2:01:21background intensity Lambda 0 which was
2:01:23mu in the lecture and then you have the
2:01:26term uh with the kernel that as I say I
2:01:29write it as Alpha sum over all the
2:01:33events that happened before the
2:01:35particular time T that we are looking at
2:01:37of some kernel
2:01:39at T minus TI and as it was said in the
2:01:43lecture already we can rewrite this
2:01:46discrete sum in the following form so as
2:01:50an integral from time let me say zero to
2:01:53time t 0 minus infinity it doesn't
2:01:56change much of the kernel so as of
2:02:01P minus Tau and then you have the
2:02:03differential of your stochastic process
2:02:06that counts how many events you have at
2:02:10times out to help you uh Tau plus beta
2:02:14that I denote in this way here
2:02:17and let me stress that this is a
2:02:19condition and intensity because it is
2:02:21conditioned to the history of the
2:02:23process so you assume that you know what
2:02:26is the history up to time T so uh which
2:02:29events occurred and at which times and
2:02:32then once you know this you can write
2:02:34down what is if you want the probability
2:02:36to have an event at a later time D plus
2:02:39DT that is controlled by this Lambda
2:02:42team
2:02:44okay then there were several things that
2:02:46were already introduced so in particular
2:02:50what we are gonna look at in here is the
2:02:53clustering ratio so in the lecture it
2:02:56was discussed the value of the
2:02:58clustering ratio sometimes somehow at
2:03:00infinite time so the stationary value
2:03:03but here let me introduce some
2:03:06more General time dependent row of T
2:03:09that we are going to compute and let me
2:03:12Define this
2:03:13similarly to the lecture as the variance
2:03:16by this V I mean variance
2:03:19of the number of events at times t plus
2:03:23now minus
2:03:24the number of events next time p
2:03:28yes
2:03:31there was a question
2:03:32okay
2:03:34divided by the average of uh of this
2:03:38difference and you see that what I have
2:03:41on the right hand side is something
2:03:42which in principle depends on T but I'm
2:03:46assuming that this the clustering ratio
2:03:49only depends on the time difference and
2:03:51this is some Assumption of stationarity
2:03:53that holds in the phase and we will see
2:03:57whenever Alpha is smaller than one so
2:04:00whenever you have the process is
2:04:02eventually ergotic
2:04:04so this is the
2:04:07clustering
2:04:09ratio
2:04:12and what we're going to do in here is to
2:04:14compute this classing ratio for any
2:04:16value of time Tau for one specific
2:04:20kernel that was already introduced in
2:04:22the lecture which is the exponential
2:04:25kernel so in here
2:04:27we are going to focus on a kernel that
2:04:30is of the form beta e to the minus
2:04:35and we're going to compute this through
2:04:37uh an equation which relates the
2:04:40clustering ratio to correlation
2:04:42functions so that gives you the
2:04:45correlation between the number of events
2:04:47at different times and this is something
2:04:49that is defined in the today in equation
2:04:53four I think so let me not rewrite
2:04:56equation for an equation five but this
2:04:58is what we are going to use to compute
2:05:00this object in here
2:05:03and so just as a comment
2:05:07actually uh let me introduce another
2:05:09quantity first
2:05:11so uh we are gonna first of all compute
2:05:14uh what I call in the CB G of t
2:05:19and G of T is defined so as it was
2:05:22stressed already in the lecture this
2:05:24quantity this conditional rate is a
2:05:27random variable itself and it is a
2:05:29random variable because it depends on
2:05:31the history of the process so once you
2:05:33know these three you know what is Lambda
2:05:35of T but in principle the history is a
2:05:38stochastic process so you have
2:05:40fluctuations depending on the different
2:05:42realizations and what we're going to
2:05:44introduce is the average of this
2:05:46quantity with respect to
2:05:48um
2:05:49to these histories or if you want the
2:05:52Ensemble leverage
2:05:53each of this conditional
2:05:57rate Lambda t
2:06:00and we're going to solve the equation
2:06:01for this G of T and what was discussed
2:06:05uh in the lecture was essentially the
2:06:08solution of this equation in the long
2:06:10time limit so the quantity Lambda bar
2:06:13was
2:06:14what I call G Infinity so it's the limit
2:06:18D go into Infinity of this G of t
2:06:23and it was given a self-consistent
2:06:25equation for this Lambda Barrow or G
2:06:27infinity and we will recover uh in here
2:06:30the results for these uh which is of the
2:06:32form so if you remember the infinity
2:06:34will be equal to in this notation Lambda
2:06:370 over 1 over alpha or in the notation
2:06:40of the lecture it was uh divided by 1
2:06:44minus
2:06:46r0 yes
2:06:50no okay
2:06:53uh good
2:06:56and the other quantity that was given in
2:06:58the lecture was the infinite time limit
2:07:00of this clustering ratio that was in our
2:07:04rotation 1 over 1 minus Alpha to the
2:07:07power 2 and this is something that we
2:07:08will derive uh right now
2:07:11for the exponential curve
2:07:14okay so let's do it so the first point
2:07:17we see this yes
2:07:20uh the first point is to use so this is
2:07:23going to be an exercise on a Laplace
2:07:25transforms basically
2:07:27and what we have to do is to solve the
2:07:29equation for this G of T and the
2:07:32equation reads as follows so I can
2:07:34directly derive it from up there
2:07:39hopefully yeah
2:07:41so if you see I have the equation for
2:07:43Lambda T up there and what I can do is
2:07:45to take the expectation value on the
2:07:48left hand side and then I take the
2:07:50expectation value on the right hand side
2:07:52and this I bring the expectation into
2:07:54inside the integral there and I will
2:07:57have the expectation value of the
2:07:59differential of my stochastic process
2:08:02and then I remember that Lambda of T was
2:08:06defined uh so that if you want the
2:08:08proper definition of Lambda T is of a
2:08:11conditional expectation value of
2:08:16having events in a small interval TD
2:08:19plus BT conditions to the history of the
2:08:22process up to time T that I will denote
2:08:26as h of t
2:08:28okay so if I have this Lambda T and then
2:08:32I take also the expectation with respect
