Lecture10. Epidemics on Networks II

recording uh all right guys so today is actually the last lecture for this module um and we're going to talk about the gamex on networks so last time we discussed um epidemics and we talked about the compartmental models of epidemics um and this is just really the quick summaries the essence for those models um what you see on this slide are three curves um this is actually the number of infected people at the function of time um so one of those compartments right we talked about for three different models and when you have a si model uh you get on this sort of only one process which is susceptible people becoming infected s-i-s model is when susceptible becoming infected and then uh can recover and again be susceptible and sir model it's a model where susceptible becoming infected and then either recover or getting removed from the system so that's where we left off on the previous lecture and i guess the most important part here the most important um fact is that at the beginning of epidemics any of those models give us um exponential growth and in fact um there for the model s i you know there is always growth and eventually everybody is get sick but for the models sis and sir there is there exists an epidemic threshold so the ratio of um the the number of people infected per unit time divided by um the you know the the the the coefficient that creates that that corresponds to um the recovery so the rate of infection uh over the rate of recovery and if it is one if it is less than one uh we don't get epidemics if it is greater than one um we get epidemics and so the idea is of course sort of some somewhat obvious right even people get his infection spreads faster than people recover well you know you get epidemics and if you know otherwise um it just dies out um and uh previously on the previous lecture we talked about what's called compartmental model and the idea and it's also called full mixing model um where we would assume that every person can interact with every other person and can't get um infected or in fact any other person within the the susceptible uh within the susceptible people now obviously you know in real life that's not the case right we do have our um interaction network um and you know you can think about your friendship network or just like literally um on the daily occasions you interact you meet with the people you know at work you meet other people if you travel to work etc etc and so that the actual network of interaction and that's um where you can get infected right infection doesn't just jump from one person to another randomly it's uh when you interact and so um that means uh we could try to you know to be more realistic we should add this network structure interaction network structure to the model and that's what we're going to do today so in order for to to make this work we're going to change a little bit um the way we approach the modeling so we're gonna look first of all we'll we'll we'll look at the network of the contacts and it will be given by adjacency matrix and uh it's uh undirected network right for the simplicity of modeling we're going to also change a little bit the model itself so instead of looking at the compartments we're going to look at each node of the network independently and individually and for each and every node we will have it in one of three states susceptible state infected state and recovered state and so nodes could transition from those states and then um we'll also add this probabilistic nature we'll actually model it as a probability of node being infected or being susceptible or being recovered at the moment at the moment t okay so now those three variables that gives us they give us probabilities we'll also change a bit notations um on the beta and gamma so beta now is going to be probability that disease is transmitted on a particular contact right so you know if you think about networks it's transmitted along the edge during the time delta t now if we have for example an average k contacts and coming back to the previous lecture um the beta and i call it beta sub seed beta compartmental model is linked to this beta by the number of connections right so if you know on average you have 10 connections well it means during the unit time of the probability of transmission on one connection is beta probability of transmission on 10 is 10 data gamma is recovery rate the same thing as before so we actually moving from this deterministic description to probabilistic description um and we're gonna you know write equations that deal now with probabilities um the other thing is we're going to look at the connected component of the network so we believe that we'll consider on the networks where all nodes are reachable and as i said the network is unsymmetric so these are the conditions for the modeling so following along there are two processes that we look after process number one is node infection so let's say reds are those nodes that are currently infected and we're going to look at this node so we can calculate the probability of this node being infected the following way so first of all the note has to be susceptible so there's a probability that it is susceptible then one of the neighbors should be infected and here we take a sum over all the neighbors and this is a probability um that they're infected and instead of i we use x simply because you know it's probabilities we don't want to mix it with with indexes so this is the probability that one of them is infected at least one of them is infected or more of them and beta is a probability um that on this contact the transmission will occur right and so that on during the time step delta t and that's the probability of a node to get infected during the time uh period delta t again in its beta probability transmission probability that the node is susceptible and this is the sum of probabilities that the neighbors are infected we're not going to look um if we write it this way yes uh you can have node infected from you know two simultaneously that's fine um node recovery that's a second process this is you know the recovery of a