Labeling Algorithm and Compact Routing Scheme for a Small World Network Model

This paper defines the toroidal small world labeling problem that asks for a labeling of the vertices of a network such that the labels possess information that allows a compact routing scheme in the network. We consider the problem over a small world network model we propose. Both the model and the compact routing scheme have applications in peer-to-peer networks. The proposed model is based on the model of Kleinberg (2000), and generates an undirected two-dimensional torus with one random long-range edge per vertex. These random edges create forbidden cycles that mimic the underlying torus topology, and this behavior confuses attempts for extracting the routing information from the network. We show that such forbidden cycles happen with small probability, allowing us to use a breadth-first search that finds the vertices positions on the torus. The positions are pairs of integer numbers that provides routing information to a greedy routing algorithm that finds small paths of the network. We present a linear time labeling algorithm that detects and removes the random edges, finds the underlying torus and labels almost all vertices through a breadth-first search. The labeling algorithm is then used by a compact routing scheme for the proposed small world model.