Low-order spectral analysis of the Kirchhoff matrix for a probabilistic graph with a prescribed expected degree sequence

Authors:
Victor M. Preciado;George C. Verghese
Affiliations:
Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA;Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA
Venue:
IEEE Transactions on Circuits and Systems Part I: Regular Papers
Year:
2009

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Abstract

We study the eigenvalue distribution of the Kirchhoff matrix of a large-scale probabilistic network with a prescribed expected degree sequence. This spectrum plays a key role in many dynamical and structural network problems such as synchronization of a network of oscillators. We introduce analytical expressions for the first three moments of the eigenvalue distribution of the Kirchhoff matrix, as well as a probabilistic asymptotic analysis of these moments for a graph with a prescribed expected degree sequence. These results are applied to the analysis of synchronization in a large-scale probabilistic network of oscillators.