Differentiable Tripartite Modularity for Clustering Heterogeneous Graphs

TL;DR

This work tackles unsupervised clustering of heterogeneous tripartite graphs where interactions are mediated by a pivot node. It defines a differentiable tripartite modularity based on mediated co-paths $X \rightarrow Y \rightarrow Z$, with an exact factorization that avoids dense third-order tensors. A pivot-flow normalization term bounds the influence of high-degree nodes, enabling stable end-to-end optimization with a graph encoder and achieving linear-time complexity in the number of edges. Validation on a large-scale urban cadastral dataset demonstrates robust convergence and spatially coherent partitions, establishing tripartite modularity as a generic building block for unsupervised clustering on complex relational data.

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.

Figures (3)

• Figure 1: Comparison of graph construction strategies on a residential area. Left (Method A): Standard k-NN graph (k=8) based on Euclidean distance. Note that edges (red dotted lines) arbitrarily cross the roadway, creating artificial connections between distinct urban blocks. Right (Method B): The proposed Dual Topological Projection. Orange lines represent common parcel boundaries (weighted by length, Eq. \ref{['eq:wZZ']}). Green lines represent the resulting connectivity between buildings. The topology strictly respects the street layout: no edge connects separated blocks, despite their geographic proximity.
• Figure 2: Diagram of the factorized modularity calculation. The pivot $j$ aggregates the assignments of its neighbors ($A_{j,l}, C_{j,n}$), weights them by its own assignment ($S_Y$) and its correction factor ($\omega_j$).
• Figure 3: Illustrative mapping of communities detected by the DMoN-3p model at the departmental scale (Hauts-de-Seine). The visualization relies on a dilation-erosion of classified buildings to reveal continuous zones associated with learned communities. In light gray: majority community corresponding to the diffuse urban background. In color: secondary communities highlighted by the model. This figure aims to illustrate the spatial coherence induced by the topological constraint of the graph, without claiming a definitive morphological typology.