2:08:34to the to the history what I get out is
2:08:38is simply The Ensemble average now
2:08:40averaging over all times of my ideas
2:08:43that is what will appear in the right
2:08:46hand side if I take the expectation of
2:08:49of this equation so these were many
2:08:51words but somehow you can easily realize
2:08:54that the equation for this
2:08:56GLT is nothing but Lambda 0 Plus
2:09:00integral there was an alpha in front
2:09:03integrated from 0 to T in the Tau
2:09:07of
2:09:08PSI of T minus Tau times
2:09:11the very same function G of Tau
2:09:15so this is a self-consistent equation an
2:09:18integral equation for G that we want to
2:09:20solve
2:09:21and as you see in this equation what you
2:09:25have on the right hand side is a
2:09:26convolution which is
2:09:28uh
2:09:30some sort of convolution
2:09:34so there is a Theta of
2:09:38so you're asking here that tau is
2:09:40smaller than T so there is a Theta of uh
2:09:42T mining style so I can rewrite this
2:09:45if you prefer
2:09:47in this way
2:09:49and then you recognize that we have a
2:09:51convolution
2:09:52and we know what we have to do when we
2:09:55have convolutions so there is a very
2:09:57useful instrument or trick that we can
2:10:02use to solve this type of equation and
2:10:04this is either Fourier or or Laplace
2:10:07transforms so in here I will use Laplace
2:10:10transforms and I will integrate only
2:10:12over positive times
2:10:15and Laplace transforms are useful
2:10:16because anytime you have a convolution
2:10:18they allow you to somehow rewrite name
2:10:22in terms of Laplace transform the
2:10:24equation in terms of a product so let me
2:10:26introduce the notation first
2:10:29so the Laplace transform
2:10:32of a function G
2:10:34that will be a function of s now is
2:10:37simply
2:10:38the integral from 0 to Infinity in DT e
2:10:43to the minus s t times
2:10:46High function G of t
2:10:50and so if I take the Laplace transform
2:10:52of this equation here so now we will
2:10:54call this G hat
2:10:57of s this is a function of s let me
2:11:01apply a such a transform to this
2:11:03equation and if you do this
2:11:05you will realize that this gives you G
2:11:08hat of s equal to
2:11:10the Laplace transform of a constant that
2:11:14we I will compute in a minute that's
2:11:16very simple
2:11:17plus as I said so this is an exercise
2:11:20that if if you want we can do at the end
2:11:23if you have questions but it's very easy
2:11:25to say to see that here you will end up
2:11:27with a product of the Laplace transforms
2:11:30of these two quantities uh in here so
2:11:35I will simply have
2:11:37the Laplace transform of my kernel
2:11:40evaluated at s times
2:11:44my function transforms evaluated
2:11:48foreign
2:11:52and this is nice because now this is
2:11:54just an algebraic equation for my
2:11:56function in LaPlace space so what I have
2:11:59to do is to compute these two terms that
2:12:03I have in here so the first one is the
2:12:05Laplace transform of a constant but this
2:12:07is very easy so if you just put a
2:12:09constant in here and you integrate the
2:12:12exponential you just get a factor of 1
2:12:14over s
2:12:15times the constant so this means
2:12:18that this is number 0 over s
2:12:22plus Alpha
2:12:24jihat of Ash
2:12:26and then we plug our assumption that the
2:12:30kernel is exponential so let me compute
2:12:33uh this c bar
2:12:36that's assuming that Phi as the form up
2:12:40there so this is again just an
2:12:42exponential integral so this will be the
2:12:45integral from 0 to Infinity
2:12:49of e to the minus s t times my function
2:12:53P which is e to the minus beta
2:12:56t okay
2:12:59and so you see that again doing the same
2:13:02exponentially integral you have B
2:13:05divided this time
2:13:07by a factor B plus s
2:13:10so I plug it in here so I have beta
2:13:13sorry not B but beta divided by beta
2:13:17plus s
2:13:20okay and now let me solve for a G of s
2:13:25I hope that you see yes
2:13:28so then I I have this G of s
2:13:32I collect all of the terms so I will
2:13:34have one minus alphabeta over
2:13:38beta plus s this is equal to
2:13:41Lambda 0 over s
2:13:44and therefore
2:13:48G hat of s
2:13:50is now very simple it's number zero
2:13:53times
2:13:54beta plus s
2:13:56divided by S times
2:14:00uh bit I have beta minus Alpha Beta plus
2:14:03s so I can write this as beta
2:14:071 minus Alpha plus s
2:14:12okay so this is the solution
2:14:14in terms of Laplace transforms
2:14:19and now of course once we have this what
2:14:22we have to do is to do the inverse
2:14:24Laplace transform to get the solution as
2:14:28a function of time
2:14:29so let me briefly recall how you do or
2:14:32how you define the inverse transform and
2:14:36what is the usual trick to compute it
2:14:39which is the residue CRM so uh first of
2:14:43all what is the definition of the
2:14:44inverse
2:14:46applied
2:14:48to my function G hat of my laplacians 4
2:14:51so this will be a function of t
2:14:54again
2:14:56and this is
2:14:57defined as you have a factor of 1 over 2
2:15:02pi I
2:15:03and then you have to perform an integral
2:15:05in general in the complex plane
2:15:08a longer Contour which is usually called
2:15:11the Bromwich Contour
2:15:14then I will just write and then I will
2:15:16comment so it's an integral along a
2:15:19vertical axis in my complex plane where
2:15:22the real part is equal to some constant
2:15:24gamma and then I integrate over all the
2:15:27axis so from gamma minus I Infinity to
2:15:30gamma Plus
2:15:31I Infinity
2:15:33of what while the integral is in DS now
2:15:37Ash is a complex variable in general and
2:15:40then I have e to the HT so the
2:15:43reciprocal of the factor that I had in
2:15:46the direct transform times G hat
2:15:49avash
2:15:52now how do you choose gamma well you can
2:15:55choose it more or less arbitrarily but
2:15:58with a constraint that if you are doing
2:16:01the inverse transform of a function
2:16:03which has some singularities on the
2:16:05complex plane you have to choose gamma
2:16:08in such a way that you are always at the
2:16:10right of the singularity so let me give
2:16:12an example so now this is my complex
2:16:15plane for the variable s real and
2:16:18imaginary part
2:16:20and let me assume that we want to do the
2:16:22inverse transform of some function which
