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Hi folks. So, we're back again and let's talk a little bit

Â more about solving this SIS model and getting

Â explicit expressions for the infection rates

Â when there exists a positive steady state infection rate.

Â And so, if you recall when we were looking at solutions,

Â we could find the steady state in terms of the theta which is

Â the fraction of people that you would meet randomly who are infected.

Â Then it was given by this expression and we can simplify this

Â by dividing each side by theta and we end up with an expression that looks like this.

Â And so solving this for the theta would then give us a solution.

Â So, let's first look at

Â a regular example with regular network so everybody has exactly the same degree.

Â Then P(d) is just going to put weight on

Â some particular degree which is just the expected degree.

Â And so then things simplify and we end up with

Â 1 = Î£ P(d) Î»d^2 /[ (Î»Î¸d+ 1) E[d]].

Â So, that's going to simplify and we'll end up with an expression which looks like this,

Â 1= Î»E[d]/ (Î»Î¸E[d] + 1).

Â So, just plugging in that everybody

Â has the same degree which is just the expected degree,

Â then we can get rid of this, Î£ P(d),

Â this is going to become E[d]^2,

Â and this is the expected degree here, expected degree here.

Â We can get rid of one of these,

Â get rid of that and we end up with this expression here, okay?

Â So, when we solve that we end up with a very simple expression.

Â Now, we can rearrange this in terms of theta and we

Â end up with Î¸ = 1 - 1/ Î»E[d]).

Â So, if everybody had the same degree then we can solve explicitly for what

Â the steady state expression is going to be for the infection rate.

Â And we notice now that this is increasing in Î»E(d) and in order for this thing to

Â be at least zero then we need Î»E(d)

Â to be greater than

Â one or greater than equal to one or greater than one for it to be positive.

Â So, that was what we found in regular networks and

Â then this thing just scales up with lambda and with

Â the expected degree so it's actually just linear in the level of Î»E(d),

Â which is effectively just this infection rate.

Â So this is one where it goes back to the original model we looked at where we just had

Â random meetings and this is just increasing the rate of random meetings

Â and things are just proportional to the lambda parameter.

Â Okay. So, let's have a look at a more interesting degree distribution,

Â one where we have this power law,

Â and if you plug in the power law

Â and integrate this out

Â then you'll end up with an expression that you can solve for theta.

Â And, in that case if you want to go through and verify,

Â you can just integrate this and then solve for

Â theta or you can take my word for it and theta

Â will come out to be Î¸ = 1 /(Î»(e_1/Î» - 1) ).

Â So, what do we end up with?

Â We end up with theta having an expression which

Â depends on lambda and we can plot that out.

Â So, if you plot that function out,

Â so we're just looking at this function right here,

Â and plotting this function as a function of lambda.

Â So, how does it vary with lambda?

Â And what we see is it's very rapidly increasing.

Â So, as lambda increases we get a very rapid increase and then eventually it asymptotes.

Â It can't go above one,

Â but we're getting a very high neighbor infection rate

Â as lambda increases because then we've got these very high degree nodes.

Â They become infected, they infect others and so forth,

Â and as lambda is increasing,

Â we get a very rapid infection increase.

Â Okay, what can we say about how these things change with the degree distribution?

Â So, if we want to do comparisons and say okay,

Â if we go from regular to power law or a regular to Erdos Renyi, sort of, graph?

Â How is that going to change?

Â And one way we can do that is we can look at this expression

Â and ask how this right hand side changes with P(d).

Â Right, because remember the way that we're solving this,

Â we look at theta here,

Â we have this right hand side,

Â which is H(Î¸) and we're looking for

Â the solution to this thing and if we can say that H(Î¸) goes up,

Â right, so if we do something that changes H(Î¸) in a way that goes up then

Â that's going to move the solution to this equation upwards.

Â So, any kind of comparative static where we're making changes

Â that change the distribution in a way that

Â increases this overall expression on the right hand side for each theta gets

Â a higher value then we can say something about what the resulting change is in theta.

Â So, let's see what we can say about how this right hand side moves.

Â Okay. So first thing,

Â if you look at this function,

Â what we're doing is we're weighting it by

Â different degrees and then we've got some function

Â here that we're taking an expectation over with respect to the different degrees.

Â And, what can we say about this function right here?

Â How does it behave?

Â And one thing we notice,

Â so we're taking expectations with respective degrees,

Â this thing is increasing in degrees, okay.

Â So higher degree nodes are going to have

Â higher relative expected infection rates and basically that's what we're getting here.

