Inference of a Rumor's Source in the Independent Cascade ModelDownload PDF

Published: 08 May 2023, Last Modified: 26 Jun 2023UAI 2023Readers: Everyone
Keywords: Patient Zero, Independent Cascade Model, Phase-Transition
Abstract: We consider the so-called \emph{Independent Cascade Model} for rumor spreading or epidemic processes popularized by Kempe et al.\ (2003). In this model, a node of a network is the source of a rumor -- it is \emph{informed}. In discrete time steps, each informed node ``infects'' each of its uninformed neighbors with probability $p$. While many facets of this process are studied in the literature, less is known about the inference problem: given a number of infected nodes in a network, can we learn the source of the rumor? In the context of epidemiology this problem is often referred to as \emph{patient zero problem}. It belongs to a broader class of problems where the goal is to infer parameters of the underlying spreading model. In this work we present a maximum likelihood estimator for the rumor's source, given a snapshot of the process in terms of a set of active nodes $X$ after $t$ steps. Our results show that, for acyclic graphs, the likelihood estimator undergoes a phase transition as a function of $t$. We provide a rigorous analysis for two prominent classes of acyclic network, namely $d$-regular trees and Galton-Watson trees, and verify empirically that our heuristics work well in various general networks.
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