Facundo Mémoli, Guilherme Vituri F. Pinto
Motivated by the concept of network motifs we construct certain clustering methods (functors) which are parametrized by a given collection of motifs (or representers).
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Parametrized Power-Iteration Clustering for Directed Graphs
针对有向图顶点聚类问题，提出基于参数化可逆随机游走的ParPIC方法，通过自动调优扩散时间和高效截断嵌入，在保持聚类精度的同时提升可扩展性。
Generalized Dirichlet Energy and Graph Laplacians for Clustering Directed and Undirected Graphs
针对有向图聚类中边不对称导致经典谱方法失效的问题，提出广义狄利克雷能量（GDE）框架与广义谱聚类（GSC）方法，利用参数化顶点测度编码方向与密度，无需传送即可实现弱连通有向图聚类，在真实数据集上准确率与鲁棒性均优于现有方法。
Graph clustering is a basic technique in machine learning, and has widespread applications in different domains. While spectral techniques have been successfully applied for clustering undirected graphs, the performance of spectral clustering algorithms for directed graphs (digraphs) is not in general satisfactory: these algorithms usually require symmetrising the matrix representing a digraph, and typical objective functions for undirected graph clustering do not capture cluster-structures in which the information given by the direction of the edges is crucial. To overcome these downsides, we…
How do vertices exert influence in graph data? We develop a framework for edge clustering, a new method for exploratory data analysis that reveals how both vertices and edges collaboratively accomplish directed influence in graphs, especially for directed graphs. In contrast to the ubiquitous vertex clustering which groups vertices, edge clustering groups edges. Edges sharing a functional affinity are assigned to the same group and form an influence subgraph cluster. With a complexity comparable to that of vertex clustering, this framework presents three different methods for edge spectral clu…
Directed networks are broadly used to represent asymmetric relationships among units. Co-clustering aims to cluster the senders and receivers of directed networks simultaneously. In particular, the well-known spectral clustering algorithm could be modified as the spectral co-clustering to co-cluster directed networks. However, large-scale networks pose great computational challenges to it. In this paper, we leverage sketching techniques and derive two randomized spectral co-clustering algorithms, one \emph{random-projection-based} and the other \emph{random-sampling-based}, to accelerate the c…
We provide a complete taxonomic characterization of robust hierarchical clustering methods for directed networks following an axiomatic approach. We begin by introducing three practical properties associated with the notion of robustness in hierarchical clustering: linear scale preservation, stability, and excisiveness. Linear scale preservation enforces imperviousness to change in units of measure whereas stability ensures that a bounded perturbation in the input network entails a bounded perturbation in the clustering output. Excisiveness refers to the local consistency of the clustering out…
Spectral Clustering for Directed Graphs via Likelihood Estimation on Stochastic Block Models
本文针对有向图聚类问题，基于有向随机块模型的最大似然估计推导全局目标函数，并提出一种自适应谱聚类方法，在合成与真实数据上显著优于现有基线。
This paper presents a concise tutorial on spectral clustering for broad spectrum graphs which include unipartite (undirected) graph, bipartite graph, and directed graph. We show how to transform bipartite graph and directed graph into corresponding unipartite graph, therefore allowing a unified treatment to all cases. In bipartite graph, we show that the relaxed solution to the $K$-way co-clustering can be found by computing the left and right eigenvectors of the data matrix. This gives a theoretical basis for $K$-way spectral co-clustering algorithms proposed in the literatures. We also show …
Cut-based directed graph (digraph) clustering often focuses on finding dense within-cluster or sparse between-cluster connections, similar to cut-based undirected graph clustering methods. In contrast, for flow-based clusterings the edges between clusters tend to be oriented in one direction and have been found in migration data, food webs, and trade data. In this paper we introduce a spectral algorithm for finding flow-based clusterings. The proposed algorithm is based on recent work which uses complex-valued Hermitian matrices to represent digraphs. By establishing an algebraic relationship …
