Dynamic networks and directed percolation

TL;DR

This paper models dynamic networks with changing links, mapping them to directed percolation, revealing universal critical behavior and fundamental changes in network laws, including giant component size and path length scaling.

Contribution

It introduces a novel model for dynamic networks, analyzing their phase transition behavior and fundamental properties through a directed percolation framework.

Findings

Percolation threshold decreases with link change rate r.

Giant component size at criticality scales with N for all r.

Optimal path length scales as N^{1/2} in dynamic networks.

Abstract

We introduce a model for dynamic networks, where the links or the strengths of the links change over time. We solve the model by mapping dynamic networks to the problem of directed percolation, where the direction corresponds to the evolution of the network in time. We show that the dynamic network undergoes a percolation phase transition at a critical concentration $p_c$, which decreases with the rate $r$ at which the network links are changed. The behavior near criticality is universal and independent of $r$. We find fundamental network laws are changed. (i) For Erd\H{o}s-R\'{e}nyi networks we find that the size of the giant component at criticality scales with the network size $N$ for all values of $r$, rather than as $N^{2/3}$. (ii) In the presence of a broad distribution of disorder, the optimal path length between two nodes in a dynamic network scales as $N^{1/2}$, compared to $N^{1/3}$ in a static network.