2013/10/15 by Piet Van Mieghem, Van Mieghem, Piet
Mathematics · Physics and Astronomy · Psychology · #COVID-19 epidemiological studies #Complex Network Analysis Techniques #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Mental Health Research Topics #Physics and Society (physics.soc-ph) #Probability (math.PR) #Social and Information Networks (cs.SI) #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.1310.3980
openalex publication_date 2013/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The decay rate of SIS epidemics on the complete graph KN is computed analytically, based on a new, algebraic method to compute the second largest eigenvalue of a stochastic three-diagonal matrix up to arbitrary precision. The latter problem has been addressed around 1950, mainly via the theory of orthogonal polynomials and probability theory. The accurate determination of the second largest eigenvalue, also called the decay parameter, has been an outstanding problem appearing in general birth-death processes and random walks. Application of our general framework to SIS epidemics shows that the maximum average lifetime of an SIS epidemics in any network with N nodes is not larger (but tight for KN) than E[ T] ∼\frac1δ\frac\fracττc√(2π)% ( \fracττc-1) 2\fracexp( N\ log\fracττc+\fracτcτ-1\ ) √ N=O( e^Nln\fracττc) for large N and for an effective infection rate τ=\fracβδ above the epidemic threshold τc. Our order estimate of E[ T] sharpens the order estimate E[ T] =O( e^bNa) of Draief and Massoulié \citeDraiefMassoulie. Combining the lower bound results of Mountford et al. \citeMountford2013 and our upper bound, we conclude that for almost all graphs, the average time to absorption for τ>τc is E[ T] =O( e^cGN) , where cG>0 depends on the topological structure of the graph G and τ.