Cavity approach for the approximation of spectral density of graphs with heterogeneous structures

Grover E. C. Guzman, Peter F. Stadler, and Andre Fujita

Phys. Rev. E 109, 034303 – Published 7 March 2024

Abstract

Graphs have become widely used to represent and study social, biological, and technological systems. Statistical methods to analyze empirical graphs were proposed based on the graph's spectral density. However, their running time is cubic in the number of vertices, precluding direct application to large instances. Thus, efficient algorithms to calculate the spectral density become necessary. For sparse graphs, the cavity method can efficiently approximate the spectral density of locally treelike undirected and directed graphs. However, it does not apply to most empirical graphs because they have heterogeneous structures. Thus, we propose methods for undirected and directed graphs with heterogeneous structures using a new vertex's neighborhood definition and the cavity approach. Our methods' time and space complexities are $O (| E | h_{max}^{3} t)$ and $O (| E | h_{max}^{2} t)$ , respectively, where $| E |$ is the number of edges, $h_{max}$ is the size of the largest local neighborhood of a vertex, and $t$ is the number of iterations required for convergence. We demonstrate the practical efficacy by estimating the spectral density of simulated and real-world undirected and directed graphs.

Received 13 August 2023
Revised 10 January 2024
Accepted 30 January 2024

DOI:https://doi.org/10.1103/PhysRevE.109.034303

Physics Subject Headings (PhySH)

Network structure

Directed networks

Monte Carlo methods Network Models Topological tools

Networks

Authors & Affiliations

Grover E. C. Guzman ^*

Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, São Paulo - SP 05508-090, Brazil

Peter F. Stadler ^†

Bioinformatics Group, Department of Computer Science, Interdisciplinary Center for Bioinformatics, School of Excellence in Embedded Composite AI Dresden/Leipzig (SECAI), Leipzig University, Härtelstraße 16-18, D-04107 Leipzig, Germany; German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Puschstraße 4, 04103 Leipzig, Germany; Competence Center for Scalable Data Services and Solutions Dresden-Leipzig, Humboldtstraße 25, 04105 Leipzig, Germany; Leipzig University, Härtelstraße 16-18, D-04107 Leipzig, Germany; Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, D-04103 Leipzig, Germany; Institute for Theoretical Chemistry, University of Vienna, Währingerstraße 17, A-1090 Wien, Austria; Facultad de Ciencias, Universidad Nacional de Colombia, Sede Bogotá 111321, Colombia; and The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

Andre Fujita ^‡

Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, São Paulo - SP 05508-090, Brazil and Division of Network AI Statistics, Medical Institute of Bioregulation, Kyushu University, Fukuoka 812-8582, Japan

^*grover@usp.br
^†Peter.Stadler@bioinf.uni-leipzig.de
^‡andrefujita@usp.br

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Vol. 109, Iss. 3 — March 2024

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Images

Figure 1
Panel (a) shows the near neighborhood of node $i$ without removing the edge's directionality. Panel (b) shows the neighborhood of node $i$ , removing the edge's directionality. $N_{i}^{(r)}$ is the neighborhood of vertex $i$ , and $N_{j ∖ i}^{(r)}$ is the neighborhood of $N_{j}^{(r)}$ minus the neighborhood of $i$ when $r = 2$ .
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Figure 2
Spectral densities of the adjacency and Laplacian matrices of the coauthorship network. The red line corresponds to the spectral density obtained using the diagonalization method. The other colors correspond to the spectral densities obtained by our proposal. The green line corresponds to the spectral density when considering the length-two cycles ( $r = 0$ ). In other words, for $r = 0$ , our proposal becomes the same as that proposed by Rogers et al. The blue line corresponds to the spectral density when we consider cycles of length three or fewer ( $r = 1$ ). The violet line corresponds to the spectral density when we consider cycles of length four or fewer ( $r = 2$ ). Notice that as the parameter $r$ increases, the similarity with the spectral density obtained using the diagonalization method increases.
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Figure 3
Spectral densities of the adjacency and Laplacian matrices of the PGP network. Spectral densities of the adjacency and Laplacian matrices of the coauthorship network. The red line corresponds to the spectral density obtained using the diagonalization method. The other colors correspond to the spectral densities obtained by our proposal. The green line corresponds to the spectral density when considering the length-two cycles ( $r = 0$ ). In other words, for $r = 0$ , our proposal becomes the same as that proposed by Rogers et al. The blue line corresponds to the spectral density when we consider cycles of length three or fewer ( $r = 1$ ). The violet line corresponds to the spectral density when we consider cycles of length four or fewer ( $r = 2$ ). Notice that as the parameter $r$ increases, the similarity with the spectral density obtained using the diagonalization method increases.
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Figure 4
Heatmap of the difference between the exact spectral distribution $[ρ (z)]$ obtained by the diagonalization and the approximated spectral distribution $[\hat{ρ} (z)]$ obtained by our proposed method with $r = 0, 1, 2$ . Each panel shows the $L_{1}$ norm between the exact and approximated spectral distributions. We simulated a Watts-Strogatz graph comprising $45 000$ nodes, $p = 0.05$ , and mean degree 8. The edges have integer weights ( $- 1$ , 0, or 1). We used $η = 0.05$ .
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Figure 5
Heatmap of the difference between the exact spectral distribution $[ρ (z)]$ obtained by the diagonalization and the approximated spectral distribution $[\hat{ρ} (z)]$ obtained by our proposed method with $r = 0, 1, 2$ . Each panel shows the $L_{1}$ norm between the exact and approximated spectral distributions. We simulated a Watts-Strogatz graph comprising $45 000$ nodes, $p = 0.05$ , and mean degree 8. The edges have complex weights. We used $η = 0.1$ .
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Figure 6
Spectral distribution obtained by our proposed method with $r = 0, 1, 2$ of Gnutella peer-to-peer sharing network from the August 2002 graph with $62 586$ vertices and $147 878$ edges. We do not include the diagonalization method results because its execution stopped due to memory allocation problems. We used $η = 0.1$ .
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