Network data often represent multiple types of relations, which can also denote exchanged quantities, and are typically encompassed in a weighted multiplex. Such data frequently exhibit clustering structures, however, traditional clustering methods are not well-suited for multiplex networks. Additionally, standard methods treat edge weights in their raw form, potentially biasing clustering towards a node's total weight capacity rather than reflecting cluster-related interaction patterns. To address this, we propose transforming edge weights into a compositional format, enabling the analysis of connection strengths in relative terms and removing the impact of nodes' total weights. We introduce a multiplex Dirichlet stochastic block model designed for multiplex networks with compositional layers. This model accounts for sparse compositional networks and enables joint clustering across different types of interactions. We validate the model through a simulation study and apply it to the international export data from the Food and Agriculture Organization of the United Nations.
Community detection is the task of clustering objects based on their pairwise relationships. Most of the model-based community detection methods, such as the stochastic block model and its variants, are designed for networks with binary (yes/no) edges. In many practical scenarios, edges often possess continuous weights, spanning positive and negative values, which reflect varying levels of connectivity. To address this challenge, we introduce the heterogeneous block covariance model (HBCM) that defines a community structure within the covariance matrix, where edges have signed and continuous weights. Furthermore, it takes into account the heterogeneity of objects when forming connections with other objects within a community. A novel variational expectation-maximization algorithm is proposed to estimate the group membership. The HBCM provides provable consistent estimates of memberships, and its promising performance is observed in numerical simulations with different setups. The model is applied to a single-cell RNA-seq dataset of a mouse embryo and a stock price dataset. Supplementary materials for this article are available online.
This paper addresses domain adaptation challenges in graph data resulting from chronological splits. In a transductive graph learning setting, where each node is associated with a timestamp, we focus on the task of Semi-Supervised Node Classification (SSNC), aiming to classify recent nodes using labels of past nodes. Temporal dependencies in node connections create domain shifts, causing significant performance degradation when applying models trained on historical data into recent data. Given the practical relevance of this scenario, addressing domain adaptation in chronological split data is crucial, yet underexplored. We propose Imposing invariance with Message Passing in Chronological split Temporal Graphs (IMPaCT), a method that imposes invariant properties based on realistic assumptions derived from temporal graph structures. Unlike traditional domain adaptation approaches which rely on unverifiable assumptions, IMPaCT explicitly accounts for the characteristics of chronological splits. The IMPaCT is further supported by rigorous mathematical analysis, including a derivation of an upper bound of the generalization error. Experimentally, IMPaCT achieves a 3.8% performance improvement over current SOTA method on the ogbn-mag graph dataset. Additionally, we introduce the Temporal Stochastic Block Model (TSBM), which replicates temporal graphs under varying conditions, demonstrating the applicability of our methods to general spatial GNNs.
Graph embeddings have emerged as a powerful tool for understanding the structure of graphs. Unlike classical spectral methods, recent methods such as DeepWalk, Node2Vec, etc. are based on solving nonlinear optimization problems on the graph, using local information obtained by performing random walks. These techniques have empirically been shown to produce ''better'' embeddings than their classical counterparts. However, due to their reliance on solving a nonconvex optimization problem, obtaining theoretical guarantees on the properties of the solution has remained a challenge, even for simple classes of graphs. In this work, we show convergence properties for the DeepWalk algorithm on graphs obtained from the Stochastic Block Model (SBM). Despite being simplistic, the SBM has proved to be a classic model for analyzing the behavior of algorithms on large graphs. Our results mirror the existing ones for spectral embeddings on SBMs, showing that even in the case of one-dimensional embeddings, the output of the DeepWalk algorithm provably recovers the cluster structure with high probability.
Fair graph clustering is crucial for ensuring equitable representation and treatment of diverse communities in network analysis. Traditional methods often ignore disparities among social, economic, and demographic groups, perpetuating biased outcomes and reinforcing inequalities. This study introduces fair graph clustering within the framework of the disparate impact doctrine, treating it as a joint optimization problem integrating clustering quality and fairness constraints. Given the NP-hard nature of this problem, we employ a semidefinite relaxation approach to approximate the underlying optimization problem. For up to medium-sized graphs, we utilize a singular value decomposition-based algorithm, while for larger graphs, we propose a novel algorithm based on the alternative direction method of multipliers. Unlike existing methods, our formulation allows for tuning the trade-off between clustering quality and fairness. Experimental results on graphs generated from the standard stochastic block model demonstrate the superiority of our approach in achieving an optimal accuracy-fairness trade-off compared to state-of-the-art methods.