Â This is remember our old P(d) and so,

Â this thing is going to tend to be higher for higher degree nodes.

Â So, more connected nodes are going to tend to have more contact,

Â they're going to tend to be more infected so this overall function is increasing

Â in D. So that tells us that any distribution which puts weight,

Â puts more weight on higher degree nodes,

Â is going to have relatively higher infection rates.

Â That's going to move this whole function up.

Â That's going to give us a higher solution and a higher steady state.

Â Okay. So one thing we can say is that if we take a distribution and then shift it,

Â so that we put more weight on higher degree nodes,

Â that's known as first order stochastic dominance

Â when you are comparing two distributions.

Â So, if we have two distributions and we move it towards

Â higher degree weights then that right hand side of this thing,

Â this H(Î¸) is going to increase that

Â every theta we're basically gonna have higher infection rates.

Â That's going to lead to a higher steady state.

Â Okay. So, basically putting weights on

Â higher degree nodes is going to increase infection.

Â Okay. So, that's relatively straightforward and

Â basically we're just shifting the weight towards higher degree nodes.

Â So when we do that we end up with shifting this H function

Â higher at every theta and that leads us from a steady state with P. So,

Â if we go to P prime,

Â which is increase the weight on higher degree nodes,

Â we're going to end up with an increased steady state

Â so the theta that solves this is going to be higher.

Â Okay. So, let's take another look at this.

Â So that just says that if we are all shifting weight towards

Â higher degree nodes in a very well-defined sense

Â this notion of first order stochastic dominance.

Â And just to, sort of,

Â give you a feeling for first order stochastic dominance,

Â if you're dealing with a frequency distribution

Â where you've got different degrees down here,

Â you know, say one, two, etc.,

Â three, so you've got some degree distribution.

Â First order stochastic dominance shifts are ones where we're

Â essentially moving the distribution to the right so we're putting

Â more weights on higher degree nodes and that's

Â what's moving us up and having us have more interactions,

Â higher infection rates, everybody gets a higher steady state of infection.

Â So, if you have, you know,

Â when you look at the world and you have increased travel or

Â increased contact with people,

Â you're going to have increased spread and things like the flu or other.

Â In this particular model,

Â it's something you can catch repeatedly but you're going to have increased contacts

Â and increased number of contacts per individual are going to

Â increase the steady state infection rate.

Â Okay, that's fairly intuitive, fairly simple.

Â Let's do a little more nuanced calculation now and again we're looking at this function,

Â this RHO of D function,

Â so the infection rate for different degrees.

Â And if we look at that function,

Â it's also a function which is convex in D, okay.

Â So if you look at this function,

Â it's a function of d squared over something which is linear in D. This

Â is actually a convex function in D. So, in fact,

Â when you look at what this RHO of D function looks like,

Â it is not only increasing,

Â so this function is increasing and convex, okay.

Â So, that tells us, okay, first of all,

Â if we put more weight on higher degrees we're going to end up with

Â higher values for this are going to come out.

Â Right, so as we, we put on higher degrees,

Â we get, here is degree,

Â here is this RHO of D function,

Â and as we put weight on higher degrees we're getting higher values.

Â That was what we just showed but also even if you took a mean preserving spread,

Â so suppose instead of putting all your weight on some particular value E of D. So,

Â we start with a regular network and instead we spread it out,

Â so that now we have half our weight on something lower,

Â half our weight on something higher,

Â but we move these in equal distance.

Â If we keep the same mean then when we take the expectation over the higher and lower,

Â we're going to end up with a higher expected value than what we started with.

Â Okay. So, the idea is if you're taking an expectation of

Â a convex function and you do a mean preserving spread,

Â so you move some weight higher and some weight lower,

Â when you're moving lower, well,

Â the rate at which you slow down decreases,

Â but here you get an increasing rate at which you get higher values.

Â So the convexity of this function means that the expectation is higher.

Â So, if you take a mean preserving spread,

Â so if you start with some P and you

Â take some expectation with respect to P of some function,

Â in this case our row of D,

Â and now you take a mean preserving spread, P prime,

Â and you take the same expectation of RHO of D. If this is a mean preserving spread,

Â so you've kept the same mean but you've spread out and put more weight on the extremes,

Â then what you're going to end up with is a higher expectation

Â and that's going to lead then to a higher value of the right hand side here, okay.

Â So this is a form of what's known as

Â a second order stochastic dominance where you fix the mean.

Â So, taking mean preserving spreads on convex functions gets you a higher expectation.

