From Local Measurements to Network Spectral Properties: Beyond Degree Distributions

TL;DR

This paper establishes a connection between local network motifs and the global spectral properties of the network, enabling inference of eigenvalue spectra from local measurements to understand network dynamics.

Contribution

It introduces explicit relationships between spectral moments and small subgraphs, and proposes a decentralized method to compute spectral moments from local data.

Findings

Spectral moments can be inferred from motif counts.

A decentralized scheme for spectral moment computation is proposed.

Global spectral properties relate to local topological measurements.

Abstract

It is well-known that the behavior of many dynamical processes running on networks is intimately related to the eigenvalue spectrum of the network. In this paper, we address the problem of inferring global information regarding the eigenvalue spectrum of a network from a set of local samples of its structure. In particular, we find explicit relationships between the so-called spectral moments of a graph and the presence of certain small subgraphs, also called motifs, in the network. Since the eigenvalues of the network have a direct influence on the network dynamical behavior, our result builds a bridge between local network measurements (i.e., the presence of small subgraphs) and global dynamical behavior (via the spectral moments). Furthermore, based on our result, we propose a novel decentralized scheme to compute the spectral moments of a network by aggregating local measurements of the network topology. Our final objective is to understand the relationships between the behavior of dynamical processes taking place in a large-scale complex network and its local topological properties.