Exact Solution for the Time Evolution of Network Rewiring Models

TL;DR

This paper derives an exact, comprehensive solution for the time evolution of a bipartite network rewiring model combining random and preferential attachment, applicable across various related models and real-world phenomena.

Contribution

It provides the first exact solution for the degree distribution dynamics of a bipartite network rewiring process with mixed attachment mechanisms.

Findings

Exact solutions match numerical simulations closely.

Model encompasses various existing models like Urn, Balls-in-Boxes, and Watts-Strogatz.

Applicable to diverse fields such as cultural transmission, genetics, and wealth distribution.

Abstract

We consider the rewiring of a bipartite graph using a mixture of random and preferential attachment. The full mean field equations for the degree distribution and its generating function are given. The exact solution of these equations for all finite parameter values at any time is found in terms of standard functions. It is demonstrated that these solutions are an excellent fit to numerical simulations of the model. We discuss the relationship between our model and several others in the literature including examples of Urn, Backgammon, and Balls-in-Boxes models, the Watts and Strogatz rewiring problem and some models of zero range processes. Our model is also equivalent to those used in various applications including cultural transmission, family name and gene frequencies, glasses, and wealth distributions. Finally some Voter models and an example of a Minority game also show features described by our model.