Finding community structure in networks using the eigenvectors of matrices

Abstract

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as "modularity" over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a new centrality measure that identifies those vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

22 pages, 8 figures, minor corrections in this version

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• The Atlas for the Aspiring Network Scientist
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• On the Consistency of the Likelihood Maximization Vertex Nomination Scheme: Bridging the Gap Between Maximum Likelihood Estimation and Graph Matching
• How to Measure Significance of Community Structure in Complex Networks
• Multivariate Spatiotemporal Hawkes Processes and Network Reconstruction
• Finding the proper node ranking method for complex networks
• Coarse-Grained Topology Estimation via Graph Sampling
• Maximum Likelihood Latent Space Embedding of Logistic Random Dot Product Graphs
• Comprehensive spectral approach for community structure analysis on complex networks
• Community detection by resistance distance: automation and benchmark testing
• Opinion formation at Ising social networks
• Smoothness Sensor: Adaptive Smoothness-Transition Graph Convolutions for Attributed Graph Clustering
• Hierarchical clustering of bipartite data sets based on the statistical significance of coincidences
• Discovering and Visualizing Hierarchy in Multivariate Data
• Weblog Clustering in Multilinear Algebra Perspective
• A Model for Image Segmentation in Retina
• Sequential locality of graphs and its hypothesis testing
• Transforming complex network to the acyclic one
• Central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs
• A fast multilevel algorithm for graph clustering and community detection
• Stability of Maximum likelihood based clustering methods: exploring the backbone of classifications (Who is keeping you in that community?)
• D-chain tomography of networks: a new structure spectrum and an application to the SIR process
• Linking Through Time: Memory-Enhanced Community Discovery in Temporal Networks
• RDSA: A Robust Deep Graph Clustering Framework via Dual Soft Assignment
• Metrics for Community Analysis: A Survey
• Steering Macro-Scale Network Community Structure by Micro-Scale Features
• Decompositions of Triangle-Dense Graphs
• On consistent vertex nomination schemes
• Community Detection Using Slow Mixing Markov Models
• A new method for community detection in social networks based on message distribution
• Community Detection Clustering via Gumbel Softmax
• Edge based stochastic block model statistical inference
• Structural Diversity and Homophily: A Study Across More than One Hundred Big Networks
• Predicting missing links via correlation between nodes
• Network Null Model based on Maximal Entropy and the Rich-Club
• Spectral Anomaly Detection in Very Large Graphs: Models, Noise, and Computational Complexity
• Entropic Spectral Learning for Large-Scale Graphs
• Botnet Detection using Social Graph Analysis
• Systematic comparison of graph embedding methods in practical tasks
• Revisiting Spectral Graph Clustering with Generative Community Models
• Herding Positive, Complex Networks
• Multilateration of Random Networks with Community Structure
• Diffusion and Superposition Distances for Signals Supported on Networks
• The emergence of unshared consensus decisions in bottlenose dolphins
• The ubiquity of small-world networks
• Uniqueness of communities in regular stochastic block models
• Systematic assessment of the quality of fit of the stochastic block model for empirical networks
• A decaying factor accounts for contained activity in neuronal networks with no need of hierarchical or modular organization
• Cluster Synchronization of Networks via a Canonical Transformation for Simultaneous Block Diagonalization of Matrices
• Costly bilingualism model in a population with one zealot
• An analysis of community structure in Brazilian political topic-based Twitter networks
• NEW: A Generic Learning Model for Tie Strength Prediction in Networks
• Landscape Correspondence of Empirical and Population Risks in the Eigendecomposition Problem
• Testing for the Network Small-World Property
• Network Refinement: A unified framework for enhancing signal or removing noise of networks
• Correlation Clustering with Constrained Cluster Sizes and Extended Weights Bounds
• Community Structure aware Embedding of Nodes in a Network
• Directed Network Laplacians and Random Graph Models
• A New Analysis Method for Simulations Using Node Categorizations
• Large algebraic connectivity fluctuations in spatial network ensembles imply a predictive advantage from node location information
• Fault Tolerance of Random Graphs with respect to Connectivity: Mean-field Approximation for Semi-dense Random Graphs
• Detecting hierarchical and overlapping network communities using locally optimal modularity changes
• Strength of Connections in a Random Graph: Definition, Characterization, and Estimation
• A new measure of modularity density for community detection
• Modeling Graphs Using a Mixture of Kronecker Models
• Overlapping community detection in networks via sparse spectral decomposition
• Dynamic Ising Model: Reconstruction of Evolutionary Trees
• Maximizing spreading in complex networks with risk in node activation
• Spectral community detection in heterogeneous large networks
• Using Sampled Network Data With The Autologistic Actor Attribute Model
• Dynamics of center-periphery patterns in knowledge networks - the case of China's biotech science and technology system
• Dynamical thermalization and turbulence in social stratification models
• Corrected Bayesian information criterion for stochastic block models
• Numerical Performance of the Implicitly Restarted Arnoldi Method in OFP8, Bfloat16, Posit, and Takum Arithmetics
• Optimization of convergence rate via algebraic connectivity
• The Census-Stub Graph Invariant Descriptor
• Spectral Graph Analysis: A Unified Explanation and Modern Perspectives
• Network compression with configuration models and the minimum description length
• Contribution of directedness in graph spectra
• Real-Time Community Detection in Large Social Networks on a Laptop