Â So, since this is a convex function,

Â we can say that mean preserving spread is also

Â going to increase things and this is why, you know,

Â so even though you're losing some degree on some nodes,

Â you're increasing it on other nodes.

Â The fact that those are hub's could actually increases the expectation overall.

Â So, if P prime is a mean preserving spread of P,

Â then the right hand side increases at every theta and so what happens?

Â Well, it increases everywhere.

Â We end up with a higher steady state, okay.

Â So, either way that we went through things, mean preserving spread,

Â more high degree nodes and low degree nodes,

Â but the higher degree nodes are more prone to infection.

Â Neighbors are more likely to be high degree.

Â So, either first order stochastic dominance or mean preserving spreads,

Â both of those lead to increases in the infection rate.

Â So here we are now able to say something about

Â the degree distributions of interactions and how infection rates,

Â so it's a nice model in terms of allowing us to be

Â able to do these kinds of calculations.

Â 13:24

Okay, what about average rates?

Â So, what we've been talking about is theta, right?

Â So, theta is the chance when you're meeting somebody in the population,

Â that they're infected and that has been what we showed was increased,

Â as we increased in the senses of first or second order stochastic dominance.

Â But what about the actual average,

Â the RHO in the population?

Â Okay. So, if we take expectations over all degrees.

Â So, the higher degree people are going to be infected at higher rates.

Â So, when you're meeting them at higher values that means

Â that people you're going to meet are more infected but

Â if we're somebody who just cares about the average level in the population.

Â So, if I'm a government and I care about how infected my population is, ultimately,

Â what I care about is what Rho is,

Â not what theta is.

Â So, theta is very important in determining what

Â the steady state is going to be but the thing I might be

Â interested in in terms of my policies is

Â what fraction of my population ends up being infected?

Â Okay. So, in work with Brian Rogers we looked at

Â this question in more detail and interestingly the things can

Â reverse themselves when you get to looking at Rho compared to theta.

Â So, if you look for instance at what we just

Â did where you take a mean preserving spread of

Â a distribution then the highest steady state of theta, that goes up.

Â So, we end up with a new theta prime which is higher than what it was before.

Â But, the corresponding Rho,

Â it goes up if lambda is relatively low,

Â but it actually goes down if lambda is very high.

Â So, if you've got a very high infection rate, lambda,

Â then the corresponding Rho prime goes

Â up in the case where lambda is low but it actually goes down if lambda is high.

Â Okay, and what's the intuition behind this?

Â So, let's sort of go through the intuition and then we

Â can take a quick look at why this is true.

Â The intuition is that,

Â that in situations where lambda is very high,

Â the high interaction nodes are already going to be very infected.

Â And so, actually increasing,

Â in putting weight on higher degrees

Â isn't going to matter that much because those nodes are

Â going to be infected at such a high rate that they're going to already be infected.

Â And so you're not changing that much but putting

Â more weight on low degree nodes can actually decrease the,

Â so now you've got some people who have very few interactions.

Â Those people can actually end up being infected with lower rates.

Â So, the actual overall infection rate in

Â the society can be balanced by the fact that you are increasing some of

Â the higher degree nodes but those people are already going to

Â be infected even without this increase and

Â the low degree nodes as you move them towards lower part they can actually end

Â up with lower infection rates and so that counterbalances it.

Â So, when you average across the population,

Â not with the relative frequency of meetings,

Â you actually end up with a decrease in the overall rate.

Â So, in terms of pictures,

Â here's what the picture looks like.

Â So, here's our, this is actually log of lambda

Â in this picture and then this is the largest steady state RHO.

Â So, this is the positive Rho in equilibrium and the blue figure,

Â these are regular networks,

Â and these all have the same mean.

Â Regular network, this one is a power law,

Â the green one and the red one here is one for an exponential.

Â So basically, the growing version of

Â the Erdos-Renyi random network all with the same mean,

Â it's what you see is the framed is relatively low then

Â the regular one has everybody right at the mean is the lowest.

Â Spread it out in terms of exponential, it increases things.

Â Spread it out even further in terms of going to a power distribution so here,

Â as you increase the variance,

Â you end up increasing the relative Rhos so that's what's happens for low lambda

Â but once you get to very high Lambda the picture completely flips itself and

Â the regular one has

Â the highest infection rate and the power one actually has the lowest infection rate.

Â So these things can reverse themselves as you get to

Â higher lambdas and that's because Rho

Â is actually something you're weighting things by

Â their frequency in the population not by meeting rates.

Â And so, if you actually want to go through a proof of this,

Â I'll just go through a quick proof.

