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We propose a dynamical process for network evolution, aiming at explainingthe emergence of the small world phenomenon, i.e., the statistical observationthat any pair of individuals are linked by a short chain of acquaintancescomputable by a simple decentralized routing algorithm, known as greedyrouting. Previously proposed dynamical processes enabled to demonstrateexperimentally (by simulations) that the small world phenomenon can emerge fromlocal dynamics. However, the analysis of greedy routing using the probabilitydistributions arising from these dynamics is quite complex because of mutualdependencies. In contrast, our process enables complete formal analysis. It isbased on the combination of two simple processes: a random walk process, and anharmonic forgetting process. Both processes reflect natural behaviors of theindividuals, viewed as nodes in the network of inter-individual acquaintances.We prove that, in k-dimensional lattices, the combination of these twoprocesses generates long-range links mutually independently distributed as ak-harmonic distribution. We analyze the performances of greedy routing at thestationary regime of our process, and prove that the expected number of stepsfor routing from any source to any target in any multidimensional lattice is apolylogarithmic function of the distance between the two nodes in the lattice.Up to our knowledge, these results are the first formal proof that navigabilityin small worlds can emerge from a dynamical process for network evolution. Ourdynamical process can find practical applications to the design of spatialgossip and resource location protocols.

Networks become navigable as nodes move and forget