# Random Matrix Theory Applied to Small-World Networks

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.

### DETAILS

Snapshot 1: Poisson distribution in the (almost) regular graph regime, very small rewiring probability
Snapshot 2: network graph corresponding to Snapshot 1
Snapshot 3: critical distribution at the onset of small-world regime (), rewiring probability
Snapshot 4: network graph corresponding to Snapshot 3
Snapshot 5: Wigner distribution in random graph regime, rewiring probability
Snapshot 6: network graph corresponding to Snapshot 5
Reference
[1] J. N. Bandyopadhyay and S. Jalan, "Universality in Complex Networks: Random Matrix Analysis," Physical Review E, 76(2), 2007 026109. doi:10.1103/PhysRevE.76.026109.

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