Random Matrix Theory Applied to Small-World Networks

​
rewiring probability
p
0.7
show plot
eigenvalue spacings
network graph
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
p
" 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
p
is zero or very small (
p<0.001
), 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
p=1.0
, 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 (
0<p<1.0
), the eigenvalue spacing distribution can be modeled by a critical semi-Poisson-like function, whose onset occurs at
p=0.1
. For intermediate values in the range
0.1<p<1.0
, 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
0<p<0.1
Snapshot 2: network graph corresponding to Snapshot 1
Snapshot 3: critical distribution at the onset of small-world regime (
0.1<p<1.0
), rewiring probability
p=0.1
Snapshot 4: network graph corresponding to Snapshot 3
Snapshot 5: Wigner distribution in random graph regime, rewiring probability
p=1.0
Snapshot 6: network graph corresponding to Snapshot 5

References

[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.

Permanent Citation

Jessica Alfonsi
​
​"Random Matrix Theory Applied to Small-World Networks"​
​http://demonstrations.wolfram.com/RandomMatrixTheoryAppliedToSmallWorldNetworks/​
​Wolfram Demonstrations Project​
​Published: September 10, 2020