Â So, one way you can do this is remember in a steady state you have to

Â have these changes in Rhos be equal to zero.

Â So, zero is equal to this expression,

Â which is the change in Rho that we got before, which, you know,

Â remember our expression for what the change in

Â the infection rate of anybody's given degree was per unit of time

Â and, again, that was some recovery.

Â This is the rate at which they recover.

Â This is the fraction that aren't infected.

Â This is the rate at which they become infected depending on theta.

Â And if you just take,

Â so we've got an expression which looks like this thing is equal to zero.

Â Now just, let's take an expectation over D.

Â So we'll just take an expectation over D. What do we end up with?

Â We're going to end up with V theta D,

Â times to one take an expectation of that,

Â you get this expression,

Â take an expectation of this,

Â you get that expression,

Â take an expectation of this you get the actual overall Rho times Delta and

Â so what do we end up with is an expression where this first one looks like V times theta.

Â This, if you take that expectation,

Â is just our definition of what theta was.

Â So, we get theta squared times E of D and then the minus Rho of D.

Â And so, you can solve this for how Rho behaves as a function of theta.

Â So now, what we get is Rho is equal to something which is proportional to theta.

Â But also, times one minus theta.

Â So this, so what we can do is as we look,

Â change theta, how do,

Â how did that change Rho.

Â Well, the right hand side here is increasing in theta,

Â theta is less than a half but

Â it's decreasing in theta if theta's bigger than a half, right?

Â So, once theta's bigger than a half then this thing actually starts

Â decreasing in theta so this is proportional to theta times one minus theta.

Â This is an increasing function once,

Â up to one half and then decreasing thereafter and so,

Â what happens is initially as you're increasing theta you're

Â increasing Rho up to theta equals a half

Â and then beyond that,

Â as theta continues to increase,

Â Rho actually decreases above this level of a half.

Â And so what we get is,

Â initially we get an increasing part of Rho and then actually

Â then increases in theta lead to decreases in Rho, okay.

Â And overall we also have that theta as

Â increasing in lambda so then you can do comparative statics of Rho in

Â terms of lambda because the lamda's going to change

Â the theta which then changes the Rho, okay.

Â So, what we've done is gone through the SIS diffusion model,

Â our most useful model just to get

Â some comparative statics out and to study some techniques.

Â So, these techniques of actually looking at

Â distributions and talking about changes in degree distributions,

Â doing stochastic dominance, actually turns out to be quite useful.

Â It's been used now in other areas besides in

Â the original paper with Brian Rogers but now been used in games,

Â on networks and other areas as well.

Â And the SIS model is a very simple and tractable model.

Â So, it's very stylized.

Â It's nice because it brings in relative meeting rates which has

Â some elements of network structure.

Â And then we can order infections by properties of

Â networks in terms of degree distributions.

Â What are some of the limitations of this model?

Â Well, important limitations are, first of all,

Â just in terms of this SIS we lose the fact that a lot of

Â applications are ones where you become infected but then if you

Â recover you're actually immune to catching the disease again,

Â which is actually true of some flus and other kinds of things where if, you know,

Â if you add new virus protection software to your computer then you won't

Â get certain viruses again and you get a new virus but not the same old one.

Â So it limits, it's somewhat limited in terms of the applications.

Â Also the interactions that we talked about were completely random meeting processes.

Â So, it was not as if we'd drawn out

Â a network and actually had people located on the network.

Â We just had people bumping into each other and meeting each other and so

Â that's a special kind of process which gives rise to special kinds of conditions.

Â Now we, more generally what we'd have to do if we start

Â to work with things where the network architecture is given,

Â then it's going to be more important to use simulations and so forth.

Â So, we did some calculations before where we talked about

Â component science and so forth and that gives us some insights.

Â But, more generally if we actually want to study

Â these processes a lot of it's going to be done by simulation.

Â So, if you give me a particular network and ask what's going to happen on it then

Â I might have to write down a program and actually simulate what's going to happen there.

Â So, the next thing we'll look at is a simple model

Â of diffusion where we'll do some calculations and just

Â simulate that model and see exactly how it

Â works and that will allow us to actually fit something directly to data.

Â And there's a large amount of that that goes on in

Â epidemiology and marketing and

Â other kinds of areas where you're trying to make predictions.

Â If you know something about the network you're working with you can actually simulate

Â things and that's going to go a long way towards improving your accuracy.

Â So, the SIS model gives us nicer intuitions,

Â simple ideas, but it's not one you're going to easily take to data.

Â We're going to have to enrich the model to fit it to networks.

Â And that's what we'll talk about next.

Â