#### Title

Spectra of adjacency and Laplacian matrices of inhomogeneous Erdos-Rényi random graphs

#### Article Type

Research Article

#### Publication Title

Random Matrices: Theory and Application

#### Abstract

This paper considers inhomogeneous Erdos-Rényi random graphs N on N vertices in the non-sparse non-dense regime. The edge between the pair of vertices {i,j} is retained with probability Nf(i N, j N), 1 ≤ ij ≤ N, independently of other edges, where f:[0, 1] × [0, 1] → [0,∞) is a continuous function such that f(x,y) = f(y,x) for all x,y [0, 1]. We study the empirical distribution of both the adjacency matrix AN and the Laplacian matrix ΔN associated with N, in the limit as N →∞ when limN→∞N = 0 and limN→∞NN = ∞. In particular, we show that the empirical spectral distributions of AN and ΔN, after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where f(x,y) = r(x)r(y) with r:[0, 1] → [0,∞) a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, we apply our results to constrained random graphs, Chung-Lu random graphs and social networks.

#### DOI

10.1142/S201032632150009X

#### Publication Date

1-1-2021

#### Recommended Citation

Chakrabarty, Arijit; Hazra, Rajat Subhra; Den Hollander, Frank; and Sfragara, Matteo, "Spectra of adjacency and Laplacian matrices of inhomogeneous Erdos-Rényi random graphs" (2021). *Journal Articles*. 2269.

https://digitalcommons.isical.ac.in/journal-articles/2269