Differentiable Tripartite Modularity for Clustering Heterogeneous Graphs

TL;DR

This paper introduces a differentiable tripartite modularity method for clustering complex heterogeneous graphs with three node types, enabling end-to-end learning and stable optimization without dense tensor computations.

Contribution

It extends differentiable modularity to tripartite graphs, defining community structure via weighted co-paths and introducing normalization for degree heterogeneity.

Findings

Robust convergence on large-scale urban data

Produces spatially coherent partitions

Maintains linear complexity in edges

Abstract

Clustering heterogeneous relational data remains a central challenge in graph learning, particularly when interactions involve more than two types of entities. While differentiable modularity objectives such as DMoN have enabled end-to-end community detection on homogeneous and bipartite graphs, extending these approaches to higher-order relational structures remains non-trivial. In this work, we introduce a differentiable formulation of tripartite modularity for graphs composed of three node types connected through mediated interactions. Community structure is defined in terms of weighted co-paths across the tripartite graph, together with an exact factorized computation that avoids the explicit construction of dense third-order tensors. A structural normalization at pivot nodes is introduced to control extreme degree heterogeneity and ensure stable optimization. The resulting objective can be optimized jointly with a graph neural network in an end-to-end manner, while retaining linear complexity in the number of edges. We validate the proposed framework on large-scale urban cadastral data, where it exhibits robust convergence behavior and produces spatially coherent partitions. These results highlight differentiable tripartite modularity as a generic methodological building block for unsupervised clustering of heterogeneous graphs.