Graph energy as a measure of community detectability in networks

TL;DR

It is shown that the difference in graph energy between a PPM and an Erd\H{o}s--R\'enyi (ER) network has a distinct transition at the detectability threshold even for the adjacency matrices of the underlying networks.

Abstract

A key challenge in network science is the detection of communities, which are sets of nodes in a network that are densely connected internally but sparsely connected to the rest of the network. A fundamental result in community detection is the existence of a nontrivial threshold for community detectability on sparse graphs that are generated by the planted partition model (PPM). Below this so-called ``detectability limit'', no community-detection method can perform better than random chance. Spectral methods for community detection fail before this detectability limit because the eigenvalues corresponding to the eigenvectors that are relevant for community detection can be absorbed by the bulk of the spectrum. One can bypass the detectability problem by using special matrices, like the non-backtracking matrix, but this requires one to consider higher-dimensional matrices. In this paper, we show that the difference in graph energy between a PPM and an Erdős--Rényi (ER) network has a distinct transition at the detectability threshold even for the adjacency matrices of the underlying networks. The graph energy is based on the full spectrum of an adjacency matrix, so our result suggests that standard graph matrices still allow one to separate the parameter regions with detectable and undetectable communities.

Figures (3)

• Figure 1: Structural and spectral properties of networks that we generate using the planted-partition model (PPM) for different inter-community degrees. In (a)--(c), we show networks with $N = 1000$ nodes, mean degree $k = 50$, and two equal-sized communities (orange and blue) for different values of the inter-community degree parameter $k_{\rm a b}$. We consider (a) high inter-community connectivity ($k_{\rm ab} = k$), (b) moderate inter-community connectivity ($k_{\rm ab} = k/2$), and (c) no inter-community edges ($k_{\rm ab} = 0$). In (d)--(f), we show the means of the corresponding eigenvalue distributions of the networks' adjacency matrices of 100 realizations of the PPMs. The dash-dotted black and dashed blue curves, respectively, indicate the Wigner approximations for ER networks with mean degree $k = 50$ and $N = 1000$ nodes and for corresponding PPM networks with two equal-sized communities and the same mean degree and network size. As we decrease $k_{\rm ab}$, the second-largest eigenvalue (blue) increasingly separates from the bulk of the spectrum and moves towards the largest eigenvalue (red), indicating stronger community structure. (To help visibility, we rescale the sizes of the bars for the largest and second-largest eigenvalues.) For $k_{\rm ab} = 0$, a PPM network consists of two disjoint communities that have the same largest eigenvalue on average. The graph energy decreases as we decrease $k_{\rm ab}$. That is, $E(G; k_{\rm ab} = k) > E(G; k_{\rm ab} = k/2) > E(G; k_{\rm ab} = 0)$.
• Figure 2: The second-largest eigenvalue $\lambda_2$ of the adjacency matrix as a function of the difference $k_{\mathrm{aa}} - k_{\mathrm{ab}}$ between intra-community and inter-community degree parameters. Each panel has results for a different mean degree $k \in \{5, 10, 20, 50\}$ in networks with $N\in\{10^3,10^4,10^5\}$ nodes. The gray markers give our theoretical predictions, and the colored markers give our simulation results. The theoretical predictions use the assumption that the second-largest eigenvalue equals $2 \sqrt{k}$ at and below the theoretical detectability threshold. This assumption breaks down for sparse networks. The dashed vertical line indicates the theoretical detectability threshold, and the dashed horizontal line indicates the value of $\lambda_2$ below the detectability threshold. Both values equal $2\sqrt{k}$. The dash-dotted vertical lines in panels (a) and (b) mark the effective detectability threshold; below this threshold, $\lambda_2$ is absorbed by the bulk of the spectrum. The error bars are smaller than the markers.
• Figure 3: Graph-energy difference $\Delta E(G;k_{\rm ab}) \coloneqq E(G;k_{\rm ab}) - E(G;k_{\rm ab} = k)$ between the PPM graph energy $E(G;k_{\rm ab})$ and the ER graph energy $E(G;k_{\rm ab} = k)$ for graphs with the same size $N$ and same mean degree $k$. Each panel has results for a different mean degree $k \in \{5, 10, 20, 50\}$ in graphs with $N\in\{500,1000\}$ nodes. The gray markers give our theoretical predictions, and the colored curves give our simulation results. The dashed gray lines indicate the theoretical detectability threshold $2\sqrt{k}$, and the solid black curves show the shifted negative second-largest eigenvalue, which we compute for PPMs with $N = 10^5$ nodes. Specifically, each black curve is a plot of $-(\lambda_2(k_{\rm aa} - k_{\rm ab}) - \lambda_2(0))$, which equals $0$ when $k_{\rm aa} - k_{\rm ab} = 0$. The dash-dotted vertical lines in panels (a) and (b) mark the effective detectability threshold; below this threshold, $\lambda_2$ is absorbed by the bulk of the spectrum. Observe that $\Delta E(G;k_{\rm ab}) \approx 0$ below the detectability threshold and that it decreases as community structure becomes more pronounced (i.e., as we increase $k_{\mathrm{aa}} - k_{\mathrm{ab}}$). The error bars are smaller than the line widths.