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Efficient Laplacian spectral density computations for networks with arbitrary degree distributions

Published online by Cambridge University Press:  07 October 2021

Grover E. C. Guzman ,
Peter F. Stadler Open the ORCID record for Peter F. Stadler [Opens in a new window]  and
André Fujita Open the ORCID record for André Fujita [Opens in a new window]
Grover E. C. Guzman
Affiliation:
Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010 - São Paulo - SP, 05508-090 Brazil (e-mail: grover@ime.usp.br) and
Peter F. Stadler
Affiliation:
Bioinformatics Group, Department of Computer Science, and Interdisciplinary Center for Bioinformatics, Universität Leipzig, Härtelstraße 16-18, Leipzig, D-04107, Germany (e-mail: studla@bioinf.uni-leipzig.de)
André Fujita*
Affiliation:
Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010 - São Paulo - SP, 05508-090 Brazil (e-mail: grover@ime.usp.br) and
*
*Corresponding author. Email: andrefujita@usp.br
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Abstract

The network Laplacian spectral density calculation is critical in many fields, including physics, chemistry, statistics, and mathematics. It is highly computationally intensive, limiting the analysis to small networks. Therefore, we present two efficient alternatives: one based on the network’s edges and another on the degrees. The former gives the exact spectral density of locally tree-like networks but requires iterative edge-based message-passing equations. In contrast, the latter obtains an approximation of the spectral density using only the degree distribution. The computational complexities are 𝒪(|E|log(n)) and 𝒪(n), respectively, in contrast to 𝒪(n3) of the diagonalization method, where n is the number of vertices and |E| is the number of edges.

Keywords

spectral densityLaplacian matrixnormalized Laplacian matrixconfiguration model tree-like network
Type
Research Article
Information
Network Science , Volume 9 , Issue 3 , September 2021 , pp. 312 - 327
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

Action Editor: Ulrik Brandes

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