2:16:24has some pole poles or singularities on
2:16:28the complex plane so for instance let's
2:16:30go back to the function that we have as
2:16:32you see you have two simple poles of
2:16:35this function one at s equal to zero so
2:16:38you will have one pole here and another
2:16:41one at s equal to minus beta times 1
2:16:45minus Alpha that we assume uh
2:16:49but you can take arbitrary sign but in
2:16:52the drawing let me assume that one minus
2:16:54size is positive so that the pole is is
2:16:58negative so you have two singularities
2:17:00and you have to choose gamma to the
2:17:02right of this Singularity so whatever
2:17:04vertical line
2:17:06that is to the right of zero in this
2:17:08example would be a good contour for me
2:17:11to do to perform this integral and since
2:17:13the function is analytic I can move it
2:17:15back and forth provided that I do not
2:17:19hit any singularity
2:17:22and then once I have this Contour the
2:17:25usual trick to to perform this
2:17:28integration is to close the Contour
2:17:31or one way if you want to perform the
2:17:34integration is to close the Contour at
2:17:36Infinity
2:17:37and this is good because it allows me to
2:17:40use the so-called residue theorem which
2:17:43tells me how to compute
2:17:45integrass over close Contours on the
2:17:48complex plane by summing the residues of
2:17:52the function add to the singularities
2:17:54which are inside the Contour so first
2:17:57let me comment why it is it's not
2:18:00dangerous to close the Contour at
2:18:02Infinity well this is so whenever you
2:18:04have a function which decays
2:18:06sufficiently fast at Infinity so in this
2:18:09case you have a Decay that is one over s
2:18:11Square so you know that the contribution
2:18:13along this big circle will go to
2:18:16Infinity if I send a will go to zero
2:18:18sorry if I send the radius of the
2:18:21Contour to Infinity so these pieces of
2:18:24the controller will not eventually
2:18:26contribute but they allow me to close my
2:18:29my Contour of integration and then I can
2:18:33use this residue
2:18:35formula that I'm sure you know so
2:18:37suppose that I want to compute an
2:18:40integral over a closed Contour
2:18:42that I call now Capital gamma of a
2:18:44function f of z d z
2:18:49so this residue theorem tells me that
2:18:52what I have to do is to sum over all the
2:18:54singularities that I have inside the
2:18:56Contour the residues of the function at
2:18:59those singularities so I will get
2:19:03there is a factor of 2 pi I which I'm
2:19:05happy about because it will cancel this
2:19:08one and then I have a sum overall The
2:19:11Singularity that I call
2:19:13zadai
2:19:16so for instance that I are eventually
2:19:18the poles of my function f of Z
2:19:22of the residue
2:19:25of the function f
2:19:27at the point that I
2:19:32so if you have never seen this before
2:19:35just let me know but otherwise
2:19:38uh let me just remind you so if you have
2:19:41a simple pulse like in here Computing
2:19:44the residues is very very simple so what
2:19:46you have to do is uh essentially to so
2:19:50if you see in here I have a poet s equal
2:19:52to zero so the residue of this function
2:19:55will be given by you multiply the
2:19:58function by S minus the value at the
2:20:01pole so in this case it would be just s
2:20:03and then you compute what remains
2:20:05exactly at the pole so okay let me write
2:20:09a formula
2:20:10just to be concrete
2:20:16here
2:20:28so if you have a simple Pole
2:20:33this means that your function will be of
2:20:35the form let me say G of Z
2:20:39divided by Z minus z i which is the pole
2:20:44and this is my f of Z
2:20:48and then the residue
2:20:51of s at I
2:20:54is
2:20:56where space
2:20:58the limit without going to the die
2:21:02of Z minus z i times
2:21:05F of Z
2:21:08so it is just essentially the value of
2:21:10what I call Gene here computed that said
2:21:13I
2:21:15okay so let's use this then to compute
2:21:18the inverse Laplace transform of my
2:21:21function G hat
2:21:26see this
2:21:28yeah
2:21:30yes
2:21:33okay so what will be G of t
2:21:37so I will have a factor of 1 over 2 pi I
2:21:41which I cancel with the 2 pi I coming
2:21:44from the residue theorem and then I have
2:21:46the sum over my two singularities
2:21:52as I so s i is either 0 or minus beta
2:21:56times 1 minus Alpha
2:21:59of the residue now what is the function
2:22:02of which I have to compute the residue
2:22:04so if you see
2:22:05from up there I have G hat times the
2:22:08exponential factor which I don't have to
2:22:10forget which G hat
2:22:13of s e to the SP
2:22:17at the point h i
2:22:21okay
2:22:22so for our examples
2:22:24example up there so as I say the first
2:22:27poll is at s equal to zero so if I
2:22:31compute the residues there I just get
2:22:34Lambda 0 times beta from the numerator
2:22:38the exponential computed at s equal to 0
2:22:40gives me one
2:22:42and then below I have beta
2:22:46times 1 minus Alpha
2:22:48because X is equal to zero
2:22:51plus the contribution of the second pole
2:22:54so the second pole is at minus beta
2:22:571 minus Alpha so I will have at the
2:23:00denominator just
2:23:02beta times 1 minus Alpha
2:23:05which comes from the factor one whereas
2:23:07and then I have up here Lambda 0 then I
2:23:11have beta Plus
2:23:13uh s computed as a pole so this gives me
2:23:16beta minus beta
2:23:18so this is just Alpha
2:23:22I think and then the exponential which
2:23:25this time is no zero is e to the minus
2:23:29Alpha times t
2:23:31okay
2:23:35so beta here simplifies so let me write
2:23:39it as Lambda 0 1 minus Alpha and then I
2:23:43have 1 minus
2:23:44Alpha times this exponential Factor
2:23:51okay
2:23:53and this gives me for any time the
2:23:57average value
2:23:58of of the condition and intensity
2:24:03so from here you can now connect with
2:24:06what was said already in the lecture so
2:24:09the first thing that you can see is that
2:24:13uh for this simple case of the
2:24:15exponential kernel
2:24:17you have so let's assume now that one
2:24:20minus five is positive so Alpha is
2:24:22smaller than one
2:24:23and remember that Alpha was row zero I
2:24:27think in the lecture so in this case the
2:24:30process is uh stationary and it will be
2:24:33ergotic this factor in the long time
2:24:36limit will Decay to zero and you will
