Spectral Analysis of Critical Erdös-Renyi Graphs

27.09.2019 11:15 – 12:15

RESEARCH CENTER FOR STATISTICS SEMINAR / ABSTRACT

The Erdös-Rényi graph G(N,p) is the simplest model of a random graph, where each edge of the complete graph on N vertices is open with probability p, independently of the others. If p = p_N is not too small then the degrees of the graph concentrate with high probability and the graph is homogeneous. On the other hand, for p of order (log N) / N and smaller, the degrees cease to concentrate and the graph is with high probability inhomogeneous, containing isolated vertices, leaves, hubs, etc. I present results on the eigenvalues and eigenvectors of the adjacency matrix of G(N,p) at and below the critical scale. I show a rigidity estimate for the locations of the eigenvalues and explain a transition from localized to delocalized eigenvectors at a specific location in the spectrum.

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