The Bethe-Hessian matrix, introduced by Saade, Krzakala, and Zdeborov\'a (2014), is a Hermitian matrix designed for applying spectral clustering algorithms to sparse networks. Rather than employing a non-symmetric and high-dimensional non-backtracking operator, a spectral method based on the Bethe-Hessian matrix is conjectured to also reach the Kesten-Stigum detection threshold in the sparse stochastic block model (SBM). We provide the first rigorous analysis of the Bethe-Hessian spectral method in the SBM under both the bounded expected degree and the growing degree regimes. Specifically, we demonstrate that: (i) When the expected degree $d\geq 2$, the number of negative outliers of the Bethe-Hessian matrix can consistently estimate the number of blocks above the Kesten-Stigum threshold, thus confirming a conjecture from Saade, Krzakala, and Zdeborov\'a (2014) for $d\geq 2$. (ii) For sufficiently large $d$, its eigenvectors can be used to achieve weak recovery. (iii) As $d\to\infty$, we establish the concentration of the locations of its negative outlier eigenvalues, and weak consistency can be achieved via a spectral method based on the Bethe-Hessian matrix.
Community detection is a significant challenge in analysing real graphs, including social, citation, biological, and many other networks. This article proposes three new, general, hierarchical frameworks to deal with this task. The introduced approach supports various linkage-based clustering algorithms, vertex proximity matrices, and graph representation learning models. We compare 364 module combinations on Stochastic Block Model graphs and real-life datasets. We observe that our best pipelines (Wasserman-Faust and the mutual information-based PPMI proximity, as well as the deep learning-based DNGR representations) perform competitively with the state-of-the-art Leiden and Louvain algorithms. At the same time, unlike the latter, they remain hierarchical. Thus, they output a series of nested partitions of all possible cardinalities which are compatible with each other. This feature is crucial when the number of correct partitions is unknown in advance, and we show that it is obtained at near-zero practical cost in terms of partition quality. On graphs composed of many small communities, our pipelines are more accurate than the reference solutions. Nevertheless, the downside of our approach is its time complexity, which confines the framework to networks of some 10000 vertices on personal machines.
Knowledge graphs (KGs) are useful for analyzing social structures, community dynamics, institutional memberships, and other complex relationships across domains from sociology to public health. While recent advances in large language models (LLMs) have improved the scalability and accessibility of automated KG extraction from large text corpora, the impacts of extraction errors on downstream analyses are poorly understood, especially for applied scientists who depend on accurate KGs for real-world insights. To address this gap, we conducted the first evaluation of KG extraction performance at two levels: (1) micro-level edge accuracy, which is consistent with standard NLP evaluations, and manual identification of common error sources; (2) macro-level graph metrics that assess structural properties such as community detection and connectivity, which are relevant to real-world applications. Focusing on affiliation graphs of person membership in organizations extracted from social register books, our study identifies a range of extraction performance where biases across most downstream graph analysis metrics are near zero. However, as extraction performance declines, we find that many metrics exhibit increasingly pronounced biases, with each metric tending toward a consistent direction of either over- or under-estimation. Through simulations, we further show that error models commonly used in the literature do not capture these bias patterns, indicating the need for more realistic error models for KG extraction. Our findings provide actionable insights for practitioners and underscores the importance of advancing extraction methods and error modeling to ensure reliable and meaningful downstream analyses.
The Popularity Adjusted Block Model (PABM) provides a flexible framework for community detection in network data by allowing heterogeneous node popularity across communities. However, this flexibility increases model complexity and raises key unresolved challenges, particularly in effectively adapting spectral clustering techniques and efficiently achieving strong consistency in label recovery. To address these challenges, we first propose the Thresholded Cosine Spectral Clustering (TCSC) algorithm and establish its weak consistency under the PABM. We then introduce the one-step Refined TCSC algorithm and prove that it achieves strong consistency under the PABM, correctly recovering all community labels with high probability. We further show that the two-step Refined TCSC accelerates clustering error convergence, especially with small sample sizes. Additionally, we propose a data-driven approach for selecting the number of communities, which outperforms existing methods under the PABM. The effectiveness and robustness of our methods are validated through extensive simulations and real-world applications.