node this is infected node is going to be independent from whatever is happening with the neighbors um it's just uh the probability of recovery during the time delta t so that's our model and now we're ready to write equations and uh we're going to do the same sort of we follow the same path as we did previously we're going to go through s-i-s-i-s and s-i-r models so in s-i model it's susceptible becoming infected now our little answers and x's now are probabilities but for every node the sum of the probabilities is equal to 1. notice there is an index i here so it's not one equation it's you know number n equations it's as many equations it's equation per node right so this is n equations um then we're going to write the differential equation for the change of the state or the change of the probability of the state of the node the probability of node to be infected at time t plus delta t is equal to the probability of the node being infected at the previous moment of time plus the probability of the change and probability of the change is as i said before on the previous slide its probability of transmission probability that not was susceptible and some of the neighbors on the previous slide i just wrote it as a summation over nearest neighbors uh but instead of that we can actually put here the sum over all j's and put here adjacent matrix a i j because it's non-zero for the node j only for the nearest neighbors right okay so that's pretty much it and now we can write down um this system again notice um it does look actually very similar to um the equations we we had before except for now on the right hand side we have the summation and we have adjacency matrix and um there is the indices i and i here so there is n equations and n constraints all right so n differential equations all right so um now we can solve them uh you know we actually follow exactly the same way as as we did before we you know substitute s i by 1 minus x i and remember x i of t it is the probability that the node that e node is infected and uh we're going to consider early times early times in infection so when the times are early majority of the nodes are not infected so let's see what the behavior of the system would be in that case so we're going to look um then you know if x i of t is much less than one we kind of forget about this and we get the following system dxi dt beta sum and so we can take this and write it in the form of matrices right in the matrix format and left hand side is a derivative of of of the vector of the vector of probabilities right and again um every element of this vector is a node in the network and on the right hand side uh well you know we get the the dot product now this is a differential equation actually the system of differential equations written in the matrix form um the way to solve the system the standard way to solve the system is to look for the solution of the system in the basis of eigenvectors and eigenvalues so what we do is this is the basis is we we can write an eigenvalue of inductor equation and then represent the solution um as as a combination as a linear combination of eigenvectors and so it's it's sort of typical separation variable separation process where we take the time and put it here as a coefficients in a linear system does this make sense i'm sure you should have done this type of you know this you should have used this approach uh in differential equation class guys does this make sense should i slow down yes yes what i mean speed up slow down make sense make sense okay all right all right okay um moving on then so then uh you know again the way you do this is you know we write the solution as a linear combination in terms of eigenvectors and then we take that and plug it in right and then we'll get an equation for coefficients so we do it we plug it in um so that's the derivative right and we get uh you know linear we get sort of the first linear and then differential equation for each of the coefficient again this is absolutely sort of standard way of doing it boom so that's the differential equation now here we get easily separation of variables and we can write for each coefficient the time dependence of it as some initial value times exponent y exponent well because you know if you do separation variables this goes here dt goes there d a over a is d of logarithm you know you you exponentiate it's going to be exponent here so plugging everything back along the chain line we get the solution as the solution for that system as a linear combination of eigenvectors of the matrix a with this initial conditions and then with a time factor and for us this time factor is the most interesting thing look here is a sum over all the eigenvectors and k are all eigenvectors when you take when you calculate eigenvectors eigenvalues of the matrix the eigenvalues have different values right you can sort them and notice here is an exponent now and there is a sum so when we look at the sum the this exponent growth at the different rates for different lambdas and the large is a lambda you know the largest exponent so if we want to make an approximation out of all the sum we can keep the one term which is the largest and this largest term corresponds to the largest eigenvalue and uh you know if if we sort in in in decreasing order well that's lambda so to proxy approximate and you know i shouldn't put here equal i should put here you know approximately x as a function of t approximately is given to us by um eigenvector u1 and this is how probability infection changes with time so it's you know it's exponent it's growing and by the way we shouldn't be surprised that it is exponent because that's how fast things grow but what's critical here is that growth rate is determined not only by the coefficient beta or beta but also by an eigenvalue an eigenvalue in the matrix it actually contains information about the structure of the matrix well in fact there are for example um some bounds that allow you to that shows you that lambda max is usually less than uh you know the largest node degree i'll call it d max so d max bounds um the the eigenvalue the largest eigenvalue