This paper studies community detection for a nonlinear opinion dynamics model from its equilibria. It is assumed that the underlying network is generated from a stochastic block model with two communities, where agents are assigned with community labels and edges are added independently based on these labels. Agents update their opinions following a nonlinear rule that incorporates saturation effects on interactions. It is shown that clustering based on a single equilibrium can detect most community labels (i.e., achieving almost exact recovery), if the two communities differ in size and link probabilities. When the two communities are identical in size and link probabilities, and the inter-community connections are denser than intra-community ones, the algorithm can achieve almost exact recovery under negative influence weights but fails under positive influence weights. Utilizing fixed point equations and spectral methods, we also propose a detection algorithm based on multiple equilibria, which can detect communities with positive influence weights. Numerical experiments demonstrate the performance of the proposed algorithms.
Detection of correlation in a pair of random graphs is a fundamental statistical and computational problem that has been extensively studied in recent years. In this work, we consider a pair of correlated (sparse) stochastic block models $\mathcal{S}(n,\tfrac{\lambda}{n};k,\epsilon;s)$ that are subsampled from a common parent stochastic block model $\mathcal S(n,\tfrac{\lambda}{n};k,\epsilon)$ with $k=O(1)$ symmetric communities, average degree $\lambda=O(1)$, divergence parameter $\epsilon$, and subsampling probability $s$. For the detection problem of distinguishing this model from a pair of independent Erd\H{o}s-R\'enyi graphs with the same edge density $\mathcal{G}(n,\tfrac{\lambda s}{n})$, we focus on tests based on \emph{low-degree polynomials} of the entries of the adjacency matrices, and we determine the threshold that separates the easy and hard regimes. More precisely, we show that this class of tests can distinguish these two models if and only if $s> \min \{ \sqrt{\alpha}, \frac{1}{\lambda \epsilon^2} \}$, where $\alpha\approx 0.338$ is the Otter's constant and $\frac{1}{\lambda \epsilon^2}$ is the Kesten-Stigum threshold. Our proof of low-degree hardness is based on a conditional variant of the low-degree likelihood calculation.
Network data are observed in various applications where the individual entities of the system interact with or are connected to each other, and often these interactions are defined by their associated strength or importance. Clustering is a common task in network analysis that involves finding groups of nodes displaying similarities in the way they interact with the rest of the network. However, most clustering methods use the strengths of connections between entities in their original form, ignoring the possible differences in the capacities of individual nodes to send or receive edges. This often leads to clustering solutions that are heavily influenced by the nodes' capacities. One way to overcome this is to analyse the strengths of connections in relative rather than absolute terms, expressing each edge weight as a proportion of the sending (or receiving) capacity of the respective node. This, however, induces additional modelling constraints that most existing clustering methods are not designed to handle. In this work we propose a stochastic block model for composition-weighted networks based on direct modelling of compositional weight vectors using a Dirichlet mixture, with the parameters determined by the cluster labels of the sender and the receiver nodes. Inference is implemented via an extension of the classification expectation-maximisation algorithm that uses a working independence assumption, expressing the complete data likelihood of each node of the network as a function of fixed cluster labels of the remaining nodes. A model selection criterion is derived to aid the choice of the number of clusters. The model is validated using simulation studies, and showcased on network data from the Erasmus exchange program and a bike sharing network for the city of London.
We investigate the classification performance of graph neural networks with graph-polynomial features, poly-GNNs, on the problem of semi-supervised node classification. We analyze poly-GNNs under a general contextual stochastic block model (CSBM) by providing a sharp characterization of the rate of separation between classes in their output node representations. A question of interest is whether this rate depends on the depth of the network $k$, i.e., whether deeper networks can achieve a faster separation? We provide a negative answer to this question: for a sufficiently large graph, a depth $k > 1$ poly-GNN exhibits the same rate of separation as a depth $k=1$ counterpart. Our analysis highlights and quantifies the impact of ``graph noise'' in deep GNNs and shows how noise in the graph structure can dominate other sources of signal in the graph, negating any benefit further aggregation provides. Our analysis also reveals subtle differences between even and odd-layered GNNs in how the feature noise propagates.
Clustering algorithms frequently require the number of clusters to be chosen in advance, but it is usually not clear how to do this. To tackle this challenge when clustering within sequential data, we present a method for estimating the number of clusters when the data is a trajectory of a Block Markov Chain. Block Markov Chains are Markov Chains that exhibit a block structure in their transition matrix. The method considers a matrix that counts the number of transitions between different states within the trajectory, and transforms this into a spectral embedding whose dimension is set via singular value thresholding. The number of clusters is subsequently estimated via density-based clustering of this spectral embedding, an approach inspired by literature on the Stochastic Block Model. By leveraging and augmenting recent results on the spectral concentration of random matrices with Markovian dependence, we show that the method is asymptotically consistent - in spite of the dependencies between the count matrix's entries, and even when the count matrix is sparse. We also present a numerical evaluation of our method, and compare it to alternatives.
