Changepoint Detection over Graphs with the Spectral Scan Statistic

TL;DR

This paper introduces a spectral scan statistic for change-point detection on graphs, leveraging graph spectra to improve detection performance over naive methods, with theoretical analysis and simulations supporting its effectiveness.

Contribution

It proposes a novel spectral scan statistic based on the graph Laplacian, providing a tractable approach for change detection that outperforms traditional methods.

Findings

Spectral scan statistic's performance depends on the graph's spectrum.

The method outperforms naive edge thresholding and chi-squared tests.

Asymptotic properties are derived for key graph topologies.

Abstract

We consider the change-point detection problem of deciding, based on noisy measurements, whether an unknown signal over a given graph is constant or is instead piecewise constant over two connected induced subgraphs of relatively low cut size. We analyze the corresponding generalized likelihood ratio (GLR) statistics and relate it to the problem of finding a sparsest cut in a graph. We develop a tractable relaxation of the GLR statistic based on the combinatorial Laplacian of the graph, which we call the spectral scan statistic, and analyze its properties. We show how its performance as a testing procedure depends directly on the spectrum of the graph, and use this result to explicitly derive its asymptotic properties on few significant graph topologies. Finally, we demonstrate both theoretically and by simulations that the spectral scan statistic can outperform naive testing procedures based on edge thresholding and $\chi^2$ testing.