Hierarchical Clustering of Networks via Hierarchical Distance Matrices

2026-07-31

AI Summary

This work addresses the challenge of simultaneously identifying community structure and recovering its underlying hierarchical organization in networks, a task for which existing methods often rely on pre-specified parameters. The authors propose NHC-TST, a top-down, fully data-driven hierarchical clustering algorithm that formalizes hierarchy via a hierarchical distance matrix and recursively partitions the network using spectral clustering. An adaptive stopping criterion based on the graph-based two-sample test eliminates the need to predefine the number of clusters or tree depth, enabling reconstruction of unbalanced hierarchical trees. Theoretical analysis establishes the algorithm’s statistical consistency and structural recovery accuracy. Experiments demonstrate that NHC-TST precisely recovers both cluster memberships and hierarchical relationships across diverse synthetic networks and uncovers multi-scale dynamic structures in global migration data that flat clustering approaches fail to capture.

Abstract

Clustering populations of networks while recovering their latent hierarchical organization is a fundamental yet largely unexplored problem in network analysis. To formalize this, we introduce the Hierarchical Distance Matrix, a specific class of population-level distance matrices that encodes latent hierarchical organization through recursively nested distance separation, accommodating unbalanced tree depths. Building on this framework, we propose a fully data-driven top-down procedure: network hierarchical clustering based on two-sample testing (NHC-TST). The algorithm recursively splits networks via spectral clustering and uses a graph-based two-sample stopping rule. The procedure adaptively determines the branching structure without requiring prior knowledge of the number of clusters or tree depth. Theoretically, we establish exact recovery of the population-level hierarchical structure and statistical consistency in the empirical procedure. Simulation studies demonstrate highly accurate recovery of both cluster memberships and hierarchical relationships across a wide range of settings. Applied to a global migration dataset, NHC-TST uncovers interpretable multi-resolution temporal structures that are not revealed by conventional flat clustering approaches.

Problem

• hierarchical clustering
• network populations
• latent hierarchical organization
• distance matrices
• cluster recovery

Innovation

• Hierarchical Distance Matrix
• Network Clustering
• Two-sample Testing
• Spectral Clustering
• Hierarchical Structure Recovery