Blind Community Detection From Low-Rank Excitations of a Graph Filter

Abstract

This paper considers a new framework to detect communities in a graph from the observation of signals at its nodes. We model the observed signals as noisy outputs of an unknown network process, represented as a graph filter that is excited by a set of unknown low-rank inputs/excitations. Application scenarios of this model include diffusion dynamics, pricing experiments, and opinion dynamics. Rather than learning the precise parameters of the graph itself, we aim at retrieving the community structure directly. The paper shows that communities can be detected by applying a spectral method to the covariance matrix of graph signals. Our analysis indicates that the community detection performance depends on a 'low-pass' property of the graph filter. We also show that the performance can be improved via a low-rank matrix plus sparse decomposition method when the latent parameter vectors are known. Numerical experiments demonstrate that our approach is effective.

Estimating Centrality Blindly From Low-Pass Filtered Graph Signals

“…This condition is also known as the 1-low pass in the author's prior work [19] which implies that the vector c eig = v1 remains the top eigenvector of the graph filter H(A). This condition is common for processes on graphs in practice.…”

“…Graph Filter Robust Estimation. To design a centrality estimation method agnostic to the underlying graph filtering process, we consider the boosted graph filter [19] defined as:…”

“…Particularly, the corollary indicates that we can reliably estimate the node centralities from analyzing the singular vectors of L, especially when c boost 1. Our final task is to compute the decomposition (19) in order to retrieve L. To this end, we observe L is close to rank-one if c boost 1 and S := ρB is sparse under the considered influence model. The matrix decomposition problem can be effectively tackled through the convex problem [25]:…”

“…Note a number of recent works [19][20][21][22] have worked on the related problem of blind community detection. In contrary, this paper proposes to estimate node centrality in a similarly blind manner.…”

“…In contrary, this paper proposes to estimate node centrality in a similarly blind manner. Specificatlly this paper studies a model where the graph signals are generated from exciting a low pass graph filter [19] via a sparse influence matrix. Our model is motivated by a diffusion process in stock market.…”