Detecting Central Nodes from Low-rank Excited Graph Signals via Structured Factor Analysis

TL;DR

This paper introduces a method to detect central nodes in a graph from filtered signals using low-rank and sparse factor models, with a two-stage algorithm combining NMF and robust PCA, outperforming previous approaches.

Contribution

It proposes a structured factor analysis approach for blind detection of central nodes under low pass graph filter conditions, with theoretical identifiability analysis and practical algorithms.

Findings

PCA accurately detects central nodes in strong low pass filter cases.

The structured factor model enables detection in general low pass filter scenarios.

Numerical experiments show significant performance improvements over prior methods.

Abstract

This paper treats a blind detection problem to identify the central nodes in a graph from filtered graph signals. Unlike prior works which impose strong restrictions on the data model, we only require the underlying graph filter to satisfy a low pass property with a generic low-rank excitation model. We treat two cases depending on the low pass graph filter's strength. When the graph filter is strong low pass, i.e., it has a frequency response that drops sharply at the high frequencies, we show that the principal component analysis (PCA) method detects central nodes with high accuracy. For general low pass graph filter, we show that the graph signals can be described by a structured factor model featuring the product between a low-rank plus sparse factor and an unstructured factor. We propose a two-stage decomposition algorithm to learn the structured factor model via a judicious combination of the non-negative matrix factorization and robust PCA algorithms. We analyze the identifiability conditions for the model which lead to accurate central nodes detection. Numerical experiments on synthetic and real data are provided to support our findings. We demonstrate significant performance gains over prior works.