We present a novel federated multi-task learning method that leverages cross-client similarity to enable personalized learning for each client. To avoid transmitting the entire model to the parameter server, we propose a communication-efficient scheme that introduces a feature anchor, a compact vector representation that summarizes the features learned from the client's local classes. This feature anchor is shared with the server to account for local clients' distribution. In addition, the clients share the classification heads, a lightweight linear layer, and perform a graph-based regularization to enable collaboration among clients. By modeling collaboration between clients as a dynamic graph and continuously updating and refining this graph, we can account for any drift from the clients. To ensure beneficial knowledge transfer and prevent negative collaboration, we leverage a community detection-based approach that partitions this dynamic graph into homogeneous communities, maximizing the sum of task similarities, represented as the graph edges' weights, within each community. This mechanism restricts collaboration to highly similar clients within their formed communities, ensuring positive interaction and preserving personalization. Extensive experiments on two heterogeneous datasets demonstrate that our method significantly outperforms state-of-the-art baselines. Furthermore, we show that our method exhibits superior computation and communication efficiency and promotes fairness across clients.
In speaker diarization, traditional clustering-based methods remain widely used in real-world applications. However, these methods struggle with the complex distribution of speaker embeddings and overlapping speech segments. To address these limitations, we propose an Overlapping Community Detection method based on Graph Attention networks and the Label Propagation Algorithm (OCDGALP). The proposed framework comprises two key components: (1) a graph attention network that refines speaker embeddings and node connections by aggregating information from neighboring nodes, and (2) a label propagation algorithm that assigns multiple community labels to each node, enabling simultaneous clustering and overlapping community detection. Experimental results show that the proposed method significantly reduces the Diarization Error Rate (DER), achieving a state-of-the-art 15.94% DER on the DIHARD-III dataset without oracle Voice Activity Detection (VAD), and an impressive 11.07% with oracle VAD.
We consider the problem of detecting a community of densely connected vertices in a high-dimensional bipartite graph of size $n_1 \times n_2$. Under the null hypothesis, the observed graph is drawn from a bipartite Erd\H{o}s-Renyi distribution with connection probability $p_0$. Under the alternative hypothesis, there exists an unknown bipartite subgraph of size $k_1 \times k_2$ in which edges appear with probability $p_1 = p_0 + \delta$ for some $\delta > 0$, while all other edges outside the subgraph appear with probability $p_0$. Specifically, we provide non-asymptotic upper and lower bounds on the smallest signal strength $\delta^*$ that is both necessary and sufficient to ensure the existence of a test with small enough type one and type two errors. We also derive novel minimax-optimal tests achieving these fundamental limits when the underlying graph is sufficiently dense. Our proposed tests involve a combination of hard-thresholded nonlinear statistics of the adjacency matrix, the analysis of which may be of independent interest. In contrast with previous work, our non-asymptotic upper and lower bounds match for any configuration of $n_1,n_2, k_1,k_2$.
Semi-supervised local community detection aims to leverage known communities to detect the community containing a given node. Although existing semi-supervised local community detection studies yield promising results, they suffer from time-consuming issues, highlighting the need for more efficient algorithms. Therefore, we apply the "pre-train, prompt" paradigm to semi-supervised local community detection and propose the Pre-trained Prompt-driven Semi-supervised Local community detection method (PPSL). PPSL consists of three main components: node encoding, sample generation, and prompt-driven fine-tuning. Specifically, the node encoding component employs graph neural networks to learn the representations of nodes and communities. Based on representations of nodes and communities, the sample generation component selects known communities that are structurally similar to the local structure of the given node as training samples. Finally, the prompt-driven fine-tuning component leverages these training samples as prompts to guide the final community prediction. Experimental results on five real-world datasets demonstrate that PPSL outperforms baselines in both community quality and efficiency.
