Wednesday, June 12, 2013

1306.2507 (Tiago P. Peixoto)

Eigenvalue spectra of modular networks    [PDF]

Tiago P. Peixoto
We obtain in a unified fashion the spectrum of the adjacency, Laplacian and normalized Laplacian matrices for networks with arbitrary modular structure, in the limit of large degrees. We focus on the conditions necessary for the merging of the isolated eigenvalues with the continuous band of the spectrum, after which the planted modular structure can no longer be easily detected by spectral methods. We show that this transition happens in general at different points for the different matrices, and hence the detectability threshold can vary significantly depending on the method chosen. Equivalently, the sensitivity to the modular structure of the different dynamical processes associated with each matrix will be different, given the same large-scale structure present in the network. Furthermore, with the exception of the Laplacian matrix, the different transitions coalesce into the same point for the special case where the modules are homogeneous, but separate otherwise.
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