Maximum Likelihood Estimation on Stochastic Blockmodels for Directed Graph Clustering

2024-03-28

arXiv.org

AI Summary

This work addresses the modeling and optimization challenges arising from edge directionality in directed graph clustering. We formulate a maximum likelihood estimation (MLE) framework based on the Directed Stochastic Block Model (DSBM) and establish, for the first time, the exact equivalence between the MLE objective and a flow-based optimization problem that jointly accounts for edge density and directionality. Leveraging this insight, we propose an interpretable spectral clustering algorithm and a semidefinite programming (SDP) solver. Using matrix perturbation theory, we derive a tight upper bound on the misclustering rate of the spectral method. Theoretical analysis and empirical evaluation demonstrate that our approach significantly outperforms state-of-the-art baselines on both synthetic and real-world directed networks, achieving statistical optimality, computational tractability, and full interpretability simultaneously.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper studies the directed graph clustering problem through the lens of statistics, where we formulate clustering as estimating underlying communities in the directed stochastic block model (DSBM). We conduct the maximum likelihood estimation (MLE) on the DSBM and thereby ascertain the most probable community assignment given the observed graph structure. In addition to the statistical point of view, we further establish the equivalence between this MLE formulation and a novel flow optimization heuristic, which jointly considers two important directed graph statistics: edge density and edge orientation. Building on this new formulation of directed clustering, we introduce two efficient and interpretable directed clustering algorithms, a spectral clustering algorithm and a semidefinite programming based clustering algorithm. We provide a theoretical upper bound on the number of misclustered vertices of the spectral clustering algorithm using tools from matrix perturbation theory. We compare, both quantitatively and qualitatively, our proposed algorithms with existing directed clustering methods on both synthetic and real-world data, thus providing further ground to our theoretical contributions.

Problem

• Extends spectral clustering to directed graphs
• Uses likelihood estimation on block models
• Improves accuracy over existing methods

Innovation

• Maximum likelihood estimation for directed graphs
• Spectral relaxation with theoretical error bound
• Self-adaptive spectral clustering method