Scale-free Loopy Structure is Resistant to Noise in Consensus Dynamics in Complex Networks

TL;DR

This paper demonstrates that scale-free, loopy networks exhibit strong resistance to noise in consensus dynamics, with coherence approaching a constant as networks grow, highlighting their robustness due to topology.

Contribution

It provides analytical and numerical evidence that scale-free loopy networks maintain low coherence under noise, revealing their inherent robustness in consensus processes.

Findings

Coherence approaches a constant in large scale-free networks.

Noise impact on consensus is negligible in power-law networks.

Analytical expressions for coherence in specific deterministic networks.

Abstract

The vast majority of real-world networks are scale-free, loopy, and sparse, with a power-law degree distribution and a constant average degree. In this paper, we study first-order consensus dynamics in binary scale-free networks, where vertices are subject to white noise. We focus on the coherence of networks characterized in terms of the $H_2$-norm, which quantifies how closely agents track the consensus value. We first provide a lower bound of coherence of a network in terms of its average degree, which is independent of the network order. We then study the coherence of some sparse, scale-free real-world networks, which approaches a constant. We also study numerically the coherence of Barab\'asi-Albert networks and high-dimensional random Apollonian networks, which also converges to a constant when the networks grow. Finally, based on the connection of coherence and the Kirchhoff index, we study analytically the coherence of two deterministically-growing sparse networks and obtain the exact expressions, which tend to small constants. Our results indicate that the effect of noise on the consensus dynamics in power-law networks is negligible. We argue that scale-free topology, together with loopy structure, is responsible for the strong robustness with respect to noisy consensus dynamics in power-law networks.