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

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This 15 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 25 views.

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Date Created: 02/06/15

Interplay between topology and dynamics 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 contact 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 Susceptible s S I lnfective i s i r sir1 Removed r N a N a rate of contact 3 rate of removal y ds di dr lS lS dt dt 74 it 74 o the number of infected individuals decreases if Bylt1 no epidemic R0 IBy is the basic reproductive ratio the number of 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 kPkxquot391 G 2k 1 ZkPk 631 k 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 H0x ZPsmxs les1 s0 Generating function for the marked cluster we reach by following a randomly chosen edge ma H1xT xG1H1xTT H0xT xG0H1xTT M E J Newman Phys Rev E 66 016128 2002 Existence of a giant connected cluster Average size of marked clusters ltsgt H31T 1 TG 1 1 TG11 1 Z kPk Diverges when T T k 611 2 Zkk 1Pk Giant connected cluster epidemic 1 ltk2gtltkgt 1 Tgt MTgt 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 ltkgt After the random removal of a fraction fof the nodes Breakdown threshold fc ltk2gt 0 1 A giant connected cluster exists if fltfC 1 Complete equivalence between marking edges with prob T and keeping edges with prob 1f

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