2:24:38find that the value at infinite time of
2:24:42this G is nothing but Lambda 0 divided
2:24:44by 1 minus Alpha which is precisely the
2:24:48Lambda bar that was defined in the
2:24:51lecture so we recover this this first
2:24:54result and we also know how you Decay to
2:24:57this particular stationary value so you
2:25:00Decay with a relaxation time that is of
2:25:03the form beta times 1 minus Alpha which
2:25:07is also related to
2:25:09um to the comments in the lecture so you
2:25:11see that as soon as you have Alpha which
2:25:13is smaller than one everything is fine
2:25:15as you approach this critical value
2:25:18Alpha being equal to one you have two
2:25:20divergences so you have one Divergence
2:25:23of the stationary rate in here and you
2:25:26also have the Divergence of these
2:25:29relaxation time and this tells you that
2:25:32you are approaching a regime for this
2:25:34point process which is unstable and
2:25:38indeed if you choose Alpha larger than
2:25:39one the processes is non-fictionary it
2:25:42is not well designed if you want and you
2:25:45see it from the fact that you have an
2:25:46intensity which explodes exponentially
2:25:49over time so if you have time to look at
2:25:52the homework
2:25:54on work five there there is a simple
2:25:57code or a code to implement this hoax
2:26:01process with the exponential kernel and
2:26:04you can play around with this parameter
2:26:07Alpha and you can really see what
2:26:08happens if you choose Alpha uh becoming
2:26:12closer and closer to one so you see that
2:26:13the number of events start
2:26:17increasing in a way that is somehow
2:26:20uncontrolled as you expect from here
2:26:23okay so this was the first point
2:26:27now let me do another comment which is
2:26:30good for the next exercise so we solve
2:26:34let me go back to the equation where it
2:26:36is
2:26:37I I erased it so we had an equation for
2:26:40G of T which was uh this convolution so
2:26:44GST is a constant I know it's up there
2:26:51okay
2:26:54so you see it's a constant Plus this
2:26:58convolution between the kernel and the
2:27:00constant and the function itself
2:27:02and we are saying that we want to choose
2:27:04a kernel that is exponential and we can
2:27:07solve the equation uh by Laplace
2:27:09transform and this is what you should do
2:27:11for any arbitrary kernel
2:27:13but if you have a kernel that is
2:27:15exponential you can immediately guess
2:27:17what is the form of your function G of T
2:27:20that is by the way this form in here
2:27:23that is you can guess that the function
2:27:26is itself an exponential and this is
2:27:28because if you look at the right hand
2:27:30side you have an integral of Phi which
2:27:33is an exponential function and if you
2:27:35assume that g is itself an exponential
2:27:37function the integral of the product of
2:27:40two exponentials will give you back
2:27:42another exponential so by adjusting the
2:27:45coefficient you can match the right hand
2:27:47side to the left hand side and see that
2:27:49any exponential answer is a good
2:27:51solution for for the equation up there
2:27:54so this was another possible way to uh
2:27:58to proceed and this is what we are using
2:27:59now to uh to solve another equation that
2:28:02is the point two
2:28:04of the exercise that is now the equation
2:28:07yes
2:28:09I have a question about the solution we
2:28:12just found so does this mean if we
2:28:14choose Alpha equals to one oh sorry can
2:28:16you hear me
2:28:17yes it is it's always bad but so far yes
2:28:22okay sorry
2:28:23um if we choose Alpha equals to one does
2:28:26that mean that we reduce the problem
2:28:27back to a poisson process
2:28:30in here you should know actually it's
2:28:34the limit Alpha going to zero that gives
2:28:36you the percent so the question is uh
2:28:39somehow for which Alpha I go back to a
2:28:40person process
2:28:42and the idea is that uh the limit of the
2:28:45poisson process is given by
2:28:48having no uh memory kernel if you want
2:28:51in your process so what happens in in a
2:28:54personal process is that uh the the
2:28:57probability to have
2:28:59a certain number of events in a given
2:29:01small interval is uh independent with
2:29:05respect to uh to the previous history of
2:29:07the process so to the to the number of
2:29:09events that you had before reaching that
2:29:11interval and this is something that you
2:29:14would recover setting Alpha equal to
2:29:15zero in here so killing this memory term
2:29:18that that carries memory about uh the
2:29:22previous uh realization of the process
2:29:25and indeed in the poisson case you just
2:29:28have that the rate that controls whether
2:29:32you have an event or not in a small
2:29:34interval DP is a constant rate so this
2:29:36would be poisson
2:29:40and now that we are going to compute
2:29:41this we are uh we will be able to check
2:29:45that indeed that the ratio the
2:29:48clustering ratio of the postman process
2:29:50which is equal to one is recovered for
2:29:53uh Alpha equal to zero
2:29:56is it fine
2:29:59uh yes thank you very much okay
2:30:05so let's do that
2:30:08and to do that we have to
2:30:12let's say accept another equation which
2:30:15is another integral equation this time
2:30:18for the
2:30:19correlation function
2:30:25that I am not deriving in here but I
2:30:28think you find
2:30:30some derivation in the lecture notes
2:30:34in in the last chapter that was given of
2:30:36the lecture notes
2:30:39and this equation has a name it is
2:30:42called
2:30:45this fold
2:30:49there's a name that I forgot so let me
2:30:53what is it you'll Walker equation
2:31:01okay
2:31:02and it is quite similar to the equation
2:31:04we had for G so the idea is that you get
2:31:08the correlation between the number of
2:31:11events in two interval in two times if
2:31:14you want separated by uh by a Time shift
2:31:17Tau
2:31:19this is equal to Alpha divided by
2:31:23G Infinity so G Infinity maybe I didn't
2:31:25Define it but it is
2:31:30simply the limit of what we just
2:31:33computed at large times
2:31:35so for the explanation
2:31:37currently it was just Lambda 0 1 minus
2:31:41Alpha so this was Lambda bar in the
2:31:43lecture
2:31:46so this is what appears uh in here then
2:31:49you have your kernel
2:31:52I'm rewriting equation eight if you're
2:31:55looking at it today
2:31:58and then we have another
2:32:01convolution again between
2:32:04your kernel and the correlation function
2:32:06itself
2:32:09and this correlation function is assumed
2:32:11to be symmetric
2:32:15with respect to this
2:32:17time difference now
2:32:21okay now what we can do so why we want
2:32:23to compute this because as you see from
2:32:25that today there is an equation which
2:32:28relates our clustering ratio that is uh
2:32:32naturally related to a correlation
2:32:34function because you see that it has two
2:32:36times appearing when you compute this
2:32:39variance so there is an explicit
2:32:41equation which relates this quantity to
2:32:44this correlation function so what we are
2:32:46going to do is to compute this and then
2:32:48use the result again for the exponential
2:32:50kernel to get the clustering ratio
2:32:54and we can proceed exactly as before so
2:32:56we can do the Laplace transform of this
2:32:59equation and solve it and this is what
2:33:01you find in the lecture notes or we can
2:33:05use this idea that I just mentioned that
2:33:07if Phi is an exponential then we should
2:33:10expect some sort of exponential form uh
2:33:14also for the function C so it is
2:33:16consistent to assume and this is what
2:33:18I'm gonna do in here that our sea of Tau
2:33:23has an explonation form so it is some
2:33:26constant capital c
2:33:28times e to the minus some exponents that
2:33:32we have to determine that I call gamma
2:33:34in here
2:33:35and then I I put an absolute value of
2:33:38Tau and I put an absolute value because
2:33:40I want uh to preserve a symmetry of the
2:33:43correlation function
2:33:46okay so this is an answer
2:33:51so what I can do is I take these assets
2:33:53and I plug it into into our equations
2:33:56and then I use the equation to determine
2:33:59what is the expression for C and what is
2:34:02the expression for gamma
2:34:04so the left-hand side is easy so I get c
2:34:07e to the minus gamma so now let's assume
2:34:09that tau is positive
2:34:14so I have c e to the minus gamma Tau
2:34:17then on the right hand side I have my
2:34:20exponential kernel beta e to the minus
2:34:23beta Tau
2:34:25and then I have this convolution to
2:34:28compute so this is the integral from 0
2:34:30to Infinity in the U
2:34:33Alpha Beta e to the minus beta U
2:34:37which is my fear of you
2:34:39and then C and C is what is capital c e
2:34:43to the minus gamma and now here I have
2:34:45to keep the absolute value because uh
2:34:49you as you see I'm integrating over U
2:34:51from 0 to Infinity so this can be both
2:34:54larger or smaller with respect to Tau so
2:34:56I have to split
2:34:59these two cases
2:35:02so let's split these two cases
2:35:05just a bit of algebra
2:35:07so I have alphabeta over
2:35:10G Infinity to the minus beta Tau
2:35:14plus Alpha Beta capital c
2:35:19and then
2:35:20I assume first so I do first the
2:35:23integration from zero to Tau which I
2:35:26assume to be positive
2:35:28so if I integrate from 0 to Tau I have e
2:35:31to the minus beta U
2:35:33and then U is smaller than Tau so this
2:35:35is positive so this is minus
2:35:38gamma Tau Plus
2:35:41gamma U
2:35:44okay
2:35:45and then in the remaining
2:35:48parts of the domain from Tau to Infinity
2:35:50I still have e to the minus beta U and
2:35:53then I have to flip the sign so this
2:35:55gives me minus gamma U Plus
2:35:59gamma Tau
2:36:02okay
2:36:04now where should I go maybe on the other
2:36:07side
2:36:10now I have
2:36:13to do simple exponentially integrals
2:36:16over you
2:36:27this will be equal to so this is the
2:36:30same as this
2:36:33alpha beta version Infinity to the minus
2:36:36beta u n o Tau
2:36:39Plus
2:36:41foreign
2:36:46so from here what do I get so I have a
2:36:49factor of e to the minus gamma Tau that
2:36:52I can bring outside the Integra
2:36:58and then I have the integral of e to the
2:37:00minus
2:37:02beta between brackets beta minus gamma
2:37:05times U which gives me uh well let me
2:37:09write this compactly this will be 1
2:37:11minus E to the minus
2:37:13beta minus gamma Tau
2:37:16divided by
2:37:18beta minus gamma
2:37:23plus then I have the second term
2:37:26uh let me write it here sorry that there
2:37:31is a little bit of confusion with the
2:37:33space so now in this case I have to take
2:37:36out a factor of e to the gamma Tau
2:37:40and then I have the integral
2:37:42of e to the minus this time between
2:37:44brackets beta plus gamma times U
2:37:48between infinity and Tau so this will
2:37:50give me e to the minus
2:37:53beta plus gamma Tau divided by
2:37:56beta plus gamma
2:37:58hopefully
2:38:00okay
2:38:03and so let me now collect all of the
2:38:07terms which have the same exponential in
2:38:09front
2:38:10so there are some terms which have a
2:38:12factor of e to the minus
2:38:15beta Tau in front
2:38:17so we have one here this is just Alpha
2:38:20Beta divided by
2:38:21Gene Trinity
2:38:23and then you see that if I multiply this
2:38:25times this the factor of gamma conscious
2:38:28and I just get the factor of e to the
2:38:30minus beta Tau so this will contribute
2:38:32with minus
2:38:34Alpha Beta t
2:38:37divided by
2:38:38beta minus gamma and the same from here
2:38:42so this is Plus
2:38:44Alpha Beta C divided by Beta plus gamma
2:38:50and then I have a unique term which
2:38:53depends
2:38:54at the exponents only on gamma which is
2:38:57the term that I get
2:38:59from this product so when I'm left with
2:39:04alphabetashi
2:39:06divided by Beta minus gamma
2:39:09e to the minus gamma Tau
2:39:13okay
2:39:15and I'm rewriting this in this way
2:39:16because now you see that I have an
2:39:19expression on the right hand side that I
2:39:20have to match with what we had on the
2:39:23left hand side and on the left hand side
2:39:26we only add the dependence on gamma
2:39:29so uh what this means is that I have to
2:39:32kill the factor which depends on e to
2:39:34the minus gamma Tau so I have to set to
2:39:36zero
2:39:38what is in parenthesis in here and this
2:39:41will give me an equation for C
2:39:44if I solve this and then I have to match
2:39:47the prefactoring here with the prefactor
2:39:51that I have on the left hand side which
2:39:52is just the constant C
2:39:55and this immediately tells me that I
2:39:57need
2:39:58alphabeta divided by Beta minus gamma to
2:40:02be equal to 1.
