Spectra of random networks with arbitrary degrees

M. E. J. Newman, Xiao Zhang, and Raj Rao Nadakuditi

Phys. Rev. E 99, 042309 – Published 18 April 2019

Abstract

We derive a message-passing method for computing the spectra of locally treelike networks and an approximation to it that allows us to compute closed-form expressions or fast numerical approximates for the spectral density of random graphs with arbitrary node degrees—the so-called configuration model. We find that the latter approximation works well for all but the sparsest of networks. We also derive bounds on the position of the band edges of the spectrum, which are important for identifying structural phase transitions in networks.

Received 17 January 2019

DOI:https://doi.org/10.1103/PhysRevE.99.042309

Physics Subject Headings (PhySH)

Degree distributions

Network Models

Networks

Authors & Affiliations

M. E. J. Newman^1,2, Xiao Zhang¹, and Raj Rao Nadakuditi³

¹Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
²Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA
³Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 99, Iss. 4 — April 2019

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
The values of the messages $h^{u v} (z)$ at $z = 3 + i ε$ with $ε = 0.01$ , plotted in the complex plane, for a configuration model network with nodes of two different degrees: half the nodes have degree 5 and half have degree 10. The two clouds of points correspond to nodes $v$ of the two different degrees as indicated. The plus signs mark the mean or centroid of each cloud. Our approximation consists of replacing the value of each message with the centroid for the cloud to which it belongs.
Reuse & Permissions
Figure 2
The spectral density of a configuration model network in which nodes are randomly assigned one of two different degrees, 5 and 10, with equal probability. The histogram shows the result of a direct numerical diagonalization of the adjacency matrix of a single such network with $10 000$ nodes, while the solid curve (green) shows the spectral density for the same single network computed using the message-passing approach of Eq. (13) with $z = x + i ε$ and $ε = 0.01$ . The dashed curve (blue) shows results from the degree-based approximation proposed in this paper. The two curves are quite difficult to distinguish because they coincide very closely. The dashed vertical lines on the right-hand side of the figure denote the upper and lower bounds on the position of the band edge derived in Sec. 3b.
Reuse & Permissions
Figure 3
Graphical representation of the solution of Eq. (32). The curves (in blue) represent the right side of the equation, while the three diagonal lines (green) represent three possible values of the left side. Where the curve and line cross, marked by the dots, are the real solutions of the equation. If the diagonal line intersects the leftmost curve segment, then all solutions are real. If it does not, there will be two complex solutions, which give rise to a nonzero spectral density.
Reuse & Permissions
Figure 4
Each panel shows the spectrum of a configuration model with one particular choice of degree distribution. In each case, the histogram shows the spectrum of a single realization of the model with $10 000$ nodes, the solid (green) curve shows the spectrum of the same single realization computed using the message-passing method of Sec. 2a with $ε = 0.01$ , while the dashed (blue) curve shows the degree-based approximation of Sec. 3, also with $ε = 0.01$ . The four degree distributions are (a) a Poisson degree distribution with mean 5; (b) a power-law degree distribution generated using the model of Barabási and Albert [31] with parameter $m = 3$ ; (c) nodes of degree 2, 3, and 4, with equal probability; and (d) nodes of degree 1, 2, and 3 with equal probability. For (a) and (b) we use the degree distribution of the actual network in our degree-based approximation, not the formal distribution from which the degrees were drawn.
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics