Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Year : 2019

Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime

Abstract

Network geometries are typically characterized by having a finite spectral dimension (SD), that characterizes the return time distribution of a random walk on a graph. The main purpose of this work is to determine the SD of a variety of random graphs called random geometric graphs (RGGs) in the thermodynamic regime, in which the average vertex degree is constant. The spectral dimension depends on the eigenvalue density (ED) of the RGG normalized Laplacian in the neighborhood of the minimum eigenvalues. In fact, the behavior of the ED in such a neighborhood characterizes the random walk. Therefore, we first provide an analytical approximation for the eigenvalues of the regularized normalized Laplacian matrix of RGGs in the thermodynamic regime. Then, we show that the smallest non zero eigenvalue converges to zero in the large graph limit. Based on the analytical expression of the eigenvalues, we show that the eigenvalue distribution in a neighborhood of the minimum value follows a power-law tail. Using this result, we find that the SD of RGGs is approximated by the space dimension d in the thermodynamic regime.
Fichier principal
Vignette du fichier
Spectral-Dimension.pdf (281.19 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-02397383 , version 1 (06-12-2019)

Identifiers

Cite

Konstantin Avrachenkov, Laura Cottatellucci, Mounia Hamidouche. Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime. Complex Networks 2019 - 8th International Conference on Complex Networks and Their Applications, Dec 2019, Lisbon, Portugal. ⟨10.1007/978-3-030-36687-2_80⟩. ⟨hal-02397383⟩
64 View
350 Download

Altmetric

Share

Gmail Facebook X LinkedIn More