Spectra of random graphs with arbitrary expected degrees

Abstract

We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit of large network size and large vertex degrees. We also study the effect on the spectra of hubs in the network, vertices of unusually high degree, and show that these produce isolated eigenvalues outside the main spectral band, akin to impurity states in condensed matter systems, with accompanying eigenvectors that are strongly localized around the hubs. We also give numerical results that confirm our analytic expressions.

14 pages, 5 figures

References in corpus (5)

• Modularity and community structure in networks
• Synchronization in complex networks
• Graph spectra and the detectability of community structure in networks
• Spectra of Sparse Random Matrices
• Spectral properties of the Google matrix of the World Wide Web and other directed networks

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