Commonly used priors for Vector Autoregressions (VARs) induce shrinkage on the autoregressive coefficients. Introducing shrinkage on the error covariance matrix is sometimes done but, in the vast majority of cases, without considering the network structure of the shocks and by placing the prior on the lower Cholesky factor of the precision matrix. In this paper, we propose a prior on the VAR error precision matrix directly. Our prior, which resembles a standard spike and slab prior, models variable inclusion probabilities through a stochastic block model that clusters shocks into groups. Within groups, the probability of having relations across group members is higher (inducing less sparsity) whereas relations across groups imply a lower probability that members of each group are conditionally related. We show in simulations that our approach recovers the true network structure well. Using a US macroeconomic data set, we illustrate how our approach can be used to cluster shocks together and that this feature leads to improved density forecasts.
Graph clustering is an important unsupervised learning technique for partitioning graphs with attributes and detecting communities. However, current methods struggle to accurately capture true community structures and intra-cluster relations, be computationally efficient, and identify smaller communities. We address these challenges by integrating coarsening and modularity maximization, effectively leveraging both adjacency and node features to enhance clustering accuracy. We propose a loss function incorporating log-determinant, smoothness, and modularity components using a block majorization-minimization technique, resulting in superior clustering outcomes. The method is theoretically consistent under the Degree-Corrected Stochastic Block Model (DC-SBM), ensuring asymptotic error-free performance and complete label recovery. Our provably convergent and time-efficient algorithm seamlessly integrates with graph neural networks (GNNs) and variational graph autoencoders (VGAEs) to learn enhanced node features and deliver exceptional clustering performance. Extensive experiments on benchmark datasets demonstrate its superiority over existing state-of-the-art methods for both attributed and non-attributed graphs.
Graph homophily refers to the phenomenon that connected nodes tend to share similar characteristics. Understanding this concept and its related metrics is crucial for designing effective Graph Neural Networks (GNNs). The most widely used homophily metrics, such as edge or node homophily, quantify such "similarity" as label consistency across the graph topology. These metrics are believed to be able to reflect the performance of GNNs, especially on node-level tasks. However, many recent studies have empirically demonstrated that the performance of GNNs does not always align with homophily metrics, and how homophily influences GNNs still remains unclear and controversial. Then, a crucial question arises: What is missing in our current understanding of homophily? To figure out the missing part, in this paper, we disentangle the graph homophily into $3$ aspects: label, structural, and feature homophily, providing a more comprehensive understanding of GNN performance. To investigate their synergy, we propose a Contextual Stochastic Block Model with $3$ types of Homophily (CSBM-3H), where the topology and feature generation are controlled by the $3$ metrics. Based on the theoretical analysis of CSBM-3H, we derive a new composite metric, named Tri-Hom, that considers all $3$ aspects and overcomes the limitations of conventional homophily metrics. The theoretical conclusions and the effectiveness of Tri-Hom have been verified through synthetic experiments on CSBM-3H. In addition, we conduct experiments on $31$ real-world benchmark datasets and calculate the correlations between homophily metrics and model performance. Tri-Hom has significantly higher correlation values than $17$ existing metrics that only focus on a single homophily aspect, demonstrating its superiority and the importance of homophily synergy. Our code is available at \url{https://github.com/zylMozart/Disentangle_GraphHom}.
In signal processing, exploring complex systems through network representations has become an area of growing interest. This study introduces the modularity graph, a new graph-based feature, to highlight the relationship across the graph communities. After showing an application to the random graph class known as Stochastic Block Model, we consider the brain functional connectivity network estimated from real EEG data. The modularity graph provides a quantitative framework for examining the interactions between neuron clusters within the brain's network. The modularity graph works alongside multiscale community detection algorithms, thereby enabling the identification of community structures at various scales. After introducing the modularity graph, we apply it to the brain functional connectivity network, estimated from publicly available EEG recordings of motor imagery experiments. Statistical analysis across multiple scales shows that the modularity graph differs for the distinct brain connectivity states associated with various motor imagery tasks. This work emphasizes the application of signal on graph processing techniques to understand brain behavior during specific cognitive tasks, leveraging the novel modularity graph to identify patterns of brain connectivity in different cognitive conditions. This approach sets the stage for further signal on graph analysis to devise brain network modularity, and to gain insights into the motor imagery mechanisms.
