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# 637 Class Note for PHYS 597A with Professor Albert at PSU

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Date Created: 02/06/15

Spreading processes on networks In some cases is not enough to specify the nodes components and edges interactions What are the dynamical aspects of the interaction What is the characteristic quantity changed by the interaction Social relations spreading of information knows or not Internet packet routing travel times Molecular networks chemical reactions concentrations The spread of epidemic disease Many diseases spread through populations by contact between infective and susceptible individuals The pattern of the diseasecausing contacts forms a network Questions asked what is the typical size of an outbreak will an epidemic occur outbreak affects a finite fraction of the population what determines the probability of an epidemic he fully mixed model Assume that an infective individual is equally likely to spread the disease to any other member of the population or subpopulation to which they belong Individuals can be in one of three states Susceptible S lnfective l Removed R either recovered or dead Constant rate of infection infective contacts per unit time B Constant rate of removal recoveries per unit time y Assumptions rapidly spreading disease recovered individuals become immune individuals have the same number of contacts Lowell Reed and Wade Hampton Frost 1920 Solving the fully mixed model Fraction of individuals in one of three states 1 Susceptible s S I 2lnfective i s i r sir1 3 Removed r N 9 N a N rate of contact 3 rate of removal y ds di dr 1s lS dt 396 dt 396 7 dt 7 o the number of infected individuals decreases if Bylt1 no epidemic R0 y is the basic reproductive ratio the number of secondary cases produced by an infectious individual in a totally susceptible population Spread of disease in a social network black diseased pink infected green healthy Network models Individuals are part of social networks Diseasecausing contacts are only possible along the edges of this network There is a constant disease transmission probability T along an edge Nodes can only be susceptible or infected An outbreak starts from an infected node and spreads with probability T to the first neighbors of the node then to the second Study whether the infection stops spreading or spreads to the whole network Network representation At first each node individual is susceptible mark or occupy each edge in the social network with probability T o The ultimate size of an outbreak would be the cluster of nodes that can be reached from an infective node by traversing marked edges Thus we only need to determine the sizes of the clusters formed by marked edges We know that infection of any of the nodes in that cluster will cause an outbreak equal to the size of the cluster If a marked cluster contains a large fraction of the nodes it is a giant connected component an infection of any of the nodes in that cluster will cause an epidemic This is an example of bond percolation on a network Bond percolation Start with a lattice or network Draw or mark the edges with a certain probability p The remaining edges are open unmarked At a critical nrnhahilitv n a channinn OIIIQTQI aopears kiwi 69 quotn rL Cl iquot H39 n x quot igqu39rJ 39l IEHJ Tr 1L h j u P r r xJILquotrJ1 IJJE Ei H LErquot p0515 p0525 i Hume acacuunv 894 1963 Percolation on a general random network Start with a random network with a given degree distribution Pk The network has a large connected component if 19gt Zkk 2Pkgt 0 or gt 2 k ltkgt Mark edges with probability T Disregard the unmarked edges Expectation if TgtTC there will exist a large connected component of marked edges TC depends on Pk To find the exact relation we need to use generating functions H Wilf Generatingfunctionology 1994 Generating functions in a graph Node degree generating function G0x 2mmquot x s 1 k0 Finding Pk and degree moments from the generating function Pk id G0 ltk gt Zk Pkx G0x x1 If a certain property is described by a gen function then its sum over m independent realizations is generated by the mth power of the gen function Ex The generating function for the sum of the degrees of two nodes is G0x2 Generating functions in a graph Generating function for the degree of nodes at the end of a random edge discount the edge we probability to find that node Ayned along ZkPkxk39l G x k 1 ZkPk 39 631 M E J Newman 8 H Strogatz D J Watts Phys Rev E 64 026118 2001 normalization Generating functions for marked edges Probability that m of the k edges of a node are marked CquotTm1 Tquotquot Generating function