We study the task of clustering in directed networks. We show that using the eigenvalue/eigenvector decomposition of the adjacency matrix is simpler than all common methods which are based on a combination of data regularization and SVD truncation, and works well down to the very sparse regime where the edge density has constant order. Our analysis is based on a Master Theorem describing sharp asymptotics for isolated eigenvalues/eigenvectors of sparse, non-symmetric matrices with independent entries. We also describe the limiting distribution of the entries of these eigenvectors; in the task …
Clustering is an essential technique for network analysis, with applications in a diverse range of fields. Although spectral clustering is a popular and effective method, it fails to consider higher-order structure and can perform poorly on directed networks. One approach is to capture and cluster higher-order structures using motif adjacency matrices. However, current formulations fail to take edge weights into account, and thus are somewhat limited when weight is a key component of the network under study. We address these shortcomings by exploring motif-based weighted spectral clustering me…
Node clustering is a powerful tool in the analysis of networks. We introduce a graph neural network framework, named DIGRAC, to obtain node embeddings for directed networks in a self-supervised manner, including a novel probabilistic imbalance loss, which can be used for network clustering. Here, we propose \textit{directed flow imbalance} measures, which are tightly related to directionality, to reveal clusters in the network even when there is no density difference between clusters. In contrast to standard approaches in the literature, in this paper, directionality is not treated as a nuisan…
CaSPECT: Discovering Causally Homogeneous Subgroups via Directed Spectral Clustering
该研究提出CaSPECT因果谱聚类框架，基于学习到的DAG拓扑定义相似性，从观测数据中识别因果同质亚组，无需预先指定倾向得分模型即可校正混杂，提升因果亚组发现可靠性。
Clustering is an important topic in algorithms, and has a number of applications in machine learning, computer vision, statistics, and several other research disciplines. Traditional objectives of graph clustering are to find clusters with low conductance. Not only are these objectives just applicable for undirected graphs, they are also incapable to take the relationships between clusters into account, which could be crucial for many applications. To overcome these downsides, we study directed graphs (digraphs) whose clusters exhibit further "structural" information amongst each other. Based …
Multi-view clustering integrating anchor attribute and structural information
针对传统多视图聚类仅依赖样本属性构建相似矩阵、忽略现实网络有向拓扑结构的问题，提出融合锚点属性与有向结构信息的AAS算法，采用两步邻近度方法并整合为统一优化框架，实验验证其有效性与优越性。
Round-Trip KNN Clustering: multiscale hierarchical cluster detection on directed nearest-neighbour graphs
提出往返KNN聚类方法，在有向最近邻图上多尺度检测层次簇，无需预设簇数，经细化提升聚类精度，在合成数据上表现优异。
Graph clustering is a fundamental task which discovers communities or groups in networks. Recent studies have mostly focused on developing deep learning approaches to learn a compact graph embedding, upon which classic clustering methods like k-means or spectral clustering algorithms are applied. These two-step frameworks are difficult to manipulate and usually lead to suboptimal performance, mainly because the graph embedding is not goal-directed, i.e., designed for the specific clustering task. In this paper, we propose a goal-directed deep learning approach, Deep Attentional Embedded Graph …
SDC-HSDD-NDSA: Structure Detecting Cluster by Hierarchical Secondary Directed Differential with Normalized Density and Self-Adaption
针对现有基于密度的聚类算法无法检测未被低密度区域分隔的高密度区域中多簇结构的问题，提出SDC-HSDD-NDSA算法，通过层次化二级有向微分等技术实现该类结构检测，实验证明其在多场景下优于现有算法。
We utilize the PageRank vector to generalize the $k$-means clustering algorithm to directed and undirected graphs. We demonstrate that PageRank and other centrality measures can be used in our setting to robustly compute centrality of nodes in a given graph. Furthermore, we show how our method can be generalized to metric spaces and apply it to other domains such as point clouds and triangulated meshes
This paper considers networks where relationships between nodes are represented by directed dissimilarities. The goal is to study methods that, based on the dissimilarity structure, output hierarchical clusters, i.e., a family of nested partitions indexed by a connectivity parameter. Our construction of hierarchical clustering methods is built around the concept of admissible methods, which are those that abide by the axioms of value - nodes in a network with two nodes are clustered together at the maximum of the two dissimilarities between them - and transformation - when dissimilarities are …
