Self-stabilizing Power-law Networks

Abstract

Power-law graphs model the interconnections in various types of large-scale networks ranging from physical and biological systems to man-made social networks and web graphs. In these graphs, the degree distribution of the nodes obeys the power-law property: the fraction of nodes P(k) having a degree k closely follows the rule P(k) ∞ k−-γ. In the domain of man-made systems, if the topology of a power-law network gets altered due to failures or adversarial attacks, then remedial actions to restore the power-law property are very important. This paper presents self-stabilizing algorithms for maintaining the power-law property in a network of processes. These algorithms allow spontaneous restoration of the power-law property from any initial connected configuration. The algorithms consist of three modular components: a detection component to detect the violation of the power-law property, an interim topology creation component, and a repair component to build the final graph. We propose two different interim topologies, a clique and a linear graph. We then present two different techniques for rebuilding the power-law topology -- a probabilistic approach based on the preferential attachment model, which stabilizes in O(log n) communication rounds with a link complexity of O(n) per process, and a deterministic approach that introduces the novel data structure Bridge Tree and stabilizes in O(n) communication rounds with a much lower link complexity.