Scale-free networks without growth

TL;DR

This paper introduces a model for scale-free networks that do not grow in size, using rewiring processes driven by node degrees, and demonstrates that scale-free structures can emerge without network growth.

Contribution

The study presents an ungrowing scale-free network model with an analytic solution for degree distribution, challenging the notion that growth is necessary for scale-free properties.

Findings

Degree distribution can vary from Poisson to power-law.

Scale-free structures can form without network growth.

Rewiring based on node degree influences network topology.

Abstract

In this letter, we proposed an ungrowing scale-free network model, wherein the total number of nodes is fixed and the evolution of network structure is driven by a rewiring process only. In spite of the idiographic form of $G$, by using a two-order master equation, we obtain the analytic solution of degree distribution in stable state of the network evolution under the condition that the selection probability $G$ in rewiring process only depends on nodes' degrees. A particular kind of the present networks with $G$ linearly correlated with degree is studied in detail. The analysis and simulations show that the degree distributions of these networks can varying from the Possion form to the power-law form with the decrease of a free parameter $\alpha$, indicating the growth may not be a necessary condition of the self-organizaton of a network in a scale-free structure.