How Do Networks Become Navigable?

Abstract

Networks created and maintained by social processes, such as the human friendship network and the World Wide Web, appear to exhibit the property of navigability: namely, not only do short paths exist between any pair of nodes, but such paths can easily be found using only local information. It has been shown that for networks with an underlying metric, algorithms using only local information perform extremely well if there is a power-law distribution of link lengths. However, it is not clear why or how real networks might develop this distribution. In this paper we define a decentralized ``rewiring'' process, inspired by surfers on the Web, in which each surfer attempts to travel from their home page to a random destination, and updates the outgoing link from their home page if this journey takes too long. We show that this process does indeed cause the link length distribution to converge to a power law, achieving a routing time of O(log^2 n) on networks of size n. We also study finite-size effects on the optimal exponent, and show that it converges polylogarithmically slowly as the lattice size goes to infinity.