Eigenvalues Outside the Bulk of Inhomogeneous Erdős-Rényi Random Graphs
Abstract
In this article, an inhomogeneous Erdős-Rényi random graph on {1 ,…,N } is considered, where an edge is placed between vertices i and j with probability εNf (i /N ,j /N ) , for i ≤j , the choice being made independently for each pair. The integral operator If associated with the bounded function f is assumed to be symmetric, non-negative definite, and of finite rank k. We study the edge of the spectrum of the adjacency matrix of such an inhomogeneous Erdős-Rényi random graph under the assumption that N εN→∞ sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by √{N εN }, is compactly supported, the kth largest eigenvalue goes to infinity. It turns out that the largest eigenvalue after appropriate scaling and centering converges to a Gaussian law, if the largest eigenvalue of If has multiplicity 1. If If has k distinct non-zero eigenvalues, then the joint distribution of the k largest eigenvalues converge jointly to a multivariate Gaussian law. The first order behaviour of the eigenvectors is derived as a byproduct of the above results. The results complement the homogeneous case derived by [18].
- Publication:
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Journal of Statistical Physics
- Pub Date:
- September 2020
- DOI:
- 10.1007/s10955-020-02644-7
- arXiv:
- arXiv:1911.08244
- Bibcode:
- 2020JSP...181.1746C
- Keywords:
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- Adjacency matrices;
- Inhomogeneous Erdős-Rényi random graph;
- Largest eigenvalue;
- Scaling limit;
- Stochastic block model;
- Mathematics - Probability
- E-Print:
- An assumption in Theorem 2.5 was missing. The same has now been added