Summarizing large-scaled directed graphs into small-scale representations is a useful but less studied problem setting. Conventional clustering approaches, which based on "Min-Cut"-style criteria, compress both the vertices and edges of the graph into the communities, that lead to a loss of directed edge information. On the other hand, compressing the vertices while preserving the directed edge information provides a way to learn the small-scale representation of a directed graph. The reconstruction error, which measures the edge information preserved by the summarized graph, can be used to le…
Dual-Center Graph Clustering with Neighbor Distribution
本文提出双中心图聚类方法DCGC，利用邻居分布作为可靠监督信号挖掘困难负样本，并联合特征中心与邻居分布中心构建双目标分布进行优化，实验证明其性能优越。
Hypergraphs as Weighted Directed Self-Looped Graphs: Spectral Properties, Clustering, Cheeger Inequality
本文提出了一种基于随机游走的超图谱聚类算法HyperClus-G，解决了EDVW超图的谱理论、聚类和切比雪夫不等式问题。
Complex non-backtracking matrix for directed graphs
本文提出复非回溯矩阵，融合埃尔米特邻接矩阵与非回溯矩阵性质，揭示二者关系，并证明其对稀疏有向图聚类具有重要价值。
Graph embedding methods embed the nodes in a graph in low dimensional vector space while preserving graph topology to carry out the downstream tasks such as link prediction, node recommendation and clustering. These tasks depend on a similarity measure such as cosine similarity and Euclidean distance between a pair of embeddings that are symmetric in nature and hence do not hold good for directed graphs. Recent work on directed graphs, HOPE, APP, and NERD, proposed to preserve the direction of edges among nodes by learning two embeddings, source and target, for every node. However, these metho…
Copula Adapted Directed Acyclic Graph for Cluster Representation of Biomedical Data
针对生物医学数据中标签缺失和复杂依赖问题，提出CopDAG框架，结合copula模型与DAG因果发现，在16个数据集上聚类性能领先。
Community detection in large social networks is affected by degree heterogeneity of nodes. The D-SCORE algorithm for directed networks was introduced to reduce this effect by taking the element-wise ratios of the singular vectors of the adjacency matrix before clustering. Meaningful results were obtained for the statistician citation network, but rigorous analysis on its performance was missing. First, this paper establishes theoretical guarantee for this algorithm and its variants for the directed degree-corrected block model (Directed-DCBM). Second, this paper provides significant improvemen…
In our previous works, we proposed a physically-inspired rule to organize the data points into an in-tree (IT) structure, in which some undesired edges are allowed to occur. By removing those undesired or redundant edges, this IT structure is divided into several separate parts, each representing one cluster. In this work, we seek to prevent the undesired edges from arising at the source. Before using the physically-inspired rule, data points are at first organized into a proximity graph which restricts each point to select the optimal directed neighbor just among its neighbors. Consequently, …
It has been shown that the adjacency eigenspace of a network contains key information of its underlying structure. However, there has been no study on spectral analysis of the adjacency matrices of directed signed graphs. In this paper, we derive theoretical approximations of spectral projections from such directed signed networks using matrix perturbation theory. We use the derived theoretical results to study the influences of negative intra cluster and inter cluster directed edges on node spectral projections. We then develop a spectral clustering based graph partition algorithm, SC-DSG, an…
A Unified Framework for Structure-Aware Clustering and Heterogeneous Causal Graph Learning
本文提出了一种基于有向无环图的依赖聚类方法，通过交替方向乘子法解决结构异质性问题，实现对子群体依赖结构的鲁棒发现。
In multivariate time series systems, it has been observed that certain groups of variables partially lead the evolution of the system, while other variables follow this evolution with a time delay; the result is a lead-lag structure amongst the time series variables. In this paper, we propose a method for the detection of lead-lag clusters of time series in multivariate systems. We demonstrate that the web of pairwise lead-lag relationships between time series can be helpfully construed as a directed network, for which there exist suitable algorithms for the detection of pairs of lead-lag clus…
HeNCler: Node Clustering in Heterophilous Graphs via Learned Asymmetric Similarity
HeNCler提出基于加权核奇异值分解学习非对称相似性图，实现异质图节点聚类，克服传统方法局限并显著提升性能。
BHGNN-RT: Network embedding for directed heterogeneous graphs
本文提出了一种用于有向异构图的网络嵌入方法BHGNN-RT，通过双向消息传递和随机传送机制克服过平滑问题，在节点分类和无监督聚类任务中达到了最先进的性能。
