Network community detection via neural embeddings

TL;DR

It is shown that node2vec—shallow, linear neural network—encodes communities into separable clusters better than random partitioning down to the information-theoretic detectability limit for the stochastic block models.

Abstract

Recent advances in machine learning research have produced powerful neural graph embedding methods, which learn useful, low-dimensional vector representations of network data. These neural methods for graph embedding excel in graph machine learning tasks and are now widely adopted. However, how and why these methods work -- particularly how network structure gets encoded in the embedding -- remain largely unexplained. Here, we show that node2vec -- shallow, linear neural network -- encodes communities into separable clusters better than random partitioning down to the information-theoretic detectability limit for the stochastic block models. We show that this is due to the equivalence between the embedding learned by node2vec and the spectral embedding via the eigenvectors of the symmetric normalized Laplacian matrix. Numerical simulations demonstrate that node2vec is capable of learning communities on sparse graphs generated by the stochastic blockmodel, as well as on sparse degree-heterogeneous networks. Our results highlight the features of graph neural networks that enable them to separate communities in embedding space.

Figures (4)

• Figure 1: Performance of community detection methods for networks generated by the PPM as a function of the mixing parameter $\mu$. We generated networks with $n = 10^5$ nodes, different edge sparsity ($\langle k \rangle=5$ in A and D, $\langle k \rangle=10$ in B and E, $\langle k \rangle=50$ in C and F), and the different number of communities ($q=2$ for A--C and $q=50$ for D--F). The dashed vertical line indicates the theoretical detectability limit $\mu^*$ given by \ref{['eq:detectability_limit']}: communities are detectable (i.e., $S> 0$), in principle, below $\mu^*$. Spectral embedding methods detect communities up to the theoretical limit for dense networks (C and F), supporting the detectability limit derived by previous studies nadakuditiGraphSpectraDetectability2012RadicchiDetectabilityHeterogeneousNetworks2013. However, for sparse networks, they fall short even at low $\mu$-values (A and D). node2vec and the spectral embedding based on the non-backtracking matrix outperform other spectral methods, with the performance curves close to that of the BP algorithm. Note that even the BP algorithm falls short of the exact recovery of some easily-detectable communities in the case of $q=50$ communities, with the initial parameters set according to the ground-truth communities. The error bands represent the 90% confidence interval by a bootstrapping with $10^4$ resample.
• Figure 2: Performance of community detection methods on the LFR benchmark networks, as a function of the mixing parameter $\mu$. We generated networks with $n = 10^4$ nodes with different edge sparsity ($\langle k \rangle=5$ in A and D, $\langle k \rangle=10$ in B and E, $\langle k \rangle=50$ in C and F). The degree exponent $\tau_1=2.1$ in A, B, and C, and $\tau_1=3$ in D, E, and F. node2vec consistently performs well across different sparsity regimes for most $\mu$-values, with a larger margin for sparser networks. The BP algorithm, which is provably optimal for networks generated by the PPM, fails to identify some easily-detectable communities, even with the initial parameters set according to the ground-truth communities. The error bands represent the 90% confidence interval by a bootstrapping with $10^4$ resample.
• Figure 3: Performance of community detection methods on empirical networks. Each panel illustrates the distribution of element-centric similarities for the community detection and graph embedding methods. Each circle denotes the outcome of a single run. The boxes indicate the quartiles of this distribution. The whiskers extend to the farthest data point within 1.5 times the interquartile range from the nearest hinge.
• Figure 4: Graph kernel $\phi(\lambda_i; T)$ of node2vec matrix $\mathbf{\hat{R}}^{\text{n2v}}$ across different $T$ values. The function $\phi(\lambda_i)$ is non-negative and monotonically decreasing for $0 < \lambda_i \leq 1$ and $\phi(\lambda_i) \leq 0$ for $1 < \lambda_i \leq 2$.