so um for example the the largest uh node degree will sort of dictate the the the size of this eigenvalue so that makes somewhat sense because um the you know you have nodes with large degrees um it means they're connected to a lot of nodes which means it makes propagation infection easier right okay so the growth rate of infection depends on them the one and we haven't had that before obviously in compartmental model so this is this where the structure of the network comes in into play and then notice that it's also x of t is proportional to via v1 right so it you know it means you know we look at the probability of node one to be infected right and it's proportional to uh v1 for node 1 right um which is corresponding eigenvector and you know x2 proportional to the value of the first eigenvector for the second node um etc etc now does this make sense and and why like can we interpret this this dependency where else have we seen the depend i mean you know it's not the first time we look at eigenvectors and eigenvalues of uh adjacency matrix right so this equation av equal lambda v we have solved that equation before uh when have we done it oh come on guys i'm trying to centrality right so what's the name here for this type of centrality i can vector i can vector centrality right so what it says in fact that the probability of infection of the nodes depends literally on its eigenvector centrality and the most central note is the higher probability of it to get infected right and that's also makes sense because central nodes in in in terms by connectors are those that are connected to other important nodes and again it means they're very well connected to other very well connected nodes and it's clear that if one of those very well connected node gets infected then the re then you know neighbors have high probability of get infected plus those notes usually have high note degrees and so that's of course increases probability of not get infected so like any socialites so you know party people right or people with a lot of connections in in social interactions they are primary targets for infection which actually makes sense right um so that's the connection to um to to to the centrality which sort of makes sense okay and here uh you know this is just fraction of of susceptible effective vertices for various node degrees so it just says like look um as a function you know it's exactly what we discussed as time goes by it's the nodes with the higher node degrees uh you know if i take some time this is no degree say k2 greater than lk call it k once so this is a fraction of nodes with different node degrees um this is a fraction of nodes what some node degrees is a fraction of node with a higher node degree and at any moment of time there are more nodes of high no degrees that get infected which also makes total sense and follows our discussion all right now when we deal with networks um you know this is this this model and uh even on this simple model it was not i mean we kind of look for approximation right um you know in fact you can actually take this differential system of differential equation right this one um and and and solve it numerically right so um it's not a big deal right it's just um or or you know if i substitute things in so it is then um this right again this is a system of differential equations there are n equations like that um and you know you can just plug it in into a solver and it solves and gives you um the dependency of the probability of infection for the node as a function of time so you can just solve it uh so far what we've done is um you know we just you know calculated this approximation so you can solve it but there is also another option is to actually simulate things and to simulate things the way we do this is with every node at any time step you know node can be in two states susceptible and affected so we can initialize a few nodes in in the infected state and then uh on each time step just follow the links for the node and with the probability beta infected neighbors right and you know the way to cal to use to to do this in simulation is you know you you use random number generator um you put the threshold and that defines the probability and you know you either infect or not and then you just move on to the next time so this is a quick simulation for beta point five this is the same network we we work a lot of a lot with um karate club two nodes infected here number six number seven next time step uh number seven in fact number one and number five number six infected number eleven so next time step next time step next time step next time step okay obviously eventually network gets infected right so it is sort of very easy to simulate this is simulation and if i calculate now what i calculate here is a number of nodes as a function of times that are infected right so when we started uh there was just two nodes then four nodes etc so that's what it says here when we started um there you know there are two nodes uh then there i'm sorry this is um where is it uh there are two nodes and then next time step four nodes and and and then goes up um and the blue is a number of uh susceptible nodes right or so which drops and that reminds us a lot those smooth curved curves we saw exponential growth here well it's not that smooth because the network is small but again if we had a large network um you would have it quite smooth okay um so that's s i model the next model is sis so it's susceptible becoming infected becoming susceptible and so you know again following along the same lines the same storyline as we had before these are the derivatives with respect to time the probability of infected node this is the same term as we had before right the probability of node to get infected from its neighbors and this term is now recovery right so node i can get infected from its neighbors but can also recover with some probability gamma right and that's all happens during the time delta t so again this is you know precisely following um the last lecture uh modeling except for writing it um we're writing it in this form okay so then moving along moving along we can