Blind Extraction of Equitable Partitions from Graph Signals

TL;DR

This paper introduces methods to identify equitable partitions in graphs solely from filter output observations, without knowing the network edges, using control inputs and spectral algorithms, with theoretical error bounds and numerical validation.

Contribution

It proposes novel algorithms for blind equitable partition detection from graph filter outputs, extending existing methods to new observation settings and providing theoretical guarantees.

Findings

Successful extraction of equitable partitions using control-based and spectral methods.

Theoretical bounds on the error of the spectral detection scheme.

Numerical experiments validating the algorithms and theoretical results.

Abstract

Finding equitable partitions is closely related to the extraction of graph symmetries and of interest in a variety of applications context such as node role detection, cluster synchronization, consensus dynamics, and network control problems. In this work we study a blind identification problem in which we aim to recover an equitable partition of a network without the knowledge of the network's edges but based solely on the observations of the outputs of an unknown graph filter. Specifically, we consider two settings. First, we consider a scenario in which we can control the input to the graph filter and present a method to extract the partition inspired by the well known Weisfeiler-Lehman (color refinement) algorithm. Second, we generalize this idea to a setting where only observe the outputs to random, low-rank excitations of the graph filter, and present a simple spectral algorithm to extract the relevant equitable partitions. Finally, we establish theoretical bounds on the error that this spectral detection scheme incurs and perform numerical experiments that illustrate our theoretical results and compare both algorithms.