Differentiable Tripartite Modularity for Clustering Heterogeneous Graphs

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.

💡 Research Summary

The paper tackles the problem of unsupervised clustering on heterogeneous relational data where interactions involve three distinct entity types. Existing differentiable modularity approaches such as DMoN have proven effective for homogeneous and bipartite graphs, but they do not scale to higher‑order structures without incurring prohibitive computational costs. To fill this gap, the authors propose a differentiable formulation of tripartite modularity specifically designed for graphs composed of three node sets—addresses (X), buildings (Y), and cadastral parcels (Z)—that interact exclusively through the pivot set Y (building nodes).

The core idea is to define community quality in terms of weighted co‑paths (X → Y → Z) rather than explicit three‑way tensors. A co‑path density tensor A(i,j,k) is introduced mathematically, but the authors never instantiate it. Instead, they normalize each co‑path by the product of the weighted degrees of the pivot node (degX(j)·degZ(j))⁻¹, which they call the flow‑normalization factor ωj. This factor mitigates the well‑known bias of power‑law degree distributions that would otherwise dominate the modularity objective. By expressing the tripartite contribution as a sum over pivot nodes and factorizing the terms into upstream (addresses) and downstream (parcels) aggregations, the computation reduces to O(|EXY|+|EYZ|), i.e., linear in the number of edges.

To make the objective differentiable, each node is assigned a soft community membership vector: SX for addresses, SY for buildings, and SZ for parcels. The tripartite modularity term becomes a product of three components: a soft matching probability αm,l between address community l and building community m, a soft matching probability γm,n between building community m and parcel community n, and the normalized co‑path density elmn. The soft matchings are derived from bipartite confusion matrices EXY and EYZ and are controlled by a temperature parameter β. When β → ∞ the soft match converges to the hard one‑to‑one correspondence of Murata’s classic formulation; when β → 0 it yields uniform mixing. This temperature schedule allows gradients to flow through many candidate correspondences early in training and gradually sharpens the assignment, stabilizing convergence.

The overall loss combines the negative soft tripartite modularity (to be maximized) with a collapse‑regularization term applied to each node type, weighted by λX, λY, λZ. The regularizer discourages degenerate solutions where all nodes collapse into a single community.

The methodological pipeline, dubbed DMoN‑3p, integrates the differentiable tripartite modularity with a graph neural network encoder. The GNN produces node embeddings that are fed into linear layers to predict the soft assignment matrices SX, SY, SZ. These matrices are then used in the factorized aggregation (Equations 6‑7) to compute elmn efficiently. Gradients from the loss propagate back through the factorized computation to the GNN, enabling end‑to‑end learning.

For empirical validation, the authors construct a large‑scale urban cadastral graph from French public registries: the address base (BAN), the building repository (RNB), and the cadastral parcels. Edges between parcels are weighted by the logarithm of shared boundary length, which compresses extreme values while preserving ordering. Building‑building adjacency is obtained by projecting parcel adjacency through the building‑parcel incidence matrix, ensuring that only physically contiguous parcels induce building connections. This topological construction respects real‑world constraints such as roads and property boundaries, unlike conventional k‑NN graphs that create spurious edges across physical barriers.

Experiments on this dataset demonstrate several key findings: (1) the loss converges rapidly and stably, with the temperature β schedule proving essential for avoiding early‑stage oscillations; (2) the flow‑normalization factor ωj prevents high‑degree pivot buildings from monopolizing the modularity score, as confirmed by an ablation study where removing ωj leads to partition collapse; (3) the resulting clusters are spatially coherent, aligning with natural urban blocks and cadastral units, and outperform baseline methods (standard DMoN on bipartite projections and hypergraph‑based non‑differentiable approaches) in modularity score and qualitative visual inspection.

In summary, the paper makes three major contributions: (i) a mathematically sound, differentiable definition of tripartite modularity based on co‑paths; (ii) an exact factorized computation that avoids dense third‑order tensors and runs in linear time; (iii) an end‑to‑end learning framework (DMoN‑3p) that couples this objective with a GNN, validated on a realistic, large‑scale heterogeneous graph. The approach is broadly applicable to any domain where three‑way mediated relationships exist—urban planning, multimodal transportation networks, biological multi‑omics, etc.—and opens avenues for extending differentiable modularity to even higher‑order heterogeneous structures.