write differential equation uh you know this is differential equation did exactly the same thing plugged in um again you can take this equation and there are n of those so it's a system of differential equations plug it in into the solver numerical solver and solve it what we're going to do now is look at the approximation just again to get some analysis uh look at early time approximation where x is small so we can neglect this thing um that's the differential equation we get we can actually you know play do a little trick um and you know take this term inside of the sum by actually adding here chronic delta symbol and that delta symbol is equal to zero if i is not you know it's it's zero when i is not equal to j and it is equal to one when i is equal to j right so it's diagonal um so we can put it in uh do a little bit manipulation on the variables and then we can rewrite this um in the matrix form and notice uh that you know there is a matrix here now it's instead of just matrix a which is adjacency matrix it's adjacency matrix minus diagonal matrix of ones i is a diagonal matrix of ones and game gamma over beta these are uh you know they're the ratio of the uh infection rate and recovery rate okay so this is modified adjacency matrix so the next step then is pretty much following the same story creating and looking for eigenvector bases for that now instead of matrix a we're going to look for the eigenvector basis for the matrix m but since matrix m is just um the matrix a minus diagonal matrix uh in fact you can easily show that the eigenvectors are the same for the matrix um m and for the matrix a because it's only diagonal shift and the eigenvalues they'll be shifted it's eigenvalues of the matrix and minus this shift minus the diagonal shift um that's pretty much it so we don't need to do anything else then we just plug in into our solution and we get this is a solution the same vector form it again it's a linear combination of eigenvectors of matrix a right some initial values and the exponent and what's changed is really the exponent and that's the most interesting part because now it's beta times lambda k minus gamma uh we again look at the exponent that that growing the fastest you know exact solution will you know you'll have to take into account everything but we're looking for approximation so we're going to look for the largest term uh growing the fastest this corresponds to lambda one and notice it's beta lambda minus gamma so it is beta lambda 1 minus gamma now it's again we're looking at the lambda 1 because this corresponds to the largest fastest growth notice that this thing can be either greater than 0 or less than zero this expression now if it is greater than zero then um as time goes by we get its increases and the solution on the infection increases if it is less than zero if this difference less than zero or beta lambda less than gamma uh it's negative and as we increase time this goes to zero right and so so the probability of attraction goes to zero so again getting a critical you know sort of separating behavior um and that separating behavior happens when uh we compare beta lambda 1 and gamma where beta tells us the probability of infection transmission of infection along the edge gamma tells us the probability of recovery and lambda 1 contains information encodes the structure of the matrix or some information about the structure of the matrix so you know just a summary is beta times lambda 1 if it's greater than gamma infection survives and so there's a growth the probability of infection for every node growing and uh you know we get in epidemics and if it's less than gamma well infection dies out over time and so then uh since as previously we looked at this ratio beta of a gamma so if i just take and and divide it here and put lambda on the right hand side um i can introduce an epidemic threshold and in this case it's one over lambda one where lambda one's the largest eigenvalue of adjacency matrix and if beta of a gamma greater than threshold well um we get epidemic if it is less than threshold infection dies out so it is very similar to what we had for a compartmental model but with one important difference over there the threshold was one here it's one divided by lambda one so here threshold encodes um the information about matrix structure or graph structure network structure all right does this make sense and lambda one is an eigenvalue and so it's greater than one usually so this value is usually less than one okay um the same way as with as i model we can actually make a simulation um you know we initialize some number of nodes uh into the infected state we will keep node infected for some number of iterations and some number of steps and that number is one of a gamma which is gamma is a parameter we have and on each time step um you know there is a as as in the previous model there's a probability to infect nearest neighbors and after t sub gamma time steps node recovers right and this is the c sub gamma so the dynamics is infected plus susceptible yes we can get to infected and then in fact it can recover um again it's actually better to see once than here many times um bait is point five um tau is two which is sort of gamma is is one you know one one house let's see what happens and by the way in order to predict if this is gonna grow or if it is gonna um decay if if interaction grows or decays we will need to compare this beta and gamma uh with respect to uh lambdas of um eigenvalue eigen largest eigenvalue of this graph which actually i didn't write here so you you know it's hard to predict but anyway let's look at this six seven um next step growing neighbors but some recovers right uh other getting infected recovers either getting infected recovers are they getting infected recovers it's actually very interesting simulation notice what happens so infection sort of travels in the graph but never goes away all right and that's what often happens in the society right some people get recovered that but then get reinfected again and so this is a cis infection now you'll see here oscillations eventually you know if we had the larger um