Detectability of communities in heterogeneous networks

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Detectability of communities in heterogeneous networks

Filippo Radicchi

Phys. Rev. E 88, 010801(R) – Published 12 July 2013

Abstract

Communities are fundamental entities for the characterization of the structure of real networks. The standard approach to the identification of communities in networks is based on the optimization of a quality function known as modularity. Although modularity has been at the center of an intense research activity and many methods for its maximization have been proposed, not much is yet known about the necessary conditions that communities need to satisfy in order to be detectable with modularity maximization methods. Here, we develop a simple theory to establish these conditions, and we successfully apply it to various classes of network models. Our main result is that heterogeneity in the degree distribution helps modularity to correctly recover the community structure of a network and that, in the realistic case of scale-free networks with degree exponent $γ < 2.5$ , modularity is always able to detect the presence of communities.

Received 6 May 2013

DOI:https://doi.org/10.1103/PhysRevE.88.010801

Authors & Affiliations

Filippo Radicchi^*

Departament d'Enginyeria Quimica, Universitat Rovira i Virgili, 43007 Tarragona, Spain

^*f.radicchi@gmail.com

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Vol. 88, Iss. 1 — July 2013

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Figure 1
(a) Components of the principal eigenvector of the modularity matrix as functions of the difference between internal and external degrees for a system composed of two Poissonian networks of size $N = 1024$ , with average internal degree $c_{in}$ and average external degree $c_{out}$ such that $c_{in} + c_{out} = 64$ . We plot here only the components of one of the two modules, but analogous results (with opposite sign) are valid for the other module. Different colors and symbols correspond to different combinations of $c_{in}$ and $c_{out}$ . (b) Correlation coefficient between the eigenvector components and the difference between internal and external degrees as a function of $c_{in} - c_{out}$ . While the correlation coefficient is not significant in the region in which communities are not detectable [i.e., $c_{in} - c_{out} < \sqrt{c_{in} + c_{out}}$ , see Eq. (12)], it becomes significant in the detectability regime (i.e., $c_{in} - c_{out} \geq \sqrt{c_{in} + c_{out}}$ ).Reuse & Permissions
Figure 2
Numerical estimation of the largest eigenpair of the modularity matrix $Q$ in the case of two random regular graphs [(a) and (b)] and in the case of two Poissonian graphs [(c) and (d)]. In both cases, we consider three different network sizes $N = 256$ (blue squares), $N = 512$ (red circles) and $N = 1024$ (green triangles), and we fix $c_{in} + c_{out} = 64$ . The results presented here have been obtained by averaging over 100 different realizations of the models. (a), (c) Largest eigenvalue of $Q$ as a function of $c_{in} - c_{out}$ . The full black line corresponds to $c_{in} - c_{out}$ in (a), and to Eq. (11) in (c). The dashed black lines are given by $2 \sqrt{c_{in} + c_{out}}$ . (b), (d) Size-independent inner products $〈 v_{1} | 1 〉$ and $〈 v_{2} | 1 〉$ as functions of $c_{in} - c_{out}$ .Reuse & Permissions
Figure 3
Numerical estimation of the largest eigenpair of the modularity matrix $Q$ in the case of two random scale-free graphs with degree exponents $γ = 2.2$ [(a) and (b)] and $γ = 2.8$ [(c) and (d)]. In both cases, we consider several network sizes ranging from $N = 256$ to 8192, and we set $c_{in} + c_{out} = 64$ . The results presented here have been obtained by averaging over 100 different realizations of the models. To suppress fluctuations, we restricted to the case of networks for which the square of the largest strength is smaller than the sum of all strengths (i.e., $s_{\max}^{2} < \sum_{i} s_{i}$ ) [23, 26], although the qualitative outcome does not depend on this choice. For clarity, we placed arrows in the various panels to indicate the direction of increasing $N$ . (a), (c) Largest eigenvalue $λ_{\max}$ multiplied by $q_{1} / q_{2}$ as a function of $c_{in} - c_{out}$ . The full black line corresponds to $c_{in} - c_{out}$ . (b), (d) Size-independent inner products $〈 v_{1} | 1 〉$ and $〈 v_{2} | 1 〉$ as functions of $c_{in} - c_{out}$ .Reuse & Permissions

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