Network Topology Identification from Spectral Templates

TL;DR

This paper introduces a convex optimization approach to infer network topology from spectral templates, enabling efficient recovery of graph structures like adjacency matrices and Laplacians, with applications in brain network analysis.

Contribution

It proposes a novel method leveraging spectral templates for topology inference, including algorithms and conditions for identifying key graph shifts such as adjacency and Laplacian matrices.

Findings

Effective recovery of synthetic networks demonstrated.

Successful application to structural brain networks.

Algorithms show robustness and efficiency in various scenarios.

Abstract

Network topology inference is a cornerstone problem in statistical analyses of complex systems. In this context, the fresh look advocated here permeates benefits from convex optimization and graph signal processing, to identify the so-termed graph shift operator (encoding the network topology) given only the eigenvectors of the shift. These spectral templates can be obtained, for example, from principal component analysis of a set of graph signals defined on the particular network. The novel idea is to find a graph shift that while being consistent with the provided spectral information, it endows the network structure with certain desired properties such as sparsity. The focus is on developing efficient recovery algorithms along with identifiability conditions for two particular shifts, the adjacency matrix and the normalized graph Laplacian. Application domains include network topology identification from steady-state signals generated by a diffusion process, and design of a graph filter that facilitates the distributed implementation of a prescribed linear network operator. Numerical tests showcase the effectiveness of the proposed algorithms in recovering synthetic and structural brain networks.