Double Clustering and Graph Navigability

Abstract: Graphs are called navigable if one can find short paths through them using only local knowledge. It has been shown that for a graph to be navigable, its construction needs to meet strict criteria. Since such graphs nevertheless seem to appear in nature, it is of interest to understand why these criteria should be fulfilled. In this paper we present a simple method for constructing graphs based on a model where nodes vertices are ``similar'' in two different ways, and tend to connect to those most similar to them - or cluster - with respect to both. We prove that this leads to navigable networks for several cases, and hypothesize that it also holds in great generality. Enough generality, perhaps, to explain the occurrence of navigable networks in nature.