2:40:04okay so from here I can read
2:40:08that gamma similarly to the Decay that
2:40:12we had above
2:40:14must be equal to Beta times
2:40:161 minus Alpha
2:40:20and then plug in this expression from
2:40:22gamma in here and setting this to zero I
2:40:25get out an expression for C that I will
2:40:28directly give you
2:40:31in a minute
2:40:33and you can do it by yourself
2:40:37so with this matching condition
2:40:47I recover the full correlation function
2:40:50for the exponential kernel
2:41:09foreign
2:41:16is
2:41:17of the following form so now I use the
2:41:20expression that you can derive for the
2:41:22constant C which is
2:41:25Alpha Beta
2:41:29over two
2:41:31there is
2:41:332 minus Alpha divided 1 minus Alpha
2:41:381 over G Infinity that we know what it
2:41:41is so this is Lambda 0 divided by 1
2:41:43minus Alpha so you can plug it in if you
2:41:46want and then I have e to the minus
2:41:48beta 1 minus Alpha times
2:41:52absolute value of Tau so above I assume
2:41:56that I was positive if you assume it
2:41:57negative you get a very similar
2:42:00um
2:42:01results that gives you this final
2:42:04expression
2:42:06okay and now finally
2:42:09and perhaps I'm not doing this
2:42:11explicitly but once we have the
2:42:14correlation we can plug this
2:42:16expression into the form
2:42:18or the equation for
2:42:20rope so they clustering ratio
2:42:30so this requires doing other
2:42:33exponential integrals but that those are
2:42:36straightforwards so I will just
2:42:38comment on what you get in the end
2:42:41so from equation five uh in the today
2:42:44where row is called r
2:42:50you know that this will be 1 plus 2 this
2:42:54G Infinity integral from 0 to Tau in the
2:42:58U
2:42:591 minus U over Tau times the function of
2:43:03C that we just determined
2:43:05so C is an exponential so if you
2:43:07integrate this term will give you an
2:43:10exponential and the second integral is
2:43:12just an exponential times U so the
2:43:15fastest way to do this if you have the
2:43:17integral of x e to the minus alpha x is
2:43:21to write it as minus the derivative over
2:43:24a or Alpha of the integral
2:43:27of e to the minus ax so you just have to
2:43:31compute exponential integers and then
2:43:33eventually derive it
2:43:35with respect to the to the exponent so
2:43:38you can do this in here and you get out
2:43:40the following that now we can
2:43:43comment
2:43:46so we have a factor of 1 over
2:43:491 minus Alpha Square
2:43:52and then we have a factor that depends
2:43:55that contains 1 over Tau which comes
2:43:58from here
2:44:00some numbers to minus Alpha over Alpha
2:44:04so these are coming from the constant C
2:44:06in the correlation function
2:44:10okay
2:44:11and then we have exponentials coming
2:44:14from the integral with
2:44:16this gamma exponent
2:44:20that is beta times one minus Alpha
2:44:26okay so again what you see so now this
2:44:30is an expression for arbitrary times Tau
2:44:34so what we can do is to study
2:44:36the two limits of this Expressions so
2:44:39first we assume so we have a relaxation
2:44:41time uh in here let me call it Tau
2:44:44relaxation that is
2:44:471 over beta
2:44:50times one minus Alpha
2:44:53so this tells me that if I choose times
2:44:57which are small with respect to this
2:44:59relaxation times I can approximate this
2:45:02function uh rasi by I can do the linear
2:45:04expansion of the exponential and if
2:45:06instead I'm looking at very large times
2:45:08I can approximate be exponential with
2:45:10with zero essentially and this will give
2:45:13me the two limits so for sure times
2:45:21meaning shorter than the relaxation
2:45:24times I expand linearly this exponential
2:45:27and what I get is that
2:45:30row of tau is approximately
2:45:34well the same factor of 1 over
2:45:371 minus Alpha Square
2:45:39and then I have so let me copy this to
2:45:42minus
2:45:44Alpha
2:45:462 minus Alpha
2:45:481 minus Alpha cubed
2:45:51and then you see if I expand this this
2:45:53will give me
2:45:541 minus Tau which cancels so the one
2:45:58will cancel and the Tau coming from the
2:46:01linear term will cancel with this
2:46:03there is a factor of beta that will
2:46:05cancel with this and then there is a
2:46:06factor of 1 minus Alpha that will cancel
2:46:08with one of these Powers so if you do
2:46:10this
2:46:11you just get this
2:46:14and you see that what you have in the in
2:46:17the numerator is 1 minus two alpha plus
2:46:20Alpha Square so this is precisely
2:46:22y minus Alpha to the power 2 so you see
2:46:25that in this short time limit
2:46:27your clustering ratio is essentially
2:46:31equal to one in particular in the in the
2:46:33limit Tau going to zero this is uh well
2:46:35no
2:46:37you have this Singularity yes in the
2:46:39limit Tau equal to zero this is exactly
2:46:41equal to one because you cancel this
2:46:44Singularity with this linearized
2:46:46function
2:46:47and what is one well what is one is the
2:46:50poisson
2:46:59and why do you get the personal ratio
2:47:01well one way to to understand it
2:47:03intuitively so remember that we said
2:47:07that our Lambda
2:47:11was Lambda 0 that is what you would get
2:47:14if you just had a personal process plus
2:47:17your kernel
2:47:19and your kernel was an integral over
2:47:22time and so what you can expect is that
2:47:25in order for this kernel to be important
2:47:28you have to allow your process to uh to
2:47:32run a little bit because the process has
2:47:34to realize that there are some events at
2:47:37previous times and has to be excited by
2:47:40this previous event and therefore if you
2:47:43just look at very short times you
2:47:44basically don't see the aspect of this
2:47:47excitation kernel and you just recover
2:47:49the poisson result
2:47:51on the other hand if you've got a very
2:47:52long time or large times
2:48:00then you can eliminate this exponential
2:48:04you also have a factor of one over Tau
2:48:06that kills completely uh this second
2:48:09term uh on the right and so you see that
2:48:12your profile goes asymptotically as one
2:48:16over
2:48:17one minus Alpha Square
2:48:20which is what was given in the lecture
2:48:22foreign
2:48:29larger than one because Alpha is
2:48:32positive so the denominator is smaller
2:48:34than one
2:48:35so again just to have an example in the
2:48:37arm work if you really do the simulation
2:48:41of of the hoax process you can compute
2:48:45this clustering ratio and you do this
2:48:47taking the ratio of the variance with
2:48:49respect to the expectation value and as
2:48:52a function of Tau you should see
2:48:54this you can see in the solution
2:48:56something like this so you see that at
2:48:59zero time it starts from one and then it
2:49:01converges to some value that now we know
2:49:04it has to be equal to one minus Alpha to
2:49:07the power two so you can choose a value
2:49:09of Alpha and then you can check that
2:49:11this is the case and this is number five
2:49:15point four
2:49:17I think
2:49:21okay
2:49:22uh
2:49:24what else well maybe just one finite
2:49:27comment final which is related to
2:49:31[Music]
2:49:32um
2:49:33uh to to to
2:49:35the third point of exercise one that we
2:49:38are not doing in detail but I will just
2:49:40comment
2:49:42so in the end what uh what do we derive
2:49:45so we derive
2:49:46in particular this expression
2:49:49for the relaxation time
2:49:52and for the asymptotic
2:49:54uh
2:49:55average rate which is
2:49:58Lambda 0 1 minus Alpha
2:50:05and I just want to point out that you
2:50:07really see the aspect of this excitation
2:50:10currentness so on one end
2:50:13you see the effect because whenever
2:50:15Alpha so when Alpha is equal to zero you
2:50:16recover uh question whenever Alpha is
2:50:20different from zero you have a value of
2:50:22G which is larger so you have more
2:50:24events than what you would expect in an
2:50:26independent question process and this is
2:50:28because as soon as you have one event
2:50:30your excitation kernels promotes uh new
2:50:33events uh in your process and this is
2:50:36what you read out from from this and
2:50:39also you have a relaxation time
2:50:41which is again larger than what you
2:50:45would have in absence of the current so
2:50:47when Alpha is equal to zero because
2:50:52so if you uh or actually this is larger
2:50:55than what you would have somehow
