Cluster detection in networks using percolation

TL;DR

This paper introduces percolation-based methods for detecting salient clusters in sensor networks, offering advantages over traditional scan statistics in terms of shape independence and computational efficiency, supported by theoretical analysis and numerical experiments.

Contribution

It proposes novel percolation-based cluster detection methods and analyzes their asymptotic performance, providing a computationally feasible alternative to scan statistics.

Findings

Percolation-based methods outperform traditional scan statistics in shape independence.

The proposed methods are computationally more feasible for large networks.

Theoretical results are supported by numerical experiments.

Abstract

We consider the task of detecting a salient cluster in a sensor network, that is, an undirected graph with a random variable attached to each node. Motivated by recent research in environmental statistics and the drive to compete with the reigning scan statistic, we explore alternatives based on the percolative properties of the network. The first method is based on the size of the largest connected component after removing the nodes in the network with a value below a given threshold. The second method is the upper level set scan test introduced by Patil and Taillie [Statist. Sci. 18 (2003) 457-465]. We establish the performance of these methods in an asymptotic decision- theoretic framework in which the network size increases. These tests have two advantages over the more conventional scan statistic: they do not require previous information about cluster shape, and they are computationally more feasible. We make abundant use of percolation theory to derive our theoretical results, and complement our theory with some numerical experiments.