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Spectra of networks containing short loops

M. E. J. Newman
Phys. Rev. E 100, 012314 – Published 26 July 2019
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  • INTRODUCTION
  • NETWORKS CONTAINING SHORT LOOPS
  • MESSAGE PASSING
  • EXAMPLES
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    The spectrum of the adjacency matrix plays several important roles in the mathematical theory of networks and network data analysis, for example in percolation theory, community detection, centrality measures, and the theory of dynamical systems on networks. A number of methods have been developed for the analytic computation of network spectra, but they typically assume that networks are locally treelike, meaning that the local neighborhood of any node takes the form of a tree, free of short loops. Empirically observed networks, by contrast, often have many short loops. Here we develop an approach for calculating the spectra of networks with short loops using a message passing method. We give example applications to some previously studied classes of networks and find that the presence of loops induces substantial qualitative changes in the shape of network spectra, generating asymmetries, multiple spectral bands, and other features.

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    • Received 28 February 2019

    DOI:https://doi.org/10.1103/PhysRevE.100.012314

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    CyclesNetwork structure
    1. Physical Systems
    Random graphs
    1. Techniques
    Network Models
    Networks

    Authors & Affiliations

    M. E. J. Newman

    • Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA

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    Vol. 100, Iss. 1 — July 2019

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    Images

    • Figure 1
      Figure 1

      Top: A network built from a collection of motifs. In this case there are three motifs: Single edges, triangles, and loops of length four. The shading is intended only to highlight the motifs; the network itself consists of edges between nodes alone. Bottom: The same network represented as a factor graph, a bipartite network with two sets of nodes. One set, the filled circles, represents the nodes of the original network. The other, the open squares, represents the motifs.

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    • Figure 2
      Figure 2

      A complete excursion from a starting node (bottom, circled) consists of a closed walk within a single motif, here having four steps, plus, optionally, any number of subexcursions along the way that leave the motif and return to it some time later via the same node. Each subexcursion is itself of the same form, which allows us to write a self-consistent expression (10) for the total number of excursions.

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    • Figure 3
      Figure 3

      The spectrum of a random regular network in which every node belongs to exactly one single edge and one triangle. The histogram shows the distribution of eigenvalues calculated by numerical diagonalization of the adjacency matrix of one random realization of the network with 10002 nodes. The solid curves show the analytic solution, with the two δ-function peaks added by hand at their calculated positions.

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    • Figure 4
      Figure 4

      (a) The spectrum of a random network composed of single edges and triangles, with a Poisson distribution for both: The number of both edges and triangles is Poisson distributed at every node with mean 2. (b) The spectrum of a configuration model with the same node degrees as the network in (a). In each panel the solid curve shows the spectral density for a single randomly generated network with 10000 nodes, calculated by direct iteration of the message passing equations with Lorentzian broadening parameter η=0.01. The histogram shows the spectrum of the same network calculated by numerical diagonalization of the adjacency matrix.

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