This Demonstration shows an application of random matrix theory to complex networks, in particular, small-world network realizations according to the Watts–Strogatz model implemented in the Wolfram Language function WattsStrogatzGraphDistribution.

By changing the "rewiring probability

" slider, it is possible to explore different regimes both in complex network theory and in the eigenvalue spacing distributions known from random matrix theory (RMT). In this way, a relationship between them can be investigated dynamically.

When the rewiring probability

is zero or very small (

), the network graph is in the regular regime and the histogram of the unfolded eigenvalue nearest-neighbor spacings can be described by a Poisson distribution. In the extreme opposite case, the pure random graph regime, the rewiring probability

, and the histogram of the eigenvalue spacings follows the Wigner surmise function from the Gaussian orthogonal ensemble (GOE) statistics. In the intermediate small-world regime (

), the eigenvalue spacing distribution can be modeled by a critical semi-Poisson-like function, whose onset occurs at

. For intermediate values in the range

, the eigenvalue spacing statistics are actually intermediate between the critical and the GOE regime.