Higher-order motif structures and multi-vertex interactions are becoming increasingly important in studies that aim to improve our understanding of functionalities and evolution patterns of networks. To elucidate the role of higher-order structures in community detection problems over complex networks, we introduce the notion of a Superimposed Stochastic Block Model (SupSBM). The model is based on a random graph framework in which certain higher-order structures or subgraphs are generated through an independent hyperedge generation process, and are then replaced with graphs that are superimpos…
Transfer operators on graphs: Spectral clustering and beyond
本文在图上定义Koopman和Perron-Frobenius等转移算子并研究其谱性质，证明无向图谱聚类可由Koopman算子特征函数解释，并基于广义转移算子提出有向图新聚类算法，在多个基准问题上验证了有效性。
Clustering Time-Evolving Networks Using the Spatio-Temporal Graph Laplacian
本文提出基于CCA的时空图拉普拉斯算子，将谱聚类推广到动态图，并结合传递算子分析其谱性质，实现对有向和无向时变网络聚类演化的清晰解释。
A graph embedding algorithm embeds a graph into a low-dimensional space such that the embedding preserves the inherent properties of the graph. While graph embedding is fundamentally related to graph visualization, prior work did not exploit this connection explicitly. We develop Force2Vec that uses force-directed graph layout models in a graph embedding setting with an aim to excel in both machine learning (ML) and visualization tasks. We make Force2Vec highly parallel by mapping its core computations to linear algebra and utilizing multiple levels of parallelism available in modern processor…
This work elaborates on the important problem of (1) designing optimal randomized routing policies for reaching a target node t from a source note s on a weighted directed graph G and (2) defining distance measures between nodes interpolating between the least cost (based on optimal movements) and the commute-cost (based on a random walk on G), depending on a temperature parameter T. To this end, the randomized shortest path formalism (RSP, [2,99,124]) is rephrased in terms of Tsallis divergence regularization, instead of Kullback-Leibler divergence. The main consequence of this change is that…
This paper considers networks where relationships between nodes are represented by directed dissimilarities. The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a connectivity parameter, induced by the given dissimilarity structures. Our construction of hierarchical clustering methods is based on defining admissible methods to be those methods that abide by the axioms of value - nodes in a network with two nodes are clustered together at the maximum of the two dissimilarities between them - and transformation - when dissim…
Local Clustering on Complex Graphs and Complex Hypergraphs
本文通过扩展非近似的Andersen-Chung-Lang (ACL)聚类算法，提出了GeneralACL和HyperACL两种算法，分别适用于带权、有向、自环图以及边依赖顶点权重的超图，并证明了在温和条件下它们能识别出电导率二次最优的聚类。
MADCluster: Model-agnostic Anomaly Detection with Self-supervised Clustering Network
提出模型无关的自监督聚类异常检测框架 MADCluster，通过单向自适应损失和三模块结构缓解超球体坍塌，并在四个时间序列基准上验证其跨架构性能提升。
Online Sparsification of Bipartite-Like Clusters in Graphs
本文研究了图中的类二部聚类问题，提出了在无向图和有向图中寻找此类聚类的高效在线稀疏化算法，并在合成与真实数据集上验证了该算法能显著加速现有聚类算法且保持其有效性。
MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian
本文提出基于磁符号拉普拉斯的谱图神经网络MSGNN，可融合符号与方向信息，在节点聚类、链接预测等任务上表现领先，还提出新型合成模型与金融数据集。
Structural node embeddings, vectors capturing local connectivity information for each node in a graph, have many applications in data mining and machine learning, e.g., network alignment and node classification, clustering and anomaly detection. For the analysis of directed graphs, e.g., transactions graphs, communication networks and social networks, the capability to capture directional information in the structural node embeddings is highly desirable, as is scalability of the embedding extraction method. Most existing methods are nevertheless only designed for undirected graph. Therefore, w…
Hierarchical Clustering and Signal Denoising on Digraphs
本文提出将有向图表示为Hermitian矩阵，基于谱分解和k-means递归实现层次聚类，并利用簇的入/出度构造多尺度样条拟插值，实现有向图信号去噪，实验验证了其优越性。
Learning Bayesian Networks with Heterogeneous Agronomic Data Sets via Mixed-Effect Models and Hierarchical Clustering
针对异构层次农艺数据，本文提出将随机效应融入贝叶斯网络学习的方法，提升模型可解释性与因果发现能力，并将玉米产量预测错误率由28%降至17%。
We have proposed a model based upon flocking on a complex network, and then developed two clustering algorithms on the basis of it. In the algorithms, firstly a \textit{k}-nearest neighbor (knn) graph as a weighted and directed graph is produced among all data points in a dataset each of which is regarded as an agent who can move in space, and then a time-varying complex network is created by adding long-range links for each data point. Furthermore, each data point is not only acted by its \textit{k} nearest neighbors but also \textit{r} long-range neighbors through fields established in space…
From data to concepts via wiring diagrams
本文通过准骨架连线图方法从序列数据中提取概念，并展示了其在自主代理游戏策略识别中的应用。
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