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Title: Supervised Community Detection with Line Graph Neural Networks

Paper content:
Community detection in graphs can be solved via spectral methods or posterior inference under certain probabilistic graphical models.
Focusing on random graph families such as the stochastic block model, recent research has unified both approaches and identified both statistical and computational detection thresholds in terms of the signal-to-noise ratio.
By recasting community detection as a node-wise classification problem on graphs, we can also study it from a learning perspective.
We present a novel family of Graph Neural Networks (GNNs) for solving community detection problems in a supervised learning setting.
We show that, in a data-driven manner and without access to the underlying generative models, they can match or even surpass the performance of the belief propagation algorithm on binary and multiclass stochastic block models, which is believed to reach the computational threshold in these cases.
In particular, we propose to augment GNNs with the non-backtracking operator defined on the line graph of edge adjacencies.
The GNNs are achieved good performance on real-world datasets.
In addition, we perform the first analysis of the optimization landscape of using (linear) GNNs to solve community detection problems, demonstrating that under certain simplifications and assumptions, the loss value at any local minimum is close to the loss value at the global minimum/minima.
Graph inference problems encompass a large class of tasks and domains, from posterior inference in probabilistic graphical models to community detection and ranking in generic networks, image segmentation or compressed sensing on non-Euclidean domains.
They are motivated both by practical applications, such as in the case of PageRank, and also by fundamental questions on the algorithmic hardness of solving such tasks.From a data-driven perspective, these problems can be formulated in unsupervised, semi-supervised or supervised learning settings.
In the supervised case, one assumes a dataset of graphs with labels on their nodes, edges or the entire graphs, and attempts to perform node-wise, edge-wise and graph-wise classification by optimizing a loss over a certain parametric class, e.g. neural networks.
Graph Neural Networks (GNNs) are natural extensions of Convolutional Neural Networks to graph-structured data, and have emerged as a powerful class of algorithms to perform complex graph inference leveraging labeled data (Gori et al., 2005; BID3 (and references therein).
In essence, these neural networks learn cascaded linear combinations of intrinsic graph operators interleaved with node-wise (or edge-wise) activation functions.
Since they utilize intrinsic graph operators, they can be applied to varying input graphs, and they offer the same parameter sharing advantages as their CNN counterparts.In this work, we focus on community detection problems, a wide class of node classification tasks that attempt to discover a clustered, segmented structure within a graph.
The algorithmic approaches to this problem include a rich class of spectral methods, which take advantage of the spectrum of certain operators defined on the graph, as well as approximate message-passing methods such as belief propagation (BP), which performs approximate posterior inference under predefined graphical models (Decelle et al., 2011) .
Focusing on the supervised setting, we study the ability of GNNs to approximate, generalize or even improve upon these class of algorithms.
Our motivation is two-fold.
On the one hand, this problem exhibits algorithmic hardness on some settings, opening up the possibility to discover more efficient algorithms than the current ones.
On the other hand, many practical scenarios fall beyond pre-specified probabilistic models, requiring data-driven solutions.We propose modifications to the GNN architecture, which allow it to exploit edge adjacency information, by incorporating the non-backtracking operator of the graph.
This operator is defined over the edges of the graph and allows a directed flow of information even when the original graph is undirected.
It was introduced to community detection problems by Krzakala et al. (2013) , who propose a spectral method based on the non-backtracking operator.
We refer to the resulting GNN model as a Line Graph Neural Network (LGNN).
Focusing on important random graph families exhibiting community structure, such as the stochastic block model (SBM) and the geometric block model (GBM), we demonstrate improvements in the performance by our GNN and LGNN models compared to other methods, including BP, even in regimes within the so-called computational-to-statistical gap.
A perhaps surprising aspect is that these gains can be obtained even with linear LGNNs, which become parametric versions of power iteration algorithms.We want to mention that besides community detection tasks, GNN and LGNN can be applied to other node-wise classification problems too.
The reason we are focusing on community detection problems is that this is a relatively well-studied setup, for which different algorithms have been proposed and where computational and statistical thresholds have been studied in several scenarios.
Moreover, synthetic datasets can be easily generated for community detection tasks.
Therefore, we think it is a nice setup for comparing different algorithms, besides its practical values.The good performances of GNN and LGNN motivate our second main contribution: the analysis of the optimization landscape of simplified and linear GNN models when trained with planted solutions of a given graph distribution.
Under reparametrization, we provide an upper bound on the energy gap controlling the energy difference between local and global minima (or minimum).
With some assumptions on the spectral concentration of certain random matrices, this energy gap will shrink as the size of the input graphs increases, which would mean that the optimization landscape is benign on large enough graphs.
In this work, we have studied data-driven approaches to supervised community detection with graph neural networks.
Our models achieve comparable performance to BP in binary SBM for various SNRs, and outperform BP in the sparse regime of 5-class SBM that falls between the computationalto-statistical gap.
This is made possible by considering a family of graph operators including the power graph adjacency matrices, and importantly by introducing the line graph equipped with the non-backtracking matrix.
We also provided a theoretical analysis of the optimization landscapes of simplified linear GNN for community detection and showed the gap between the loss value at local and global minima are bounded by quantities related to the concentration of certain random matricies.One word of caution is that our empirical results are inherently non-asymptotic.
Whereas models trained for given graph sizes can be used for inference on arbitrarily sized graphs (owing to the parameter sharing of GNNs), further work is needed in order to understand the generalization properties as |V | increases.
Nevertheless, we believe our work opens up interesting questions, namely better understanding how our results on the energy landscape depend upon specific signal-to-noise ratios, or whether the network parameters can be interpreted mathematically.
This could be useful in the study of computational-to-statistical gaps, where our model can be used to inquire about the form of computationally tractable approximations.
Another current limitation of our model is that it presumes a fixed number of communities to be detected.
Other directions of future research include the extension to the case where the number of communities is unknown and varied, or even increasing with |V |, as well as applications to ranking and edge-cut problems.
A PROOF OF THEOREM 5.1For simplicity and with an abuse of notation, in the remaining part we redefine L andL in the following way, to be the negative of their original definition in the main section: DISPLAYFORM0 .
Thus, minimizing the loss function (5) is equivalent to maximizing the function L n (β) redefined here.We write the Cholesky decomposition of EX n as EX n = R n R T n , and define DISPLAYFORM1 n ) T , and ∆B n = B n − I n .
Given a symmetric matrix K ∈ R M ×M , we let λ 1 (K), λ 2 (K), ..., λ M (K) denote the eigenvalues of K in nondecreasing order.

assistant: text: We propose a novel graph neural network architecture based on the non-backtracking matrix defined over the edge adjacencies and demonstrate its effectiveness in community detection tasks on graphs.