Quickest Inference of Network Cascades with Noisy Information

TL;DR

This paper develops near-optimal methods for quickly identifying the source of a network cascade using noisy observations, with theoretical bounds on the number of observations needed for different network structures.

Contribution

It introduces Bayesian and minimax estimators for cascade source detection, providing near-optimal solutions and bounds for simple network topologies.

Findings

Optimal estimators require log log n / log (k - 1) observations on k-regular trees.

On -lattices, th root of n observations are sufficient.

Proposed methods can be extended to arbitrary graph structures.

Abstract

We study the problem of estimating the source of a network cascade given a time series of noisy information about the spread. Initially, there is a single vertex affected by the cascade (the source) and the cascade spreads in discrete time steps across the network. The cascade evolution is hidden, but one can observe a time series of noisy signals from each vertex. The time series of a vertex is assumed to be a sequence of i.i.d. samples from a pre-change distribution $Q_0$ before the cascade affects the vertex, and the time series is a sequence of i.i.d. samples from a post-change distribution $Q_1$ once the cascade has affected the vertex. Given the time series of noisy signals, which can be viewed as a noisy measurement of the cascade evolution, we aim to devise a procedure to reliably estimate the cascade source as fast as possible. We investigate Bayesian and minimax formulations of the source estimation problem, and derive near-optimal estimators for simple cascade dynamics and network topologies. In the Bayesian setting, an estimator which observes samples until the error of the Bayes-optimal estimator falls below a threshold achieves optimal performance. In the minimax setting, optimal performance is achieved by designing a novel multi-hypothesis sequential probability ratio test (MSPRT). We find that these optimal estimators require $\log \log n / \log (k - 1)$ observations of the noisy time series when the network topology is a $k$-regular tree, and $(\log n)^{\frac{1}{\ell + 1}}$ observations are required for $\ell$-dimensional lattices. Finally, we discuss how our methods may be extended to cascades on arbitrary graphs.