Spectra of adjacency and Laplacian matrices of inhomogeneous Erdős–Rényi random graphs

Chakrabarty, Arijit ; Hazra, Rajat Subhra ; den Hollander, Frank ; Sfragara, Matteo (2021) Spectra of adjacency and Laplacian matrices of inhomogeneous Erdős–Rényi random graphs Random Matrices: Theory and Applications, 10 (01). p. 2150009. ISSN 2010-3263

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Official URL: http://doi.org/10.1142/S201032632150009X

Related URL: http://dx.doi.org/10.1142/S201032632150009X


This paper considers inhomogeneous Erdős–Rényi random graphs GN on N vertices in the non-sparse non-dense regime. The edge between the pair of vertices {i,j} is retained with probability εNf(iN,jN), 1≤i≠j≤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 GN, in the limit as N→∞ when limN→∞εN=0 and limN→∞NεN=∞. 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.

Item Type:Article
Source:Copyright of this article belongs to World Scientific Publishing Co Pte Ltd.
Keywords:Adjacency Matrices; Inhomogeneous Erdős–Rényi Random Graph; Laplacian; Empirical Spectral Distribution; Constrained Random Graphs.
ID Code:121053
Deposited On:09 Jul 2021 04:32
Last Modified:09 Jul 2021 04:32

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