Contrastive learning (CL) is a prevalent technique for training embedding models, which pulls semantically similar examples (positives) closer in the representation space while pushing dissimilar ones (negatives) further apart. A key source of negatives are 'in-batch' examples, i.e., positives from other examples in the batch. Effectiveness of such models is hence strongly influenced by the size and quality of training batches. In this work, we propose 'Breaking the Batch Barrier' (B3), a novel batch construction strategy designed to curate high-quality batches for CL. Our approach begins by using a pretrained teacher embedding model to rank all examples in the dataset, from which a sparse similarity graph is constructed. A community detection algorithm is then applied to this graph to identify clusters of examples that serve as strong negatives for one another. The clusters are then used to construct batches that are rich in in-batch negatives. Empirical results on the MMEB multimodal embedding benchmark (36 tasks) demonstrate that our method sets a new state of the art, outperforming previous best methods by +1.3 and +2.9 points at the 7B and 2B model scales, respectively. Notably, models trained with B3 surpass existing state-of-the-art results even with a batch size as small as 64, which is 4-16x smaller than that required by other methods.
Community detection, a vital technology for real-world applications, uncovers cohesive node groups (communities) by leveraging both topological and attribute similarities in social networks. However, existing Graph Convolutional Networks (GCNs) trained to maximize modularity often converge to suboptimal solutions. Additionally, directly using human-labeled communities for training can undermine topological cohesiveness by grouping disconnected nodes based solely on node attributes. We address these issues by proposing a novel Topological and Attributive Similarity-based Community detection (TAS-Com) method. TAS-Com introduces a novel loss function that exploits the highly effective and scalable Leiden algorithm to detect community structures with global optimal modularity. Leiden is further utilized to refine human-labeled communities to ensure connectivity within each community, enabling TAS-Com to detect community structures with desirable trade-offs between modularity and compliance with human labels. Experimental results on multiple benchmark networks confirm that TAS-Com can significantly outperform several state-of-the-art algorithms.
Indoor positioning systems (IPSs) are increasingly vital for location-based services in complex multi-storey environments. This study proposes a novel graph-based approach for floor separation using Wi-Fi fingerprint trajectories, addressing the challenge of vertical localization in indoor settings. We construct a graph where nodes represent Wi-Fi fingerprints, and edges are weighted by signal similarity and contextual transitions. Node2Vec is employed to generate low-dimensional embeddings, which are subsequently clustered using K-means to identify distinct floors. Evaluated on the Huawei University Challenge 2021 dataset, our method outperforms traditional community detection algorithms, achieving an accuracy of 68.97\%, an F1-score of 61.99\%, and an Adjusted Rand Index of 57.19\%. By publicly releasing the preprocessed dataset and implementation code, this work contributes to advancing research in indoor positioning. The proposed approach demonstrates robustness to signal noise and architectural complexities, offering a scalable solution for floor-level localization.
This study formalizes a computational model to simulate classical Persian poets' dynamics of influence through constructing a multi-dimensional similarity network. Using a rigorously curated dataset based on Ganjoor's corpus, we draw upon semantic, lexical, stylistic, thematic, and metrical features to demarcate each poet's corpus. Each is contained within weighted similarity matrices, which are then appended to generate an aggregate graph showing poet-to-poet influence. Further network investigation is carried out to identify key poets, style hubs, and bridging poets by calculating degree, closeness, betweenness, eigenvector, and Katz centrality measures. Further, for typological insight, we use the Louvain community detection algorithm to demarcate clusters of poets sharing both style and theme coherence, which correspond closely to acknowledged schools of literature like Sabk-e Hindi, Sabk-e Khorasani, and the Bazgasht-e Adabi phenomenon. Our findings provide a new data-driven view of Persian literature distinguished between canonical significance and interextual influence, thus highlighting relatively lesser-known figures who hold great structural significance. Combining computational linguistics with literary study, this paper produces an interpretable and scalable model for poetic tradition, enabling retrospective reflection as well as forward-looking research within digital humanities.
