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

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This 7 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 21 views.

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Date Created: 02/06/15

Spreading processes on networks In some cases is not enough to specify he nodes components and edges interactions What are the dynamical aspects ofthe interaction What is the characteristic quantity changed by the interac ion 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 popula ions by contact between infective and susceptible individuals The pattern ofthe diseasecausing contacts forms a network Questions asked what is he typical size ofan outbreak will an epidemic occur outbreak affects a nite fraction ofthe population what determines the probability of an epidemic The fully mixed model Assume that an infec ive individual is equally likely to spread the disease to any 0 her member ofthe population or subpopulation to which they belong Individuals can be in one of hree states Susceptible S Infective I Removed R either recovered or dead Constant rate of infec ion infective contacts per unit time p Constant rate of removal recoveries per unit im e 39y Assumptions rapidly spreading disease recovered individuals become immune individuals have he same number of contacts Lowell Reed andWade Hampton Frost i920 Solving the fully mixed model Fraction of Individuals in one of three States i susceptioies Zinfective i 1 L 4441 3 Removed r N N N rate of contact p rate of removal y d39 Jr 7 7 x p he number of infected individuals decreases if pylt1 a no epidemic R p7 is tne oasic reproductive ratio etne number of secondary cases produced by an inrectious individual in atotally susceptioie popuiation Spread of disease in a social network mach mseased prhh rhrecteg greeh hearthy Network models hmvrguas are part ersucrat netwurks Drseasecausrhg cehtacts are my pessrme arehgthe egges erthrs HEMDrK TnErE rs a cehstart msease trahsrhrssreh pmbameT arehg ah egge Nudes can my be suscep rme urmfected Ah uuthreakstansfrum ah Wanted HEIdE ahg spreagswrth prunanmtyTtu thenrstherghnurs at he huge thehtu he SEEDnd stugywhetherthe rhrecneh stups spreamng erspreags tn the wnme HEMDM Network representation At rst each huge rhgwrguar rs susceptrme rhark er uccupy each edge m the secrar HEMDM wrth pmbabmty T Tne ur rrhate srze crr ah eu break weurg be he crustererheges hat can be reachegrrerh ah rhrectrve huge nytraversrhg marked egges Tnus we Dn y heeg tn getehhrhe the srzes urthe ctusters rurrheg by marked egges We mew that rhrecnuh cut any urthe huges m that musterth cause an uuthreak eguat tn the srze er the muster f a marked muster Euntams a argefracnun Erme heges rt rs a grant cehhecteg EDNpDnEnt ah WEE run at any erthe heges m that muster wm cause ah eprgerhrc Thrs rs ah exarhpre crr nehg perceratreh ch a HEMDM Bond percolation Start wrth a tame EIrnEIWEIrK Draw urmark the egges wrth a certarh pmbabmtyp The rEmaan egges are epeh unmarked At a chhcar nmhahmm n a shahhrhh crhsreraupears g3 Ar pants new 525 New meat was rt cmswww Percolation on a general random network Start with a random network with a given degree distribu ion Pk The network has a large connected component if k2 kk2Pk gt0 or Wgt 2 Mark edges wi h probability T Disregard he unmarked edges Expectation if TgtTE there will exist a large connected component of marked edges TE depends on Pk To nd the exact rela ion we need to use genera ing functions H WillGerieratingfuiictioriology1994 Generating functions in a graph Node degree genera ing function Gix2Pkxk i xis 1 m Finding Pk and degree moments from the generating function 1 akaquot n n d Pk k kPkX GX klwm2k1deil lfa certain propeny is described by a gen function then its sum over in independenl realizations IS generated by the in l39i power oflhe gen function Ex The generating function for the sum of the degrees of two nodes i5 cum2 Generating functions in a graph Generating function forthe degree of nodes at the end ofa random edge discount the edge we probability to aw yd alofig Z kPkxquotquot Gm t aim lame 39 651 M E i Newniaiis iri stidgatzp i Wattsphys Rev E64026ii8t200ii normalization Generating functions for marked edges Probability that m ofthe k edges ofa node are marked