Hi folks. So we're back again and let's talk a
little bit more about solving this SIS model, and getting explicit expressions
for the infection rates when there exists a positive steady state infection rate.
And so if you recall, when we were looking at solutions.
We could find this steady state in terms of the theta, which is the fraction of
people that you would meet randomly who are infected.
Then it was given by this expression. And we can simplify this by dividing each
side by theta. And we end up with an expression that
looks like this. and so solving this for the theta would
then give us a solution. So let's first look at a regular example
with regular networks so everybody has exactly the same degree.
Then p of d is just going to put weight on some particular degree, which is just
the expected degree. And so then things simplify, and we end
up with one is equal to the expected degree squared over we've got an expected
degree here and an expected degree here, so that's going to simplify.
And we'll end up with an expression which looks like this.
Okay, so just plugging in that everybody has the same degree which is just the
expected degree, then we can get rid of this.
This is going to become expected degree squared and this is expected degree here,
expected degree here. We can get rid of one of these, get rid
of that and we end up with this expression here, okay.
So when we solve that we end up with a very different expression.
Now we can rearrange this in terms of theta and we end up with theta is equal
to 1 minus 1 over lambda times E of d. So if everybody had the same degree, then
we can solve explicitly for what the steady state expression is going to be
for the infection rate. And we notice now that this is increasing
in lambda E of d. And in order for this thing to be at
least 0, then we need lambda E of d to be greater than one, greater 1 or equal to
1, or greater than 1 for it to be positive.
So that was what we found in the irregular networks.
And then this thing just scales up with lambda and with the expected degree.
So it's actually just linear in the level of lambda times E of d, which is
effectively just this infection rate. So this is one where, it's it goes back
to the original model we looked at where we just had random meetings.
And this is just increasing the rate of random meetings, and things that are just
proportional to the, to the lambda parameter.
Okay, so lets have a look at a more interesting degree distribution one where
we have this power law. And if you plug in the power law and
integrate this out, then you'll end up with an expression that you can solve
for, for theta. And in that case if you want to go
through and verify, you can just integrate this and then solve for theta.
or you can take my word for it, and theta will come out to be 1 over lambda times e
to the 1 over lambda minus 1. So what do we end up with?
We end up with theta having an expression which depends on lambda.
and we can plot that out. So if you plot that function out, so
we're just looking at this function right here, and plotting this function as a
function of lambda. So, how does it vary with lambda.
And what we see is, it's very rapidly increasing.
So as lambda increases we get a very rapid increase and then eventually the
asymptotes it can't go above 1. but we're getting a very high neighbor
infection rate as lambda increases because then we've got these very high
degree nodes. They become infected.
They infect others and so forth. And as lambda is increasing we get a very
rapid infection increase. Okay.
what can we say about how these things change with the degree distribution.
So if we want to do comparisons and say, okay, if we go from regular to power law
or regular to[UNKNOWN] sort of graph, how does that going to change.
And one way we can do that is we can look at this expression, and ask how this
right hand side changes with p of d. Right?
Because remember, the, the way that we're solving this, we look at theta here, we
have this right hand side which is H of theta.
And we're looking for the solution to this thing, and if we can say that H of
theta goes up. Right?
So if we do something that changes H of theta in a way that goes up, then that's
going to move the solution to this equation upwards.
So, any kind of comparative static where we're making changes that change the
distribution in a way that increases this overall expression on the right-hand side
for each theta gets a higher value. Then, we're going to, we can say
something about what the resulting change is in theta.
So, let's see what we can say about how this right hand side moves.
Okay. So first thing, if you look at this, this
function, what we're doing is we're weighting it by different degrees.
And then we've got some function here that we're taking an expectation over
with respect to the different degrees. And what can we say about this function
right here, how does it behave, and one thing we notice, so we're taking
expectations with respect to degrees. This thing is increasing in degrees.
Okay. So higher degree nodes are going to have
higher relative expected infection rates. And basically that's what we're getting
here, this is, remember our old rule of d.