• Unveiling Contextual Similarity of Things via Mining Human-Thing Interactions in the Internet of Things
• Ensembles based on the Rich-Club and how to use them to build soft-communities
• The Price of Differential Privacy for Low-Rank Factorization
• Reconstructing propagation networks with temporal similarity metrics
• Phase Transitions in Spectral Community Detection of Large Noisy Networks
• Towards Modularity Optimization Using Reinforcement Learning to Community Detection in Dynamic Social Networks
• Comparative study of random walks with one-step memory on complex networks
• Unified Statistical Theory of Spectral Graph Analysis
• The spreading ability of nodes towards localized targets in complex networks
• SimRank Computation on Uncertain Graphs
• Constructing Frequency Domains on Graphs in Near-Linear Time
• Learning Resolution Parameters for Graph Clustering
• Text as Data: Real-time Measurement of Economic Welfare
• A Maximum Entropy approach to Massive Graph Spectra
• Network-ensemble comparisons with stochastic rewiring and von Neumann entropy
• Spectral analysis for gene communities in cancer cells
• Generalized Centrality Aggregation and Exclusive Centrality
• Community Detection by a Riemannian Projected Proximal Gradient Method
• On the Estimation and Use of Statistical Modelling in Information Retrieval
• Temporal similarity metrics for latent network reconstruction: The role of time-lag decay
• On the use of local structural properties for improving the efficiency of hierarchical community detection methods
• Diffusion on dynamic contact networks with indirect transmission links
• Spaceland Embedding of Sparse Stochastic Graphs
• Community structures in simplicial complexes: an application to wildlife corridor designing in Central India -- Eastern Ghats landscape complex, India
• The Ties that Bind Networks: Weak Ties Facilitate the Emergence of Collective Memories
• Attack Vulnerability of Complex Networks in Center-Based Strategies
• Non-Backtracking Centrality Based Random Walk on Networks
• Correlation Analysis of Nodes Identifies Real Communities in Networks
• Semisupervised Learning on Heterogeneous Graphs and its Applications to Facebook News Feed
• A Probabilistic Ant-based Heuristic for the Longest Simple Cycle Problem in Complex Networks
• Detecting Antagonistic and Allied Communities on Social Media
• Global Network Prediction from Local Node Dynamics
• A study on the performance of similarity indices and its relationship with link prediction: a two-state random network case
• Learning with Semi-Definite Programming: new statistical bounds based on fixed point analysis and excess risk curvature
• From Community Detection to Community Deception
• Improving Community Detection by Mining Social Interactions
• GA Based Q-Attack on Community Detection
• Sublinear Column-wise Actions of the Matrix Exponential on Social Networks
• The Continuous Node Degree: a New Measure for Complex Networks
• Ground state energy of $q$-state Potts model: the minimum modularity
• Competitive Dynamics on Complex Networks
• Block-corrected Modularity for Community Detection
• Weighted Graph Nodes Clustering via Gumbel Softmax
• Exact statistical mechanics of the Ising model on networks
• Optimisation dans la détection de communautés recouvrantes et équilibre de Nash
• The Irreducible Spine(s) of Undirected Networks
• Overlapping Community Detection Optimization and Nash Equilibrium
• HomDroid: Detecting Android Covert Malware by Social-Network Homophily Analysis
• Community Structure in Interaction Web Service Networks
• Differentially Private Link Prediction With Protected Connections
• A Model of Random Industrial SAT
• Role of Weak Ties in Link Prediction of Complex Networks
• Adversarial Training Methods for Network Embedding
• Overlapping Communities in Complex Networks
• Grouping Entities in a Fleet by Community Detection in Network of Regression Models
• Towards Fast Evaluation of Unsupervised Link Prediction by Random Sampling Unobserved Links
• A Preference Random Walk Algorithm for Link Prediction through Mutual Influence Nodes in Complex Networks
• Swayed by Friends or by the Crowd?
• DSSLP: A Distributed Framework for Semi-supervised Link Prediction
• One Node at a Time: Node-Level Network Classification
• On $r$-to-$p$ norms of random matrices with nonnegative entries: Asymptotic normality and ${\ell }_{\infty }$-bounds for the maximizer
• A framework for community detection in heterogeneous multi-relational networks
• Evaluating links through spectral decomposition
• Finding Astronomical Communities Through Co-readership Analysis
• An Empirical Study of Compression-friendly Community Detection Methods
• Approximating Spatial Distance Through Confront Networks: Application to the Segmentation of Medieval Avignon
• Dual prototype attentive graph network for cross-market recommendation
• Modularity spectra, eigen-subspaces, and structure of weighted graphs
• Interplay between Network Topology and Dynamics in Neural Systems
• MODULAR: Software for the Autonomous Computation of Modularity in Large Network Sets
• WEM: A Node Importance Algorithm in Weighted Networks
• The self-consistent field iteration for p-spectral clustering
• Multiplex Structures: Patterns of Complexity in Real-World Networks
• Effects of concurrency on epidemic spreading in Markovian temporal networks
• The Universal Recommender
• Identification of Community Structures in Networks Employing a Modified Divisive Algorithm
• Approximating Sparse Matrices and their Functions using Matrix-vector products
• Universal scaling hypothesis of quantum spatial search in complex networks
• Citation graph, weighted impact factors and performance indices
• The Function of Communities in Protein Interaction Networks at Multiple Scales
• Transfer and routing of Gaussian states through quantum complex networks with and without community structure
• PHLP: Sole Persistent Homology for Link Prediction - Interpretable Feature Extraction
• Error-Correcting Decoders for Communities in Networks
• On the Approximation of Laplacian Eigenvalues in Graph Disaggregation
• Graph-Community Detection for Cross-Document Topic Segment Relationship Identification
• SECTOR: A Neural Model for Coherent Topic Segmentation and Classification