Low-Order Spectral Analysis of the Kirchhoff Matrix for a Probabilistic Graph With a Prescribed Expected Degree Sequence

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.