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

This paper presents a small world networks generative model and a labeling algorithm for networks generated by this model. In the context of routing messages in networks, labeling algorithms process a network assigning labels, that are addresses, to the nodes. Our model is based on Kleinberg model, generating a 2-dimensional torus with additional random undirected long-range edges. The Kleinberg routing algorithm can forward messages in these networks, but it needs the vertices labels, that are positions in a n× n lattice, to make routing decisions. Finding these labels when they are not known a priori is a problem of routing in small world networks and labeling algorithms are possible solutions. The design of our labeling algorithm uses the approach of searching for induced 4-cycles to find the underlying torus. However, the generated graph may have 4-cycles with edges not in this torus. We show that the probability of these cycles appearing in a vertex is O(^-1n). This property allows the long-range edges removing through the detection of lattice patterns on combinations of 4-cycles, and the running of a breadth-first search in the resulting graph. Our labeling algorithm labels almost all vertices in O(|V|) expected time, where |V|=n^2. We also present a compact routing scheme, that is a combination of a preprocessing algorithm, that generates sub-linear structures per vertex, and a routing algorithm, that uses these structures for routing messages through paths with bounded length. Our preprocessing algorithm generates structures with expected size of O( n) bits per vertex in expected time O(|V|). The Kleinberg routing algorithm uses these structures, running in expected constant time in each vertex and performing an expected number of forwards O(^2 n).

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