We investigate privacy-preserving spectral clustering for community detection within stochastic block models (SBMs). Specifically, we focus on edge differential privacy (DP) and propose private algorithms for community recovery. Our work explores the fundamental trade-offs between the privacy budget and the accurate recovery of community labels. Furthermore, we establish information-theoretic conditions that guarantee the accuracy of our methods, providing theoretical assurances for successful community recovery under edge DP.
Community detection in networks with overlapping structures remains a significant challenge, particularly in noisy real-world environments where integrating topology, node attributes, and prior information is critical. To address this, we propose a semi-supervised graph autoencoder that combines graph multi-head attention and modularity maximization to robustly detect overlapping communities. The model learns semantic representations by fusing structural, attribute, and prior knowledge while explicitly addressing noise in node features. Key innovations include a noise-resistant architecture and a semantic semi-supervised design optimized for community quality through modularity constraints. Experiments demonstrate superior performance the model outperforms state-of-the-art methods in overlapping community detection (improvements in NMI and F1-score) and exhibits exceptional robustness to attribute noise, maintaining stable performance under 60\% feature corruption. These results highlight the importance of integrating attribute semantics and structural patterns for accurate community discovery in complex networks.
Hypergraphs, capable of representing high-order interactions via hyperedges, have become a powerful tool for modeling real-world biological and social systems. Inherent relationships within these real-world systems, such as the encoding relationship between genes and their protein products, drive the establishment of interconnections between multiple hypergraphs. Here, we demonstrate how to utilize those interconnections between multiple hypergraphs to synthesize integrated information from multiple higher-order systems, thereby enhancing understanding of underlying structures. We propose a model based on the stochastic block model, which integrates information from multiple hypergraphs to reveal latent high-order structures. Real-world hyperedges exhibit preferential attachment, where certain nodes dominate hyperedge formation. To characterize this phenomenon, our model introduces hyperedge internal degree to quantify nodes' contributions to hyperedge formation. This model is capable of mining communities, predicting missing hyperedges of arbitrary sizes within hypergraphs, and inferring inter-hypergraph edges between hypergraphs. We apply our model to high-order datasets to evaluate its performance. Experimental results demonstrate strong performance of our model in community detection, hyperedge prediction, and inter-hypergraph edge prediction tasks. Moreover, we show that our model enables analysis of multiple hypergraphs of different types and supports the analysis of a single hypergraph in the absence of inter-hypergraph edges. Our work provides a practical and flexible tool for analyzing multiple hypergraphs, greatly advancing the understanding of the organization in real-world high-order systems.
We consider the problem of recovering latent information from graphs under $\varepsilon$-edge local differential privacy where the presence of relationships/edges between two users/vertices remains confidential, even from the data curator. For the class of generalized random dot-product graphs, we show that a standard local differential privacy mechanism induces a specific geometric distortion in the latent positions. Leveraging this insight, we show that consistent recovery of the latent positions is achievable by appropriately adjusting the statistical inference procedure for the privatized graph. Furthermore, we prove that our procedure is nearly minimax-optimal under local edge differential privacy constraints. Lastly, we show that this framework allows for consistent recovery of geometric and topological information underlying the latent positions, as encoded in their persistence diagrams. Our results extend previous work from the private community detection literature to a substantially richer class of models and inferential tasks.
