Diffusion in scale-free networks with annealed disorder

TL;DR

This paper investigates how diffusion behaves in scale-free networks with dynamically changing (annealed) disorder, revealing significant differences from static (quenched) disorder scenarios, including phase transitions and reduced diffusion growth.

Contribution

It introduces models of diffusion in scale-free networks with annealed disorder and compares their behavior to quenched disorder, highlighting new phase transition phenomena.

Findings

Reduced growth of visited sites over time

Existence of a diffusion-blocking phase

Structural transition where diffusion ceases

Abstract

The scale-free (SF) networks that have been studied so far contained quenched disorder generated by random dilution which does not vary with the time. In practice, if a SF network is to represent, for example, the worldwide web, then the links between its various nodes may temporarily be lost, and re-established again later on. This gives rise to SF networks with annealed disorder. Even if the disorder is quenched, it may be more realistic to generate it by a dynamical process that is happening in the network. In this paper, we study diffusion in SF networks with annealed disorder generated by various scenarios, as well as in SF networks with quenched disorder which, however, is generated by the diffusion process itself. Several quantities of the diffusion process are computed, including the mean number of distinct sites visited, the mean number of returns to the origin, and the mean number of connected nodes that are accessible to the random walkers at any given time. The results including, (1) greatly reduced growth with the time of the mean number of distinct sites visited; (2) blocking of the random walkers; (3) the existence of a phase diagram that separates the region in which diffusion is possible from one in which diffusion is impossible, and (4) a transition in the structure of the networks at which the mean number of distinct sites visited vanishes, indicate completely different behavior for the computed quantities than those in SF networks with quenched disorder generated by simple random dilution.