Fixed-Size Dynamic Scale-Free Networks: Modeling, Stationarity, and Resilience

TL;DR

This paper introduces a novel fixed-size scale-free network model based on degree fluctuations as a state-dependent random walk, demonstrating stable power-law distributions and resilience properties without node growth or preferential attachment.

Contribution

It presents an innovative model for fixed-size networks that explains how scale-free properties emerge without node addition or preferential attachment mechanisms.

Findings

Degree distribution converges to a power-law distribution under certain conditions.

The model accurately replicates real-world network data.

Resilience of network components is analyzed post-attacks.

Abstract

Many real-world scale-free networks, such as neural networks and online communication networks, consist of a fixed number of nodes but exhibit dynamic edge fluctuations. However, traditional models frequently overlook scenarios where the node count remains constant, instead prioritizing node growth. In this work, we depart from the assumptions of node number variation and preferential attachment to present an innovative model that conceptualizes node degree fluctuations as a state-dependent random walk process with stasis and variable diffusion coefficient. We show that this model yields stochastic dynamic networks with stable scale-free properties. Through comprehensive theoretical and numerical analyses, we demonstrate that the degree distribution converges to a power-law distribution, provided that the lowest degree state within the network is not an absorbing state. Furthermore, we investigate the resilience of the fraction of the largest component and the average shortest path length following deliberate attacks on the network. By using three real-world networks, we confirm that the proposed model accurately replicates actual data. The proposed model thus elucidates mechanisms by which networks, devoid of growth and preferential attachment features, can still exhibit power-law distributions and be used to simulate and study the resilience of attacked fixed-size scale-free networks.