largest set uh you know it would converge to some constant number of infected people all the time you know some constant number of healthy people all the time um but that's it's a constant number uh but you know to say the people it will be different people so people change but the total number of infected remains approximately the same right when infection grows enough and that's what we observe here well there are some oscillations but again this is due to just small size of the sample um here is another example with beta smaller it's 0.2 remember it was 0.5 here now it's 0.2 well it started growing but then growing a little bit but then dies out and so that's sort of the picture okay so modeling works in some sense or our prediction our estimation work that yes there is depending on so the values of beta you can either get infections that persist within the population but moves around or infection that starts and then dies out okay so these are two different modes pretty much is the same as um with compartmental model but the fundamental difference is the conditions uh when this happens versus when it's not now last step sir model sort of the most interesting model to us well the same story three differential equations um you know put things together um we got we got this eventually differential equation with you know x i with r um you know we can either solve it again numerically or we can actually try to look at early times uh when early times what we see is remember there is very few infected nodes this goes to zero and actually there is very few recovery recovered nodes also goes to zero and in this approximation um it behaves pretty much the same way as sis right so early in time they became very very similar you can actually do a little bit better uh approximation but we're gonna stay with this one for right now and so that means the solution also evolves the same way again there is this v1 from the matrix a so the first eigenvector which means its eigenvector centrality that dictates which node has a higher probability of getting infected the most central nodes have high probability of getting infected and the exponent that depending on the ratio of beta gamma and lambda 1 um it's either going to grow up and then you know get infection or you know dies out so um very very similar story um you know if we solve numerically that's what we get now this is again instead of just drawing one um curve for the entire population there's there's these are separate curves for different node degrees so you have some nodes with degree um you know k1 some nodes with a degree k2 some nodes with a degree k3 and this is just a fraction of nodes with those degrees and then the nodes with a higher degree they get infected faster than the nodes with a low degree so obvious sort of thing that if you're in epidemics you know stay away from people make have less connections and then of course you have less chance um to get sick so don't be socialite um simulation pretty much the same approach um we start with uh you know we initialize some number of nodes to infect the state each node stays infected some uh some type some number of steps um and then there's a probability for every node to infect it near its neighbor and after this time steps nodes node recovers right so compared to the previous model to ss model it cannot get reinfected it just recovers and again and is removed from from the model so there you go this is sir model um you know there is infection and the green color now is those that are recovered uh you know recovered recovered recovered recovered now everybody has recovered but that actually means that the entire population got sick right because uh you know the the fact that you know all of them are green means that all of them were red at some moment of time so that's actually a very very bad scenario which means epidemics went through the entire population and if it was a plug you know pretty much everybody will do that um and and you know these are the the curves you get used to see already again they're not very smooth because the size of the network is small now here's another example with smaller beta uh yeah we get some infected people some recovered eventually it dies out and that's actually the scenario we would hope for in epidemics where there is some number of people that get infected but majority of population stays blue or stays you know susceptible remains susceptible this is sir model that's what sort of we hope for right majority of population remains on this sort of blue line okay so these are uh you know sort of analysis of early stage behavior and some simulations now um it is clear that lambda one uh that that corresponds to the structure of the network you know plays a huge role and so for different networks we would expect very different behavior so now we have random network lattice which is sort of very regular we have small world um you know it's kind of uh we talked about this uh you know sort of adding adding long distance connections then we get spatial spatial it's just a network where uh you know you connect those those nodes that are close to each other connect those nodes that are less than whatever distance and then you know scale free network you know this is barbara albert type of network where yes you have small world on one hand on the other hand uh the degree distribution is a power law for this mass okay and this one is just this is random okay so looking at those five networks where do you expect um the the infection would spread the slowest where would be the most difficult for infection to spread around spatial okay maybe spatial what about this one the the grid if you get some infected nodes here it depends on their the first people if we're in different corners it will be spread very fast but if it's yeah if they are in a different corners if you have a lot of them spread around yes but if there is you as usually you know you get the source of infection it will take forever you know to reach far distances versus say you know here where there are a lot of connections or here where there's a lot of connections yes but let's see if we got