We consider the problem of finding the graph on which an epidemic cascade spreads, given only the times when each node gets infected. While this is a problem of importance in several contexts — offline and online social networks, e-commerce, epidemiology, vulnerabilities in infrastructure networks — there has been very little work, analytical or empirical, on finding the graph. Clearly, it is impossible to do so from just one cascade; our interest is in learning the graph from a small number of cascades. For the classic and popular “independent cascade” SIR epidemics, we analytically establish the number of cascades required by both the global maximum-likelihood (ML) estimator, and a natural greedy algorithm. Both results are based on a key observation: the global graph learning problem decouples into \\(n\\) local problems — one for each node. For a node of degree \\(d\\), we show that its neighborhood can be reliably found once it has been infected \\(O({d^}_{2 \\}log n)\\) times (for ML on general graphs) or \\(O(d \\log n)\\) times (for greedy on trees). We also provide a corresponding information-theoretic lower bound of \\(\\Omega (d \\log n)\\); thus our bounds are essentially tight. Furthermore, if we are given side-information in the form of a super-graph of the actual graph (as is often the case), then the number of cascade samples required — in all cases — becomes independent of the network size \\(n\\). Finally, we show that for a very general SIR epidemic cascade model, the Markov graph of infection times is obtained via the moralization of the network graph.<\/p>\n","protected":false},"excerpt":{"rendered":"
We consider the problem of finding the graph on which an epidemic cascade spreads, given only the times when each node gets infected. 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