Universal augmentation schemes for network navigability: Overcoming the √n-barrier

Augmented graphs were introduced for the purpose of analyzing the "six degrees of separation between individuals" observed experimentally by the sociologist Standley Milgram in the 60's. Formally, an augmented graph is a pair (G,ψ) where G is a graph, and ψ is a collection of probability distributions {ψ_u, u ∈ V(G)}. Every node u ∈ V(G) is given an extra link, called a long range link, pointing to some node v, called the long range contact of u. The head v of this link is chosen at random by Pr{u → v} = ψ_u(v). In augmented graphs, greedy routing is the oblivious routing process in which every intermediate node chooses among all its neighbors (including its long range contact) the one that is closest to the target according to the distance measured in the underlying graph G, and forwards to it. Roughly, augmented graphs aim at modeling the structure of social networks, while greedy routing aims at modeling the searching procedure applied in Milgram's experiment. Our objective is to design efficient universal augmentation schemes, i.e., augmentation schemes that give to any graph G a collection of probability distributions ψ such that greedy routing in (G,ψ) is fast. It is known that the uniform scheme ψ_unif is a universal scheme ensuring that, for any n-node graph G, greedy routing in (G,ψ_unif) performs in O(√n) expected number of steps. Our main result is the design of a universal augmentation scheme ψ such that greedy routing in (G,ψ) performs in Õ(n^1/3) expected number of steps for any n-node graph G. We also show that under some more restricted model, the √n-barrier cannot be overcome.