a special infection on the isolated node it's very good so yeah yeah of course yeah if there is this one if there is isolated node yeah nothing is going to happen um okay let's see um so this is this example from from a paper you can check out the paper um when they ran simulations so here is a random versus lettuce versus small world spatial so in fact you might notice that uh you know the spatial i mean i'm sorry the latin lettuce um has the kind of the slowest right this is time and so it has a slowest growing peak um and scale three number five um right number five um you know it's quite fast i'm very surprised actually with this experiment that that num that random has such a quick spread now for the random network you would i mean yes it it should go faster than lattice because random network um if you remember possesses the small world phenomenon right um but you know scale free networks this one um on one hand it possesses small world on the other hand um you know it has it has very homogenic you know you have in homogeneity so it does have a lot of nodes of high degree that should be super spreaders right hubs and so honestly i would expect that this one would spread the fastest uh so i'm sort of a little bit surprised by by by this experiment uh to me this would be the most sort of um you know i would expect this would be to the fastest spreader but anyway so that's experiment uh you can look at it and actually you can you know you can reproduce it right and you can try it on your own on your own network generate different networks and and try those experiments um now there is quite an interesting um and this is several years old uh paper i'm sure they're gonna be a lot of papers like that appearing based on cut it um this is actually based on sars um and and and the idea of course that in today's world um the physical distance um you know makes not that much sense simply because people do travel a lot right and so infection spreads around you know through travelers and you know airlines connect all the world um stars 2003 37 countries and you noticed like yes uh coveted started in china and then covered the entire world really really quickly so there is a very interesting paper and very interesting approach where um instead of using um the the physical distance um the researchers used effective distance and so the effect of this is the way introduce the the effective distance is given through this one minus log pij where pij is a fraction of travelers going from node i to node j so pretty much it's a travel network you can take just airline network and then uh calculate this effective distance based on the number of of of travelers right going through through um through that through those particular edges and so physically it can be further away but if there are a lot of people going through this effective distance will be will be short and why is this interesting well because um if you introduce those effective distances the time for infection to get somewhere is obviously the sufficient effective distance divide the some you know effective velocity of propagation but the interesting part is that the ratio of um those effective of those times of propagation uh it does not depend on the the model of propagation but just the fact that just depends on this effective distances because you know the ratio uh you take the ratio this will effective velocity goes away from the consideration and so which means um within this effective distances you can actually easily predict the time of arrival of of the disease um in in different countries depending on the literally on on the passenger flow um and they also introduce quite nice visualization you know this is a source of infection and that's how it sort of spreads if you use effective distances uh to to monitor this and this is for stars for actual sars cases i'm sure you can have a very uh convincing picture like that for covet now and this is this effect the distance stimulated on the network um the green node here is a source um of infection and then you know you get infections spread around the network but if you look from the source using effective distance right as and um then this is a picture you're gonna see so it just you know spreads from the source and then if you look at the you know pick up one node as a recipient you know from that node that what it looks like how infection kind of surrounds it and eventually yes um gets infected so actually very very nice paper um another important part is that for some networks you will actually get these fluctuations when in fact infection comes back and we saw that type of behavior on the sis system right where um you know you get sort of nodes reinfected and you get some oscillations so there'll be some more nodes infected then less nodes in fact it'll be more no infected et cetera et cetera um there is a chance that we call it right now we're observing those oscillations with kind of multiple waves um then the infection kind of dies out then comes back um and then there is modeling um you know that that explains that behavior uh finally i think it's it's very important to show this result um you can actually we we did not derive it but um you know one can derive it and it's derived in barabasha book we you know i have showed you i have shown you that r um that the threshold um is given one over lambda one well in prabhasha book he shows um using some paper and some research some some sort of iterations that you can also represent this um the same coefficient are right the same sort of the threshold value as the ratio of every of average no degree over um average squared now average squared in fact is a metric of uh network homogeneity right because if for example all nodes have the same node degree then the sigma right the fluctuation will be equal to zero right if if k is constant then this is equal to that and this is zero so for this for the lattice um for the for the lattice where every node has the same node degree the fluctuation zero for poisson distribution for example for random network um that uh is actually equal to you know that's equal to this value you can prove