Spectra of random graphs with arbitrary expected degrees

Raj Rao Nadakuditi and M. E. J. Newman

Phys. Rev. E 87, 012803 – Published 10 January 2013

Abstract

We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit of large network size and large vertex degrees. We also study the effect on the spectra of hubs in the network, vertices of unusually high degree, and show that these produce isolated eigenvalues outside the main spectral band, akin to impurity states in condensed matter systems, with accompanying eigenvectors that are strongly localized around the hubs. We give numerical results that confirm our analytic expressions.

Received 13 August 2012

DOI:https://doi.org/10.1103/PhysRevE.87.012803

Authors & Affiliations

Raj Rao Nadakuditi¹ and M. E. J. Newman²

¹Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109, USA
²Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA

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Vol. 87, Iss. 1 — January 2013

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Images

Figure 1
The spectral density $ρ (z)$ for the degree distribution described in the text, in which vertices have expected degree $d_{1} = 50$ with probability $p_{1} = \frac{1}{4}$ and $d_{2} = 100$ with probability $p_{2} = \frac{3}{4}$ . The curve gives the analytic solution, derived from Eqs. (23) and (28); the histogram shows the results of numerical calculations for actual networks with $n = 10 000$ vertices, averaged over 100 different networks.Reuse & Permissions
Figure 2
The solutions $λ_{i}$ to Eq. (63) correspond to the points where the left-hand side of the equation (solid curves) equals 1 (dashed horizontal line). This implies that the values of the $λ_{i}$ are interleaved between the eigenvalues $β_{i}$ of the modularity matrix.Reuse & Permissions
Figure 3
Graphical solution of Eq. (81). The solid curves represent the value of $h (z)$ as a function of $z$ , above and below the spectral band, and the hub eigenvalues, which are solutions of Eq. (81), fall at the points where this curve intersects the straight line $z / k_{n}$ , represented by the dashed diagonal line. The slope of this line is $1 / k_{n}$ , and hence when $k_{n}$ is large enough, the lines intersect—case (a)—and we have two hub eigenvalues, one above and one below the band (marked by dots). Case (b) is the borderline case. If $k_{n}$ is any less than this value, then there is no intersection and the highest and lowest eigenvalues will be those at the band edges.Reuse & Permissions
Figure 4
The largest eigenvalue of the modularity matrix for a Poisson random graph of mean degree $c = 100$ plus a single additional hub of expected degree $k_{n}$ . Points are numerical results, averaged over 1000 networks of $n = 10 000$ vertices each. Statistical errors on the measurements are smaller than the points in all cases. The solid curve is Eq. (82), which gives $z = k_{n} / \sqrt{k_{n} - c}$ in this case, and the horizontal dashed line represents the value $z = 2 \sqrt{c} = 20$ , which is the lower limit on the eigenvalue set by the edge of the continuous spectral band. The vertical dashed line represents the critical value $k_{n} = 200$ of the hub degree, set by Eq. (84).Reuse & Permissions
Figure 5
Values of elements of the leading eigenvector of the modularity matrix for a Poisson random graph with $n = 10 000$ vertices and mean degree $c = 100$ , with a single added hub of degree $k_{n}$ . Main figure: value of the vector element corresponding to the hub itself. Inset: average value of the elements corresponding to the hub's immediate network neighbors. Points are numerical results, averaged over 100 networks each; curves are the analytic prediction, Eq. (90). Statistical errors are smaller than the data points in all cases.Reuse & Permissions

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