Towards small world emergence

Abstract

We investigate the problem of optimizing the routing performance of a virtual network by adding extra random links. Our asynchronous and distributed algorithm ensures, by adding a single extra link per node, that the resulting network is a navigable small world, i.e., in which greedy routing, using the distance in the original network, computes paths of polylogarithmic length between any pair of nodes with probability 1-O(1/n). Previously known small world augmentation processes require the global knowledge of the network and centralized computations, which is unrealistic for large decentralized networks. Our algorithm, based on a careful multi-layer sampling of the nodes and the construction of a light overlay network, bypasses these limitations. For bounded growth graphs, i.e., graphs where, for any node u and any radius r the number of nodes within distance 2r from u is at most a constant times the number of nodes within distance r, our augmentation process proceeds with high probability in O(log n log D) communication rounds, with O(log n log D) messages of size O(log n) bits sent per node and requiring only O(log n log D) bit space in each node, where n is the number of nodes, and D the diameter. In particular, with the only knowledge of original distances, greedy routing computes, between any pair of nodes in the augmented network, a path of length at most O(log2 n log2 D) with probability 1 - O(1/n), and of expected length O(log n log2 D). Hence, we provide a distributed scheme to augment any bounded growth graph into a small world with high probability in polylogarithmic time while requiring polylogarithmic memory. We consider that the existence of such a lightweight process might be a first step towards the definition of a more general construction process that would validate Kleinberg's model as a plausible explanation for the small world phenomenon in large real interaction networks.