Modeling interacting dynamic networks: I. Preferred degree networks and their characteristics

TL;DR

This paper introduces a dynamic network model with preferred degrees, analyzes its steady-state degree distribution through simulations and theory, and explores coupled networks revealing complex behaviors like a flat distribution of inter-group links.

Contribution

The study develops a simple yet insightful model of preferred degree networks, extending it to coupled networks, and provides analytical and simulation results explaining their complex steady-state properties.

Findings

Degree distribution differs from Poisson, explained by approximate theory.

Coupled networks show total degree distributions well-predicted, but intra- and inter-group distributions are complex.

Inter-group links exhibit a flat distribution, akin to a confined random walk.

Abstract

We study a simple model of dynamic networks, characterized by a set preferred degree, $\kappa$. Each node with degree $k$ attempts to maintain its $\kappa$ and will add (cut) a link with probability $w(k;\kappa)$ ($1-w(k;\kappa)$). As a starting point, we consider a homogeneous population, where each node has the same $\kappa$, and examine several forms of $w(k;\kappa)$, inspired by Fermi-Dirac functions. Using Monte Carlo simulations, we find the degree distribution in steady state. In contrast to the well-known Erd\H{o}s-R\'{e}nyi network, our degree distribution is not a Poisson distribution; yet its behavior can be understood by an approximate theory. Next, we introduce a second preferred degree network and couple it to the first by establishing a controllable fraction of inter-group links. For this model, we find both understandable and puzzling features. Generalizing the prediction for the homogeneous population, we are able to explain the total degree distributions well, but not the intra- or inter-group degree distributions. When monitoring the total number of inter-group links, $X$, we find very surprising behavior. $X$ explores almost the full range between its maximum and minimum allowed values, resulting in a flat steady-state distribution, reminiscent of a simple random walk confined between two walls. Both simulation results and analytic approaches will be discussed.