cyrmu Tquotquot Generating function for marked edges on par 2 2 PkCquotT quot 1 T quotquot x Gquot 1 1 xT milkm Generating function for nodes we arrive at following a random edge 5106quot 511 1 XV M E J Newrnai39nphys Rev E66 i7161282002 Generating functions for clusters Generating function for clusters connected by marked edges P5T distribution ofthe marked cluster sizes HnX2PaTX l XlS 1 11 Generating function for the marked cluster we reach by following a randomly chosen edge V 5 HIXTXGIHXTT HnXTXGnHXTT M E u Newman Phys Rev Eeoolelzzztzoom Existence of a giant connected cluster Average slze of marked clusters T G51 s Hn1T 1 1TG1 1 21km r Dlverges when 11 V7 611 Zkac 1Pk k Giant connected clusterrepldernlc when TgtTE 1 T Zkz The rnore heterogeneous tne network tne smaller Tcls r e tne easler lt ls for an epldernlc to occur Breakdown transition in general random graphs Consider a random graph with arbitrary PkBJ A giant cluster exists if each node is connected to at least two olher nodes k2 After the random removal ofa frac ion fofthe nodes Breakdown threshold f A giant connected cluster exists if fltfc Parallels between epidemics on graphs and breakdown of graphs Consider a random graph with arbitrary Pk lfa fraction fltf of nodes ls lost a glant component stlll exlsts 1 1 k2 5 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 TgtTE I T Zkzlk 1 Complete equivalence between Tand 1f lstnls an obvlous parallel Example scalefree network n4 Scalefree wi h cutoff at KK II n z Pk 1 k39 equot ltgt txansmlssblhty T M E J NewmanPhys Rev EBb l lZSlZOOZl Epidemics in scale free networks Random graph with Pk x kquotequot quot Lane w Li X 2 KW k nth polylogarithm of X To decreases wi h x lim Tc 0 K w Any infection leads to epidemics in in nite scalefree networks with no cutoff Ex 1 The network representa ion discussed here assumes that each node is susceptible How should it be modi ed to include immune individuals How will the results change What is your expecta ion for the value of TE Ex 2 Considerthe emergence ofa second disease a er an epidemic swept through the population Assume that infected and recovered individuals cannot contract the second disease How would you es imate the chances ofthe second disease to not die out Dynamics of epidemic spreading on a network Susceptibleinfected model on a network S I 39 t 39t1 N N The transmission rate p depends on the number of rst neighbors De ne the transmission rate per edge A First approximation statis ically homogeneous random network topology the fraction of infected neighbors ofa susceptible node can be approximated by ltkgti mean eld approximation dx E Aki39 1kis Mean field SI timedependent behavior Mean eld approximation 3 ki39 g Mkgtis Initial spread 10 item r51 11106 EH time scale governing the growth of he infec ion in a homogeneous network Ex 1 Write the rate of change of he fraction of susceptible infected and recovered nodes in a susceptibleinfectedrecovered model in a network using the mean eld approximation Ex 2 Write the same equations for a susceptibleinfectedsuscep ible model Ex 3 What is the condition for he existence of an epidemic in ei her of he cases above Heterogeneous network Focus on nodes with given degrees I NkNPk sk1ik Nk Assumptions nodes with degree k are equivalent the fraction of infected nodes that are neighbors of a susceptible node can be written as k k and is the same for each node of degree k kka km Heterogeneous network di t d Mid gkm 6k the density fraction per edge of infected neighbors ofa vertex of degree k 2 1Plit Uncorrelated network at E 9 T Initial spread it Increase in the fraction of infected nodes 08 05 M 04 02 I I I I I 0 1000 2000 3000 4000 5000 6000 t Barthelemy etal Phys Rev Lett 92178701 2004 Time scale of the infection 100 i lt19Altkzgt The timescale or the initial spreadrng process depends inversely on the heterogeneity or the network High degree nodes catch the infection sooner lt kmm gt The average degree or newly Inrected nodes we Is nrgn at the oegrnnrng or the spreadrng process Conclusions In nite scalefree networks do not have an epidemic threshold any disease can become an epidemic Diseases are able to spread ef cien ly 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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