2:50:58if you just so this beta is what
2:51:00controls the decay of the kernel here
2:51:02right so this was
2:51:05the kernel times
2:51:07uh G of Tau itself
2:51:11so we have a natural scale at which this
2:51:14kernel decays which is one over beta in
2:51:17our exponential example and this tells
2:51:20you that if you have a mother event so
2:51:22one point which occurs then this will
2:51:26influence the future history up to times
2:51:29which are of the order of one over beta
2:51:32naively
2:51:33but then the fact that you have this
2:51:35self-excitation it tells you that
2:51:37actually the influence goes over to
2:51:39times which are larger than 1 over beta
2:51:41by a factor of uh one times uh one over
2:51:45one minus Alpha and this is again an
2:51:48effect of an effect of self-excitation
2:51:50so what happens is that you have a
2:51:52mother event
2:51:53this of course will influence all the
2:51:55history in the future but if the mother
2:51:58event gives rise to one son so think
2:52:01again in terms of this uh Carlton Watson
2:52:05birth processes the sun itself will
2:52:09influence the future I will increase the
2:52:12probability to to have more events and
2:52:15and this is what is measured by this
2:52:17factor of one over one minus Alpha and
2:52:19in particular in the third part of the
2:52:22exercise if you think about this as a
2:52:26person that actually birth process what
2:52:28you realize is that one over beta is
2:52:31like the fertility of of the mother so
2:52:34time scale over which the mother can
2:52:36make songs but this is increased by a
2:52:39factor of one over one minus Alpha which
2:52:41is the average size of the family
2:52:43generated by the mother and this
2:52:46accounts for the fact that the mother
2:52:48can generate many sounds and each of
2:52:51these phones will increase the
2:52:52probability to future events and this is
2:52:55what eventually will increase if you
2:52:57want the influence of the mother event
2:53:00by this factor of one over one minus
2:53:02Alpha so to make this a bit more precise
2:53:04you can go and have a look at the third
2:53:07exercise but somehow the idea is is what
2:53:12I just sketched
2:53:14okay so this was
2:53:16for what concerns hoax
2:53:18and now
2:53:21we don't have much time
2:53:23I just wanted so for those of you who
2:53:26want to pay
2:53:28maybe to sketch very briefly
2:53:33um
2:53:38this idea of Poker plank
2:53:41with multiplicative noise
2:53:44so if you
2:53:46had time to go back to the tedda 3 and
2:53:49the solutions you have seen how to
2:53:51derive soccer planks for instance
2:53:54in the case of additive noise and I
2:53:56think this is also discussed in other
2:53:58courses
2:54:00so the only thing I wanted to point out
2:54:02is that whenever you have multiplicative
2:54:04noise again
2:54:06you have to always specify the
2:54:08prescription of this critization that
2:54:10you use
2:54:11so whether it's Ito or shatanovic
2:54:15and you get two different forms if you
2:54:18remember of the focal Planck equation
2:54:22so now this is let's say soccer plank
2:54:27I'll try to be short
2:54:30and just to give the main ideas
2:54:33so photo plan came from launch event
2:54:37so in one dimension you have the X where
2:54:40DT equal to some drift term
2:54:43F of t plus
2:54:45our noise
2:54:48and remember that we could discretize
2:54:54these equations so we can say that over
2:54:57a small interval DT
2:54:59we increment
2:55:01so the value of the process at the time
2:55:03t plus DTS is related to the value at a
2:55:06time t
2:55:09by this F evaluated
2:55:13at some point
2:55:15within let me call it X bar of T of
2:55:19course this is
2:55:21NX
2:55:23these are functions of the process
2:55:26so this was a point to be chosen
2:55:29properly inside the interval TD plus BT
2:55:33and then we have the same
2:55:38times the noise in the interval times DT
2:55:42and remember that the noise is of order
2:55:44one over square root of DT so what we
2:55:47have in here
2:55:48is of order
2:55:50square root of DT
2:55:54so we can rewrite it and this is useful
2:55:57to derive the plank as
2:56:02so we can call this
2:56:05as each scale
2:56:09sorry now I the D is I call it capital B
2:56:16assuming that then I will take the limit
2:56:17delta T going to zero so this is our
2:56:20order capital d t so I can rewrite it as
2:56:23the order of magnitudes which is square
2:56:25root of DT times
2:56:27some random variable which is of order
2:56:29one and which is gaussian and which
2:56:32reproduces the statistics of uh of the
2:56:35White Noise which means essentially that
2:56:38is it a square has to be equal to Sigma
2:56:41Square where Sigma is is the variance of
2:56:45the noise at equal time
2:56:47so if you want Z is the little bit of
2:56:50noise which Acts in the interval t t
2:56:53plus DT rescaled by square root of pt
2:56:58okay and now this is essentially one
2:57:02what you
2:57:03mean
2:57:04to inject or a useful way to think about
2:57:08the problem when you want to derive
2:57:10differential equations like the soccer
2:57:12Planck equation
2:57:14so the soccer plant equation is an
2:57:16equation for a quantity
2:57:18p
2:57:22of x t given
2:57:25x 0 to 0. so what is this this is the
2:57:28probability in our average overall
2:57:30possible trajectories so the role
2:57:32possible realization
2:57:33of the noise
2:57:35to be
2:57:37X time T given that
2:57:40of time t 0
2:57:42I was
2:57:44in a given position at zero
2:57:47and if you remember the structure of the
2:57:50equation for this P was different
2:57:52depending on whether when you uh Vito or
2:57:56the statino which convention
2:57:58so let me give you the form for uh one
2:58:02of each
2:58:04so we as we introduce
2:58:08this B of X is just
2:58:11it must put over 2 times 3 of X which is
2:58:13the function which multiplies the noise
2:58:15in the large of an equation and then
2:58:17from strathanovic
2:58:21the form of the equation was as follows
2:58:23so now I will drop the dependence on the
2:58:25argument
2:58:26because of the form DP over DT
2:58:29equals to minus
2:58:32the derivative over X of f
2:58:35which is the function appearing in the
2:58:38larger one equation times p
2:58:40Plus
2:58:43the derivative over X of D of x times
2:58:47the derivative over X
2:58:50of B times
2:58:54so there was a first there is this
2:58:56metric structure that appears in the
2:58:59second derivative whereas if you derive
2:59:01the focal plants for Ito if you remember
2:59:04you just have a double derivative in
2:59:06here of uh
2:59:08of the Square Times p
2:59:11uh square root
2:59:15I think
2:59:16No it should be fine
2:59:21yes
2:59:23okay
2:59:35okay
2:59:37and now given that we have not much time
2:59:40I would just uh perhaps uh tell you
2:59:45the starting point or the main ideas of
2:59:48the derivation so how do you derive an
2:59:51equation for these
2:59:52so you can start from one identity which
2:59:55is in general true
2:59:58for probabilities that is that is just
3:00:01what is the probability to be at the
3:00:03point x
3:00:04time t plus DT given that your FX 0x
3:00:11times is zero
3:00:13what you can do is to take any time
3:00:15between Type P0 and t plus TT and
3:00:19write the this is equal to the
3:00:21probability
3:00:22that you were at a given value of y at
3:00:25that time so it's being
3:00:28let me write it directly and I recommend
3:00:31so the probabilities that you reach X as
3:00:34time C plus DT given that you were at Y
3:00:36at the previous time p and then you have
3:00:39the probability that you were at Y at
3:00:42time T given that you were at x 0 x 9 t
3:00:450 and you have to integrate overall
3:00:48possible conditions or positions that
3:00:51you reaches time t
3:00:53so this is a very general identity of
3:00:55conditional probabilities if you want
3:00:59and then once you have this identity
3:01:01what you do is you take uh both the
3:01:05right and the left hand side you choose
3:01:07a function s
3:01:09arbitrary
3:01:12and smooth
3:01:17you multiply both the right and the left
3:01:20hand side by this function f and then
3:01:21you integrate over X
3:01:24so on the left hand side you will the
3:01:26left hand side is Trivial you just have
3:01:28the integral over DX so
3:01:32f of x times
3:01:34C of x
3:01:37slash DT
3:01:39even
3:01:42x 0 x 0.
