Random walk in degree space and the time-dependent Watts-Strogatz model

TL;DR

This paper introduces an analytical scheme that models the evolution of network degree distributions over time by mapping the problem to a random walk in degree space, successfully applied to dynamic Erdős-Rényi and Watts-Strogatz networks.

Contribution

It presents a novel analytical approach for time-dependent degree distributions in networks, especially for the Watts-Strogatz model, extending static models to dynamic regimes.

Findings

Derived an analytical form for the dynamic Watts-Strogatz model.

The method is asymptotically exact in certain regimes.

Applied the approach to Erdős-Rényi and Watts-Strogatz networks.

Abstract

In this work, we propose a scheme that provides an analytical estimate for the time-dependent degree distribution of some networks. This scheme maps the problem into a random walk in degree space, and then we choose the paths that are responsible for the dominant contributions. The method is illustrated on the dynamical versions of the Erd\"os-R\'enyi and Watts-Strogatz graphs, which were introduced as static models in the original formulation. We have succeeded in obtaining an analytical form for the dynamics Watts-Strogatz model, which is asymptotically exact for some regimes.