Properties of dense partially random graphs

TL;DR

This paper analyzes dense partially random graphs with properties similar to Small World Graphs, providing analytical results for mean distance, clustering, eigenvalue distribution, and implications for mixing and synchronization.

Contribution

It introduces a model of dense partially random graphs with high average degree, deriving analytical expressions for spectral properties and network dynamics.

Findings

Mean distance and clustering are similar to Small World Graphs.

Eigenvalue distribution follows a semicircle law for small eigenvalues.

The spectrum includes a discrete part scaled from the substrate spectrum.

Abstract

We study the properties of random graphs where for each vertex a {\it neighbourhood} has been previously defined. The probability of an edge joining two vertices depends on whether the vertices are neighbours or not, as happens in Small World Graphs (SWGs). But we consider the case where the average degree of each node is of order of the size of the graph (unlike SWGs, which are sparse). This allows us to calculate the mean distance and clustering, that are qualitatively similar (although not in such a dramatic scale range) to the case of SWGs. We also obtain analytically the distribution of eigenvalues of the corresponding adjacency matrices. This distribution is discrete for large eigenvalues and continuous for small eigenvalues. The continuous part of the distribution follows a semicircle law, whose width is proportional to the "disorder" of the graph, whereas the discrete part is simply a rescaling of the spectrum of the substrate. We apply our results to the calculation of the mixing rate and the synchronizability threshold.