3:01:46and on the right hand side you have a
3:01:47double integral
3:01:51right and what you can do in this double
3:01:54integral is
3:01:56so let me
3:01:58write it in the following ways so
3:02:01or just to be short so you have a double
3:02:03integral where you have to integrate f
3:02:05of x in the X
3:02:08and now f of x you assume it to be
3:02:11smooth so this means that you can
3:02:13approximate
3:02:15f of x
3:02:16as
3:02:18F of Y so assuming that Y is the point
3:02:22where you are at the previous time and
3:02:24the time difference between the times
3:02:27Associated to X and the one Associated
3:02:29to Y is small then you can expand this F
3:02:32and you can write this as as Prime of Y
3:02:36times
3:02:37y Plus
3:02:39the second derivative evaluated that's
3:02:42why over 2 times x minus y square and so
3:02:46on
3:02:47so now what I do in the left hand side
3:02:49is to do the integration over DX f of x
3:02:52and then I replace f of x with the X
3:02:55function around Y and I will get
3:02:59different terms so let me write the
3:03:01first one
3:03:03so let me exchange the integration of Y
3:03:06and X so let me
3:03:09take first this first contribution which
3:03:12depends on why so we'll have integral
3:03:15over d y of f y then I have this second
3:03:19probability which depends on your why so
3:03:21I will put it here
3:03:23and now I will drop the dependence on on
3:03:27the previous condition on X Sub 0 and d0
3:03:30and then in this case I just have the
3:03:32integral over X of this
3:03:35so P of x e plus BT given that I was at
3:03:40Y at time t
3:03:43then I have the term which contains the
3:03:47first derivative of the function f
3:03:49so I will like integral over the Y of X
3:03:53Prime of Y
3:03:55B of y t
3:03:59and now inside the integral depending on
3:04:01x
3:04:03I have an extra term because you see
3:04:04that I have this x minus y so this
3:04:08increment which depends on X so let me
3:04:11write it here
3:04:13e of x
3:04:15e plus BT
3:04:17Y2
3:04:20okay
3:04:21plus I will have a term coming
3:04:24containing the second derivative which
3:04:26is also important and then I have higher
3:04:28order
3:04:30so now let me introduce three quantities
3:04:32I'm going to leave you so uh the first
3:04:34quantity the three quantities correspond
3:04:36to the integrals over X which I find at
3:04:39each order in the expansion so the order
3:04:42zero if you see is just the
3:04:44integral of this probability so this
3:04:47tells you what is the probability that
3:04:48you start at Y at time speed and you end
3:04:51up somewhere at times t plus BT well if
3:04:54you integrate over the final State this
3:04:57probability is simply equal to one
3:04:58because you know that you will end up
3:05:01somewhere at arbitrary time and if you
3:05:04impose no conditions that the
3:05:05probability has to be equal to one
3:05:08this quantity here inside it is not
3:05:10equal to one it depends on Y and I will
3:05:14call it i1
3:05:16of white
3:05:17and you will have an analogous term
3:05:19coming from the second order expansion
3:05:22so in the end
3:05:27the important things that you have to
3:05:29compute
3:05:31and with this maybe it will stop
3:05:35are
3:05:37i1 and I2
3:05:39foreign
3:05:41is what I just defined
3:05:44so this is the integral of a small
3:05:47increment
3:05:50with respect to the uh
3:05:53the probability of your process
3:05:59you can go smaller
3:06:03over y
3:06:05EP
3:06:07so this is an average increment so you
3:06:09assume that you start at Y and you ask
3:06:11what is the average
3:06:13increments that I do over a small
3:06:15intervalitative and what is also
3:06:18important that will appear in the second
3:06:20order term is
3:06:23is he Imaging
3:06:25the fluctuations of this
3:06:29increments
3:06:31that are just obtained taking the power
3:06:33to
3:06:35okay
3:06:37and now this is what uh you have to
3:06:40compute and this is where to compute
3:06:42this quantities where you have to be
3:06:44careful about the discretization so we
3:06:48will probably go back to this but let me
3:06:50just comment to all of these details in
3:06:53the three
3:06:55exercise three
3:06:58and if you do the calculation properly
3:07:00assuming either versus
3:07:02what you should find is that this term
3:07:07I2 is actually the same for both e to
3:07:10instructions
3:07:18and it's easy
3:07:20to compute
3:07:22whereas this average increment is what
3:07:24is tricky so this is what will take
3:07:26different form
3:07:29foreign
3:07:39on the type of discretization that you
3:07:43choose and so this is what you have to
3:07:45be careful about in particular if you do
3:07:47the satanovic prescription and this is
3:07:50what in the end we'll give you two
3:07:53different forms of your pocket blank
3:07:56equation depending on how you choose to
3:07:58discretize your process
3:07:59so I don't think we have time now it
3:08:03to uh to go through this but you can
3:08:05check it uh in the today and if there
3:08:07are questions as I say there will be the
3:08:10seven where we are gonna discuss an
3:08:13example of this and there we can go back
3:08:15to the details uh if there are doubts
3:08:19okay
3:08:20so I think
3:08:22we can stop
3:08:24if there are no questions
3:08:30then I wish you a happy holiday week
3:08:34and we will see each other's theme
3:08:38three weeks I think and I will write an
3:08:41email with all the information on this
3:08:42schedule
3:08:44okay
3:08:49thank you thank you very much

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