And so this thing is going to tend to be higher far for higher degree nodes, so
more connected nodes are going to tend to have more contact, they're going to tend
to be more infected. So this overall function is increasing in
d. So that tells us that any distribution
which puts weight, puts more weight on higher degree nodes, it's going to have
relatively higher infection rates. That's going to move this whole function
up. That's going to give us a higher solution
and a higher steady state, okay? So one thing we can say is that if we
take a distribution, and then shift it so that we put more weight on higher degree
nodes. That's known as first order stochastic
dominance when you're comparing two distributions.
So if we have two distributions and we move it towards higher degree weights.
then that right hand side of this thing, this h of theta is going to increase at
every theta. We're basically going to have higher
infection rates, that's going to lead to a higher steady state.
Okay? So basically putting weights on higher
degree nodes is going to increase infection.
Okay. So that's relatively straightforward and
basically we're just shifting the weight towards higher degree nodes.
So when we do that, we end up with shifting this H function higher at every
theta, and that leads us from a steady state with p.
So if we go to p prime, which is increased the weight on higher degree
nodes, we're going to end up with an increased steady state.
So the theta that solves this is going to be higher.
Okay. So, let's take another look at this.
So that just says that if we, if prefer all shifting way towards higher degree
nodes, in a very well defined sense, this notion of first order stochastic
dominance. And just to sort of give you a feeling
for first order stochastic dominance. If you're dealing with a frequency
distribution where you've got different degrees down here.
you know say one, two, etcetra, three, so you've got some degree distribution.
first order stochastic dominant shifts are ones where we're essentially moving
the distribution to the right, so we're putting more weight on higher degree
nodes. And that's what's moving us up, and
having us have more interactions, higher infection rates, everybody gets and
higher steady state of infection. So if you have you know, when you look at
the world and you have increased travel or increased contact between people.
You going to have increased spread and things like the flu or other.
In this particular model it's something you can catch repeatedly.
but, but you're going to have, increased contact and increased number of contacts
per individual are going to increase the, the, steady state infection rate.
Okay, that's fairly intuitive, fairly simple.
Let's do a little more nuanced calculation now.
And again we're looking at this function, this row of d function.
so the infection rate for different degrees, and if we look at that function
it's also a function which is convex in d.
Okay? So if you look at this function, it's a
function of d squared over something which is linear in d.
This is actually a convex function in d. so in fact, when you look at what this
row of d function looks like, it is not only increasing, so this function is
increasing and convex, okay? So that tells us okay, first of all, if
we put more weight on higher degrees, we're going to end up with higher values
for this are going to come out, right? So as we, we put on higher degrees, we
get[UNKNOWN] here's degree, here's this row of d function, and as we put weight
on higher degrees, we're getting higher values.
That was what we just showed. But also even if you took a
mean-preserving spread. So suppose instead of putting all your
weight on some particular value, e of d, so we start with a regular network.
And instead we spread it out, so that now we have half our weight on something
lower, half our weight on something higher.
But we move these an equal distance. If we keep the same mean then when we
take the expectation over the higher and lower we're going to end up with the
higher expected value than what we started with, okay.
So the idea is if you're taking the expectation of a convex function and you
do a mean preserving spread, so you move somewhere higher and somewhere lower.
When you're moving lower, well the rate at which you slow down decreases but here
you get an increasing rate at which you get higher values.
So the convexity of this function means that the expectation is higher.
So if you take a mean preserving spread, so if you start with some p and you take
some expectation with respect to p of some function in this case our, our row
of d. And now you take a mean preserving
spread, p prime, and you take the same expectation of row of d.
If this is a mean preserving spread, so you've kept the same mean but you've
spread out and put more weight on the extremes.
Then what you're going to end up with is a higher expectation and that's going to
lead then to a higher value of the right-hand side here, okay?
So this is a form of what's known as sec, second-order stochastic dominance where
you fix the mean. So, taking mean preserving spreads on
convex functions gives you a higher expectation.
So since this is a convex function, we can say that a mean preserving spread is
also going to increase things. And this is why, you know?
