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Small-world graphs, which combine randomized and structured elements, areseen as prevalent in nature. Jon Kleinberg showed that in some graphs of thistype it is possible to route, or navigate, between vertices in few steps evenwith very little knowledge of the graph itself. In an attempt to understand howsuch graphs arise we introduce a different criterion for graphs to be navigablein this sense, relating the neighbor selection of a vertex to the hittingprobability of routed walks. In several models starting from both discrete andcontinuous settings, this can be shown to lead to graphs with the desiredproperties. It also leads directly to an evolutionary model for the creation ofsimilar graphs by the stepwise rewiring of the edges, and we conjecture,supported by simulations, that these too are navigable.

Neighbor selection and hitting probability in small-world graphs