Almost Optimal Decentralized Routing in Long-Range Contact Networks

Abstract

In order to explain the ability of individuals to find short paths to route messages to an unknown destination, based only on their own local view of a social network (the small world phenomenon), Kleinberg (2000) proposed a network model based on a d-dimensional lattice of size n augmented with k long range directed links per node. Individuals behavior is modeled by a greedy algorithm that forwards the message to the neighbor of the current holder, which is the closest to the destination. This algorithm computes paths of expected length Θ(log2 n/k) between any pair of nodes. Other topologies have been proposed later on to improve greedy algorithm performance. But, Aspnes et al. (2002) shows that for a wide class of long range link distributions, the expected length of the path computed by this algorithm is always \(\Omega\big(\log^2 n/(k^2\log\log n)\big)\).We design and analyze a new decentralized routing algorithm, in which nodes consult their neighbors near by, before deciding to whom forward the message. Our algorithm uses similar amount of computational resources as Kleinberg’s greedy algorithm: it is easy to implement, visits \(O\big(\log^2n/\log^2(1+k)\big)\) nodes on expectation and requires only Θ(log2 n/log(1+k)) bits of memory ⊢ note that [1] shows that any decentralized algorithm visits at least Ω(log2 n/k) on expectation. Our algorithm computes however an almost optimal path of expected length \(O\big(\log n(\log \log n)^2/\log^2(1+k)\big)\), between any pair of nodes. Our algorithm might fit better some human social behaviors (such as web browsing) and may also have successful applications to peer-to-peer networks where the length of the path along which the files are downloaded, is a critical parameter of the network performance.