Scale-free and stable structures in complex ad hoc 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 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 only on the well-known preferential attachments of new nodes do not lead to a sufficiently heavy-tailed degree distribution in ad hoc networks. In particular, the magnitude of the power-law exponent increases rapidly (from 3) with the deletion rate, becoming infinity in the limit of equal insertion and deletion rates. We then introduce a local and universal 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. The dynamics reported 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.