Inference of a Rumor's Source in the Independent Cascade Model

Petra Berenbrink, Max Hahn-Klimroth, Dominik Kaaser, Lena Krieg, Malin Rau

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.