{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T16:42:47Z","timestamp":1759941767229},"reference-count":19,"publisher":"Oxford University Press (OUP)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Propagation of contagion in networks depends on the graph topology. This article is concerned with studying the time-asymptotic behaviour of the extended contact processes on static, undirected, finite-size networks. This is a contact process with nonzero exogenous infection rate (also known as the $\\epsilon$-susceptible-infected-susceptible model). The only known analytical characterization of the equilibrium distribution of this process is for complete networks. For large networks with arbitrary topology, it is infeasible to numerically solve for the equilibrium distribution since it requires solving the eigenvalue-eigenvector problem of a matrix that is exponential in $N$, the size of the network. We derive a condition on the infection rates under which, depending on the degree distribution of the network, the equilibrium distribution of extended contact processes on arbitrary, finite-size networks is well approximated by a closed-form formulation. We confirm the goodness of the approximation with small networks answering inference questions like the distribution of the percentage of infected individuals and the most-probable equilibrium configuration. We then use the approximation to analyse the equilibrium distribution of the extended contact process on the 4941-node US Western power grid.<\/jats:p>","DOI":"10.1093\/comnet\/cnx003","type":"journal-article","created":{"date-parts":[[2017,5,3]],"date-time":"2017-05-03T11:07:51Z","timestamp":1493809671000},"page":"712-733","source":"Crossref","is-referenced-by-count":9,"title":["Contact process with exogenous infection and the scaled SIS process"],"prefix":"10.1093","volume":"5","author":[{"given":"June","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 M.F.","family":"Moura","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"June","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, University of Hawai\u2019i at Manoa, Honolulu, HI 96822, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2017,5,15]]},"reference":[{"key":"2019071815295061800_B1","doi-asserted-by":"crossref","first-page":"016116","DOI":"10.1103\/PhysRevE.86.016116","article-title":"Epidemics in networks with nodal self-infection and the epidemic threshold","volume":"86","author":"Van Mieghem","year":"2012","journal-title":"Phys. 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