Emergence of Network Structure in Models of Collective Evolution and Evolutionary Dynamics

TL;DR

This paper models an evolving network where nodes are duplicated or deleted, analyzing the resulting degree distributions which are mostly exponential but become power-law without mutations, revealing underlying mechanisms.

Contribution

It introduces a stochastic model of network evolution inspired by biological reproduction, analyzing degree distributions with Fokker-Planck equations and identifying conditions for exponential and power-law behaviors.

Findings

Exponential degree distributions are observed for a broad parameter range.

Degree distribution becomes a power law in the absence of mutations.

The mechanism behind these distributions is explained via mean field and Fokker-Planck analysis.

Abstract

We consider an evolving network of a fixed number of nodes. The allocation of edges is a dynamical stochastic process inspired by biological reproduction dynamics, namely by deleting and duplicating existing nodes and their edges. The properties of the degree distribution in the stationary state is analysed by use of the Fokker-Planck equation. For a broad range of parameters exponential degree distributions are observed. The mechanism responsible for this behaviour is illuminated by use of a simple mean field equation and reproduced by the Fokker-Planck equation treating the degree-degree correlations approximately. In the limit of zero mutations the degree distribution becomes a power law.