Semi-supervised community detection methods are widely used for identifying specific communities due to the label scarcity. Existing semi-supervised community detection methods typically involve two learning stages learning in both initial identification and subsequent adjustment, which often starts from an unreasonable community core candidate. Moreover, these methods encounter scalability issues because they depend on reinforcement learning and generative adversarial networks, leading to higher computational costs and restricting the selection of candidates. To address these limitations, we draw a parallel between crystallization kinetics and community detection to integrate the spontaneity of the annealing process into community detection. Specifically, we liken community detection to identifying a crystal subgrain (core) that expands into a complete grain (community) through a process similar to annealing. Based on this finding, we propose CLique ANNealing (CLANN), which applies kinetics concepts to community detection by integrating these principles into the optimization process to strengthen the consistency of the community core. Subsequently, a learning-free Transitive Annealer was employed to refine the first-stage candidates by merging neighboring cliques and repositioning the community core, enabling a spontaneous growth process that enhances scalability. Extensive experiments on \textbf{43} different network settings demonstrate that CLANN outperforms state-of-the-art methods across multiple real-world datasets, showcasing its exceptional efficacy and efficiency in community detection.
This paper develops a new spectral clustering-based method called TransNet for transfer learning in community detection of network data. Our goal is to improve the clustering performance of the target network using auxiliary source networks, which are heterogeneous, privacy-preserved, and locally stored across various sources. The edges of each locally stored network are perturbed using the randomized response mechanism to achieve differential privacy. Notably, we allow the source networks to have distinct privacy-preserving and heterogeneity levels as often desired in practice. To better utilize the information from the source networks, we propose a novel adaptive weighting method to aggregate the eigenspaces of the source networks multiplied by adaptive weights chosen to incorporate the effects of privacy and heterogeneity. We propose a regularization method that combines the weighted average eigenspace of the source networks with the eigenspace of the target network to achieve an optimal balance between them. Theoretically, we show that the adaptive weighting method enjoys the error-bound-oracle property in the sense that the error bound of the estimated eigenspace only depends on informative source networks. We also demonstrate that TransNet performs better than the estimator using only the target network and the estimator using only the weighted source networks.
The rapid growth of the internet has made personalized recommendation systems indispensable. Graph-based sequential recommendation systems, powered by Graph Neural Networks (GNNs), effectively capture complex user-item interactions but often face challenges such as noise and static representations. In this paper, we introduce the Adaptive Long-term Embedding with Denoising and Augmentation for Recommendation (ALDA4Rec) method, a novel model that constructs an item-item graph, filters noise through community detection, and enriches user-item interactions. Graph Convolutional Networks (GCNs) are then employed to learn short-term representations, while averaging, GRUs, and attention mechanisms are utilized to model long-term embeddings. An MLP-based adaptive weighting strategy is further incorporated to dynamically optimize long-term user preferences. Experiments conducted on four real-world datasets demonstrate that ALDA4Rec outperforms state-of-the-art baselines, delivering notable improvements in both accuracy and robustness. The source code is available at https://github.com/zahraakhlaghi/ALDA4Rec.
Community detection, the unsupervised task of clustering nodes of a graph, finds applications across various fields. The common approaches for community detection involve optimizing an objective function to partition the nodes into communities at a single scale of granularity. However, the single-scale approaches often fall short of producing partitions that are robust and at a suitable scale. The existing algorithm, PyGenStability, returns multiple robust partitions for a network by optimizing the multi-scale Markov stability function. However, in cases where the suitable scale is not known or assumed by the user, there is no principled method to select a single robust partition at a suitable scale from the multiple partitions that PyGenStability produces. Our proposed method combines the Markov stability framework with a pre-trained machine learning model for scale selection to obtain one robust partition at a scale that is learned based on the graph structure. This automatic scale selection involves using a gradient boosting model pre-trained on hand-crafted and embedding-based network features from a labeled dataset of 10k benchmark networks. This model was trained to predicts the scale value that maximizes the similarity of the output partition to the planted partition of the benchmark network. Combining our scale selection algorithm with the PyGenStability algorithm results in PyGenStabilityOne (PO): a hyperparameter-free multi-scale community detection algorithm that returns one robust partition at a suitable scale without the need for any assumptions, input, or tweaking from the user. We compare the performance of PO against 29 algorithms and show that it outperforms 25 other algorithms by statistically meaningful margins. Our results facilitate choosing between community detection algorithms, among which PO stands out as the accurate, robust, and hyperparameter-free method.