Confidence Sets for the Source of a Diffusion in Regular Trees

TL;DR

This paper develops confidence sets for identifying the source of a diffusion process on regular trees, achieving size bounds independent of the number of infected nodes, and analyzes related probabilistic models.

Contribution

It introduces a method for constructing source confidence sets in regular trees with size bounds independent of infected nodes, inspired by root-finding in attachment trees.

Findings

Confidence sets can be constructed with size independent of infected nodes

Probabilistic analysis of Pólya urns underpins the methodology

Illustrates limitations of source estimation in asymmetric graphs

Abstract

We study the problem of identifying the source of a diffusion spreading over a regular tree. When the degree of each node is at least three, we show that it is possible to construct confidence sets for the diffusion source with size independent of the number of infected nodes. Our estimators are motivated by analogous results in the literature concerning identification of the root node in preferential attachment and uniform attachment trees. At the core of our proofs is a probabilistic analysis of P\'{o}lya urns corresponding to the number of uninfected neighbors in specific subtrees of the infection tree. We also provide an example illustrating the shortcomings of source estimation techniques in settings where the underlying graph is asymmetric.