So, even though you're, you're losing some degree on some nodes, you're
increasing it on other nodes. The fact that those are hubs, it actually
increases the expectation overall. So if p prime is a mean-preserving spread
of p, then the right hand side increases at every theta.
And so what happens? Well, it increases everywhere.
We end up with a higher steady state. Okay.
So either way that we went through things.
mean preserving spread, more high degree nodes and low degree nodes, but the
higher degree nodes are more, more prone to infection, neighbors are more likely
to be high degree. So, either first order stochastic
dominance, or mean-preserving spreads, both of those lead to increases in the
infection rates. So here, we're now able to say something
about the degree distributions of, of interactions and how infection rates.
So it's a nice model in terms of allowing us to be able to do these kinds of
calculations. Okay, what about average rates?
so what we've been talking about is theta, right?
So theta is the chance when you're meeting somebody in the population.
That they're infected, and that is been what we showed was increased, as we
increased in the senses of first or second orders stochastic dominance.
but what about the actual average the role in the population.
Okay, so if we take expectations over all degrees.
So the, the higher degree people are going to be infected at higher rates.
so when you're meeting them at higher values that means the people you're
going to meet, are more infected. But if, if we're somebody who just cares
about the average, level in the population.
So if I'm a government and I care about how infected my population is, ultimately
what I care about is what row is, not what theta is.
So theta's very important in determining what the steady state's going to b,e but
the thing I might be interested in in terms of my policies is what fraction of
my population ends up being infected. Okay.
So what in, in work with Bryan Rogers, we looked at this question in more detail.
and interestingly, the things can, can reverse themselves when you get to
looking at rho compared to theta. So if you look for instance at what we
just did, where we take a mean preserving spread of a distribution then the highest
steady state of theta, that goes up. So we end up with a new theta prime which
is higher than what it was before. But the corresponding row, it goes up if
lambda is relatively low but it actually goes down if lambda is very high.
So if you've got a very high infection rate, lambda then the corresponding row
prime goes up in the case where lamda's low but it actually goes down if lamda's
high. Okay, and what's the intuition behind
this. So let's go through the intuition and
then we can look, take a quick look at why this is true.
The intuition is that, that in situations where lamda is very high, the high
interaction nodes are already going to be very infected.
And so actually increasing and putting weight on, on higher degrees, isn't going
to matter that much because those nodes are going to be infected at such a high
rate that they're going to, to already be infected.
And so you're, you're not changing that much but putting more weight on low
degree nodes can actually decrease the so now you've got some people have very few
interactions. Those people can actually end up being
infected with lower, lower rates. So the actual overall infection rate in
this society, it can be balanced by the fact that you are increasing some of the
higher degree nodes. But those people are already going to be
infected even without this increase. And the low degree nodes as you move them
towards lower parts, they can actually end up with lower infection rates and so
that counter balances it. So when you average across a population
not with relative frequency of meetings, you actually end up with a decrease in
the overall rate. So in terms of pictures here's what the
picture looks like. so here's our this is actually log of
lambda, in this picture. And then this is the largest steady state
row, so this is the, the positive row in equilibrium.
And the blue figure, these are regular networks.
And these all have the same mean. Regular network, this one is a power law,
the green one, and the red one here, is one for an exponential.
So basically the growing version of the[UNKNOWN] random network ,all with the
same mean. And so what you see is if lambda's
relatively low, then the regular one has everybody right at the mean as the lowest
spread it out in terms of exponential. It increases things, spread it out even
further in terms of going to a power distribution.
So here, as you increase the variants you end up increasing the relative rows.
So that's what happens for a low lambda. But once you get to very high lambda,
the, the picture completely flips itself, and the regular one has the highest
infection rate. And the power one actually has the lowest
infection rate. So these things can reverse themselves as
you get to higher lambdas. and that's because row is actually
something, you're weighting things by the, their frequency in their population
and not by their meeting rates. And so if you actually want to go through
a proof of this just go through a quick proof.
so one way you can do this is remember in a steady state, you have to have these
changes in rows be equal to 0. So 0 is equal to this expression, which
is the change in row that we got before. Which you know, remember our, our
expression for what the change in the infection rate of anybody's given degree
was pertinent of time, and again that was some recovery.
