Spectral Design of Dynamic Networks via Local Operations

TL;DR

This paper introduces a distributed algorithm enabling autonomous agents to self-organize their network structure to achieve desired spectral properties, facilitating control over dynamical processes on the network.

Contribution

It presents a novel decentralized method for spectral network design using local information, with proven stability and practical effectiveness.

Findings

Algorithm effectively controls eigenvalue spectra of networks.

Successfully designs networks with small-world and power-law spectral properties.

Demonstrates stability and efficiency in simulations.

Abstract

Motivated by the relationship between the eigenvalue spectrum of the Laplacian matrix of a network and the behavior of dynamical processes evolving in it, we propose a distributed iterative algorithm in which a group of $n$ autonomous agents self-organize the structure of their communication network in order to control the network's eigenvalue spectrum. In our algorithm, we assume that each agent has access only to a local (myopic) view of the network around it. In each iteration, agents in the network peform a decentralized decision process to determine the edge addition/deletion that minimizes a distance function defined in the space of eigenvalue spectra. This spectral distance presents interesting theoretical properties that allow an efficient distributed implementation of the decision process. Our iterative algorithm is stable by construction, i.e., locally optimizes the network's eigenvalue spectrum, and is shown to perform extremely well in practice. We illustrate our results with nontrivial simulations in which we design networks matching the spectral properties of complex networks, such as small-world and power-law networks.