We study the problem of exact community recovery in general, two-community block models, in the presence of node-attributed $side$ $information$. We allow for a very general side information channel for node attributes, and for pairwise (edge) observations, consider both Bernoulli and Gaussian matrix models, capturing the Stochastic Block Model, Submatrix Localization, and $\mathbb{Z}_2$-Synchronization as special cases. A recent work of Dreveton et al. 2024 characterized the information-theoretic limit of a very general exact recovery problem with side information. In this paper, we show algorithmic achievability in the above important cases by designing a simple but optimal spectral algorithm that incorporates side information (when present) along with the eigenvectors of the pairwise observation matrix. Using the powerful tool of entrywise eigenvector analysis of Abbe et al. 2020, we show that our spectral algorithm can mimic the so called $genie$-$aided$ $estimators$, where the $i^{\mathrm{th}}$ genie-aided estimator optimally computes the estimate of the $i^{\mathrm{th}}$ label, when all remaining labels are revealed by a genie. This perspective provides a unified understanding of the optimality of spectral algorithms for various exact recovery problems in a recent line of work.
Widespread default involves substantial deadweight costs which could be countered by injecting capital into failing firms. Injections have positive spillovers that can trigger a repayment cascade. But which firms should a regulator bailout so as to minimize the total injection of capital while ensuring solvency of all firms? While the problem is, in general, NP-hard, for a wide range of networks that arise from a stochastic block model, we show that the optimal bailout can be implemented by a simple policy that targets firms based on their characteristics and position in the network. Specific examples of the setting include core-periphery networks.
We consider a recently proposed approach to graph signal processing (GSP) based on graphons. We show how the graphon-based approach to GSP applies to graphs sampled from a stochastic block model derived from a weighted Cayley graph. When SBM block sizes are equal, a nice Fourier basis can be derived from the representation theory of the underlying group. We explore how the SBM Fourier basis is affected when block sizes are not uniform. When block sizes are nearly uniform, we demonstrate that the group Fourier basis closely approximates the SBM Fourier basis. More specifically, we quantify the approximation error using matrix perturbation theory. When block sizes are highly non-uniform, the group-based Fourier basis can no longer be used. However, we show that partial information regarding the SBM Fourier basis can still be obtained from the underlying group.
We consider the two-sample testing problem for networks, where the goal is to determine whether two sets of networks originated from the same stochastic model. Assuming no vertex correspondence and allowing for different numbers of nodes, we address a fundamental network testing problem that goes beyond simple adjacency matrix comparisons. We adopt the stochastic block model (SBM) for network distributions, due to their interpretability and the potential to approximate more general models. The lack of meaningful node labels and vertex correspondence translate to a graph matching challenge when developing a test for SBMs. We introduce an efficient algorithm to match estimated network parameters, allowing us to properly combine and contrast information within and across samples, leading to a powerful test. We show that the matching algorithm, and the overall test are consistent, under mild conditions on the sparsity of the networks and the sample sizes, and derive a chi-squared asymptotic null distribution for the test. Through a mixture of theoretical insights and empirical validations, including experiments with both synthetic and real-world data, this study advances robust statistical inference for complex network data.
We investigate how to select the number of communities for weighted networks without a full likelihood modeling. First, we propose a novel weighted degree-corrected stochastic block model (DCSBM), where the mean adjacency matrix is modeled in the same way as in the standard DCSBM, while the variance profile matrix is assumed to be related to the mean adjacency matrix through a given variance function. Our method of selecting the number of communities is based on a sequential testing framework. In each step, the weighted DCSBM is fitted via some spectral clustering method. A key component of our method is matrix scaling on the estimated variance profile matrix. The resulting scaling factors can be used to normalize the adjacency matrix, from which the test statistic is then obtained. Under mild conditions on the weighted DCSBM, our proposed procedure is shown to be consistent in estimating the true number of communities. Numerical experiments on both simulated and real-world network data demonstrate the desirable empirical properties of our method.
We consider the semi-random graph model of [Makarychev, Makarychev and Vijayaraghavan, STOC'12], where, given a random bipartite graph with $\alpha$ edges and an unknown bipartition $(A, B)$ of the vertex set, an adversary can add arbitrary edges inside each community and remove arbitrary edges from the cut $(A, B)$ (i.e. all adversarial changes are \textit{monotone} with respect to the bipartition). For this model, a polynomial time algorithm is known to approximate the Balanced Cut problem up to value $O(\alpha)$ [MMV'12] as long as the cut $(A, B)$ has size $\Omega(\alpha)$. However, it consists of slow subroutines requiring optimal solutions for logarithmically many semidefinite programs. We study the fine-grained complexity of the problem and present the first near-linear time algorithm that achieves similar performances to that of [MMV'12]. Our algorithm runs in time $O(|V(G)|^{1+o(1)} + |E(G)|^{1+o(1)})$ and finds a balanced cut of value $O(\alpha)$. Our approach appears easily extendible to related problem, such as Sparsest Cut, and also yields an near-linear time $O(1)$-approximation to Dagupta's objective function for hierarchical clustering [Dasgupta, STOC'16] for the semi-random hierarchical stochastic block model inputs of [Cohen-Addad, Kanade, Mallmann-Trenn, Mathieu, JACM'19].