This is the rate at which they would recover, this is the fraction that aren't
infected. This is the rate at which they become
infected depending on theta. And if you just take, so we've got an
expression which looks like this thing is equal to 0.
Now just, let's take an expectation over d, so we'll just take an expectation over
d, what do we end up with. we're going to end up with v theta d
times a 1, take an expectation of that, you get this expression.
Take an expectation of this, you get that expression.
Take an expectation of this, you get the actual overall row times delta.
And so what do we end up with is an expression where this first one looks
like v times theta. This, if you take that expectation is
just our definition of what theta was. so we get theta squared times E of d and
then minus rho of d, and so you can solve this for how rho behaves as a function of
theta. So now what we get is rho is equal to
something which is proportional to theta. but also of times 1 minus theta.
So this so what we can do is, is as we change theta how, how does that change
rho. Well the right hand side here is
increasing theta, theta is less than a half.
but it's decreasing in theta if theta's bigger than a half, right?
So once theta's bigger than a half, then this thing actually starts decreasing in
theta. So this is proportional to theta times
one minus theta. This is an increasing function once but
up to one half and then decreasing thereafter.
And so what happens is, initially as you're increasing theta, you're
increasing rho up to theta equals a half. And then beyond that as theta continues
to increase, row actually decreases above this level of a half.
And so what we get is, initially we get an increasing part of rho and then
actually then increases and theta lead to decreases in rho.
Okay. And, and overall we also have that
theta's increasing in lambda, so then you can do comparative statics of rho in
terms of lambda because the lambda's going to change the theta which then
changes the rho. Okay.
So, what we've done is gone through the SIS diffusion model.
it was a useful model just to get some comparative statics out and to, to study
some techniques. So these techniques of actually looking
at distributions and talking about changes in degree distributions, doing
stochastic dominance, actually turns out to be quite useful.
It's been used now in other areas besides in the original paper with Brian Rogers.
But now been used in games, on networks, and other areas as well.
And, the SIS model is a very simple and tractable model.
So it's very stylized. it's nice because it brings in relative
meaning rates which has some elements of network structure.
And then we can order infections by properties of networks in terms of the
degree distributions, okay. What are some of the limitations of this
model? Well, important limitations are first of
all, just in terms of this SIS. we, we lose the fact that a lot of
applications are ones where you become infected but then if you've recovered
you're actually immune to catching that disease again.
Which is actually true of some flu's and other kinds of things.
or if you know, if you add new virus protection software to your computer,
then you won't get certain viruses. Again, you could get new viruses but not
the same old one. so it limits, it's limit, somewhat
limited in terms of the applications. Also, the interactions that we talk,
talked about were completely random meeting processes.
So it was not as if we'd drawn out a network and actually had people located
on a network. We just had people bumping into each
other and meeting each other. [COUGH] And so that's a special kind of
process. Which gives rise to special kinds of
conditions. Now we, more generally what we have to do
if we start to work with things where the network architecture is given, then it's
going to be more important to use simulations and so forth.
So we did some calculations before, where we talked about component size and so
forth and that gives us some insights. But more generally, if we actually
want to study these processes, a lot of it's going to be done by simulation.
So if you give me a particular network and ask what's going to happen on it,
then I might have to, to write down a program and actually simulate what's
going to happen there. So the next thing we'll look at is, is a
simple model of, of diffusion, where we'll do some calculations and just
simulate that model. And see exactly how it works, and that
will allow us to actually fit something directly to data.
And there's a large amount of that that goes on in epidemiology and marketing in
other kinds of area where you're trying to make predictions.
If you know something about the network you're working with, you can actually
simulate things and that's going to go a long way towards improving your accuracy.
So the SIS model gives us nice intuitions.
Simple ideas but it's not one you can easily take to data.
We're going to have to enrich the model to to fit it and that works, and that's
what we'll talk about next.