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 $p$ 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 $p$. We propose that the average distance $\ensuremath{\ell}$ 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 ${n}^{*}(p\ensuremath{\ll}1)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\tau}}$ with $\ensuremath{\tau}\ensuremath{\approx}2/3$.