Small-world networks: Evidence for a Crossover Picture

Watts and Strogatz [Nature (London) 393, 440 (1998)] have recently introduced a model for disordered networks and reported that, even for very small values of the disorder ρ in the links, the network behaves as a "small world." Here, we test the hypothesis that the appearance of small-world behavior is not a phase transition but a crossover phenomenon which depends both on the network size n and on the degree of disorder ρ. We propose that the average distance (between any two vertices of the network is a scaling function of n/n^*. The crossover size n^* above which the network behaves as a small world is shown to scale as n^*(ρ «K 1) - ρ^~T with T ≈ 2/3.