Graduate School and Research Center in Digital Sciences

Spectral bounds of the regularized normalized Laplacian for random geometric graphs

Hamidouche, Mounia; Cottatellucci, Laura; Avrachenkov, Konstantin

GSP 2019, 4th Graph Signal Processing Workshop, June 5-7th, 2019, Minnesota, Minneapolis, USA

In this work, we study the spectrum of the regularized normalized Laplacian for random geometric graphs (RGGs) in both the connectivity and thermodynamic regimes. We prove that the limiting eigenvalue distribution (LED) of the normalized Laplacian matrix for an RGG converges to the Dirac measure in one in the full range of the connectivity regime. In the thermodynamic regime, we propose an approximation for the LED and we provide a bound on the Levy distance between the approximation and the actual distribution. In particular, we show that the LED of the regularized normalized Laplacian matrix for an RGG can be approximated by the LED of the regularized normalized Laplacian for a deterministic geometric graph with nodes in a grid (DGG). Thereby, we obtain an explicit approximation of the eigenvalues in the thermodynamic regime.

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Title:Spectral bounds of the regularized normalized Laplacian for random geometric graphs
Department:Communication systems
Eurecom ref:6048
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Bibtex: @inproceedings{EURECOM+6048, year = {2019}, title = {{S}pectral bounds of the regularized normalized {L}aplacian for random geometric graphs}, author = {{H}amidouche, {M}ounia and {C}ottatellucci, {L}aura and {A}vrachenkov, {K}onstantin}, booktitle = {{GSP} 2019, 4th {G}raph {S}ignal {P}rocessing {W}orkshop, {J}une 5-7th, 2019, {M}innesota, {M}inneapolis, {USA}}, address = {{M}innesota, {UNITED} {STATES}}, month = {06}, url = {} }
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