for marked edges G0xT i i PkC T 1 Tquotquot x G01 1 xT m0km Generating function for nodes we arrive at following a random edge G1xT G11 1 xT M E J Newman Phys Rev E 66 016128 2002 Generating functions for clusters Generating function for clusters connected by marked edges PST distribution of the marked cluster sizes H0xZPsTxs IxISI s0 Generating function for the marked cluster we reach by following a randomly chosen edge ma H1xT xG1H1xTT H0xTxG0H1xTT M E J Newman Phys Rev E 66 016128 2002 Existence of a giant connected cluster Average size of marked clusters TG31 I 1 z kPk Diverges when T T k quot39 G1391 Zkac 1Pk Giant connected cluster epidemic when TgtTC 1 Tc k2gtk 1 The more heterogeneous the network the smaller TC is ie the easier it is for an epidemic to occur Breakdown transition in general random graphs Consider a random graph with arbitrary Pk0 A giant cluster exists if each node is connected to at least two other nodes 18gt 2 W After the random removal of a fraction fof the nodes Breakdown threshold f6 1 A giant connected cluster exists if fltfC Parallels between epidemics on graphs and breakdown of graphs Consider a random graph with arbitrary Pk If a fraction fltfC of nodes is lost a giant component still exists 1 To model an epidemic process on this graph assume that the edges transmit a disease with probability T A giant connected component of diseasecarrying edges exists if TgtTc T c ltk2gtkgt 1 Complete equivalence between Tand 1f Is this an obvious parallel Example scalefree network 04 Scalefree with cutoff at k1lt v Pk z k39 equotquotquot ltSgt 0 00 00 02 04 06 transmissibility T M E J Newman Phys Rev E 66 016128 2002 Epidemics in scale free networks Random graph with Pk z k ekquot Lia 1e 1K c Lia2e Lia1e39 quotgt Lin x Zk xk k nth polylogarithm of X TC decreases with K limTc0 K ND Any infection leads to epidemics in infinite scalefree networks with no cutoff Ex 1 The network representation discussed here assumes that each node is susceptible How should it be modified to include immune individuals How will the results change What is your expectation for the value of To Ex 2 Consider the emergence of a second disease after an epidemic swept through the population Assume that infected and recovered individuals cannot contract the second disease How would you estimate the chances of the second disease to not die out Dynamics of epidemic spreading on a network Susceptibleinfected model on a network S I S l si1 N N The transmission rate B depends on the number of first neighbors Define the transmission rate per edge x First approximation statistically homogeneous random network topology the fraction of infected neighbors of a susceptible node can be approximated by ltkgti mean field approximation ds di 1kis 1kis Mean field SI timedependent behavior Mean field approximation ds Ami di E s E Altkgtls Initial spread it z ioem TH 11kgt 1H time scale governing the growth of the infection in a homogeneous network Ex 1 Write the rate of change of the fraction of susceptible infected and recovered nodes in a susceptibIeinfectedrecovered model in a network using the meanfield approximation Ex 2 Write the same equations for a susceptibIeinfectedsusceptible model Ex 3 What is the condition for the existence of an epidemic in either of the cases above Heterogeneous network Focus on nodes with given degrees I ik quot Nk NPk sk 1 ik Nk Assumptions nodes with degree k are equivalent the fraction of infected nodes that are neighbors of a susceptible node k k is the same for each node of degree k dim dt I1 k Skt kt Heterogeneous network dim dt stk t k t 9k the density of infected neighbors of a vertex of degree k Uncorrelated network ZlPlil t 9 E 9 l quot ltkgt d39k 19 k2 First order 7 1M9 191 in i01 ltkgt2 ltkgt ett 1 T ltkgt Increase in the fraction of infected nodes I I I 3000 4000 5000 6000 t I I 0 1000 2000 BanhebnnetaLcondJnaU03115O12004 Time scale of the infection 100 I I I I I I I I b 3 I r I 0 1 P x vi 3 o r I I I I I I I ltkgtmltk2gt The timescale of the initial spreading process depends inversely on the heterogeneity of the network High degree nodes catch the infection sooner 60339quot nib hi I I quot5quot 40 I 1 10 tT The average degree of newly infected nodes ltkinfgt is high at the beginning of the spreading process ltkinfagt I Conclusions Infinite scalefree networks do not have an epidemic threshold any disease can become an epidemic Diseases are able to spread efficiently in highly heterogeneous networks the high degree nodes are rapidly infected This analysis focuses on the initial spread and does not describe the recovery process

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