Scale-Free and Stable Structures in Complex {\em Ad hoc} networks

TL;DR

This paper demonstrates that in ad hoc networks, a local compensatory rewiring dynamic can produce stable, scale-free structures with tunable power-law degree distributions, even under rapid node deletions.

Contribution

It introduces a universal rewiring mechanism that enables the emergence of scale-free networks in dynamic, almost constant-size environments, unlike traditional models.

Findings

Preferential attachment alone does not produce heavy-tailed distributions in ad hoc networks.

Rewiring with local compensatory dynamics results in scale-free degree distributions.

The power-law exponent can be tuned close to -2, matching observed real-world networks.

Abstract

Unlike the well-studied models of growing networks, where the dominant dynamics consist of insertions of new nodes and connections, and rewiring of existing links, we study {\em ad hoc} networks, where one also has to contend with rapid and random deletions of existing nodes (and, hence, the associated links). We first show that dynamics based {\em only} on the well-known preferential attachments of new nodes {\em do not} lead to a sufficiently heavy-tailed degree distribution in {\em ad hoc} networks. In particular, the magnitude of the power-law exponent increases rapidly (from 3) with the deletion rate, becoming $\infty$ in the limit of equal insertion and deletion rates. \iffalse ; thus, forcing the degree distribution to be essentially an exponential one.\fi We then introduce a {\em local} and {\em universal} {\em compensatory rewiring} dynamic, and show that even in the limit of equal insertion and deletion rates true scale-free structures emerge, where the degree distributions obey a power-law with a tunable exponent, which can be made arbitrarily close to -2. These results provide the first-known evidence of emergence of scale-free degree distributions purely due to dynamics, i.e., in networks of almost constant average size. The dynamics discovered in this paper can be used to craft protocols for designing highly dynamic Peer-to-Peer networks, and also to account for the power-law exponents observed in existing popular services.