it precisely you can calculate it so it's just k average squared and so then for random network this epidemic threshold is given by one over um average node degree and it's greater than zero it's not one it's less than one but it's greater than zero so there is an epidemic threshold and again guys remember epidemic threshold means if the relationship between beta gamma and that threshold if beta over gamma is less than it there'll be no epidemics if it's greater than there is epidemics and so in random network there is a threshold now what's sort of scary that if we take a scale free network remember that network is this i mean well scale free networks right k to the power minus gamma and for most of them gamma is between two and three then remember we we did this calculations k squared actually diverges when you know the network is large enough k squared goes to infinity but if we look right now at this equation when we have k squared average k squared going to infinity r will go to zero which actually means that there is no epidemic threshold which means any any small infection right small number of infected people eventually infect the entire network now this is a very dire you know prediction that there is no epidemic threshold but that's probably what we observe now is call it that um you know some set of of infection in in you know in china in fact it pretty much almost the rest of the world uh and this happens again because um within the social you know there's a society there are nodes that are hubs that are super spreaders and that's some thing also if you look right now in the literature uh on covet um the other medical literature describing the super spreader superstrata events so some people who who infect literally tens to hundreds of people right and and so the the existence of those hub nodes nodes with high degree high large degree large k they that's what's causing this no epidemic threshold um in in a scale-free network now um again this is modeling right so in real life we know that yes networks are not precisely skill-free uh you know et cetera et cetera and they're not sort of infinite right there is only um seven billion people so it's not infinity et cetera et cetera but it actually tells us that this modeling tells us that in scale free network uh epidemics spreads really really easily compared to any other arrangement of the society it also tells explains why for example computer viruses spreads easily because computer is connected in the network that is also scale free so that's actually very very powerful result and it was obtained you know i mean in probably i think publication is like 2010 2015. so it has been known before covet started um so what do you do well um of course vaccination right but if there was some epidemic threshold you could do random vaccination and stop the infection because by random vaccination you reduce um the number of people who get infected and so you kind of free you know reduce um the probability you use beta and so that means you know you reduce um you know you can go below the threshold but if the threshold is zero well you know random vaccination is not going to help so what the strategies well the strategies obviously vaccinate hubs right so you need to find those people who are sort of super spreaders potential super spreaders those who are traveling those who have lots of connections social connections and sort of vaccinate them that will of course allow to break the network into pieces right by removing this nodes vaccinated nodes but the challenge here it is actually quite hard to to evaluate yes vaccination plays somebody in in in their our state um but it is quite hard figuring out you know you don't report how many friends you have you don't report how often you socialize i mean you can guess uh you know by facebook or you can guess by sort of lifestyle or by age but still it's not sort of you don't have a really good picture so what you want so how can actually improve the chances of vaccination compared to just random process well there are a few things first if you remember the p of k is a probability of finding uh you know node with degree k now if you pick up an edge and follow it the probability of a node having degree k at the end of the edge is actually k times p k and we we we talked about this approximation so it's already higher and so that will let you find higher notes but the easiest way is to pick up a random person but then not vaccinate that person but vaccinate this person friend remember friendship paradox that tells you that on average your friends have more friends than you you do well it means you know your friends on average have higher degree than you do which actually means it makes more sense to vaccinate them than vaccinate you and so um you know the most efficient way of doing it unless you you know who the hubs are is to pick up a random node and vaccinate one of the random friends of that node okay so that's about strategies and with that actually we're we're done um i did not put together the slide um that has all the you know lectures that all the materials that we have covered but just to go over it we talked about you know uh parallel networks uh we actually talked with about statistics on the networks we talked about parallel networks we talked about centrality metrics we talked about correlation for the nodes um and uh we talked about detecting communities and we talked about detecting different structures in the network and we talked about you know disease and epidemics on the networks that pretty much what we discussed in this module next module we're going to talk about more about processor networks and we'll kind of connect the network science and machine learning by talking about you know network embeddings and other topics related to machine learning and with that uh we're done for today any questions and you guys as always very quiet um well i guess with that i will stop the recording

Channel: Leonid Zhukov

Views: 269

Rating: 5 out of 5

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Id: X3xE9oj5-3o

Length: 56min 23sec (3383 seconds)

Published: Fri Apr 09 2021