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.

翻译：异构关系数据的聚类仍然是图学习中的核心挑战，尤其在交互涉及两种以上实体类型时。虽然诸如DMoN等可微分模块度目标已实现同质图和二分图的端到端社区发现，但将这些方法扩展到高阶关系结构仍非易事。本研究针对由三种节点类型通过中介交互连接的图，提出了一种可微分的三方模块度形式化方法。社区结构通过跨三方图的加权共路径定义，并采用精确因子分解计算以避免显式构建稠密三阶张量。引入枢轴节点的结构归一化以控制极端度异质性并确保优化稳定性。所得目标可与图神经网络以端到端方式联合优化，同时保持边数量级的线性复杂度。我们在大规模城市地籍数据上验证了所提框架，其表现出稳健的收敛行为并生成空间连贯的划分结果。这些发现凸显了可微分三方模块度作为异构图无监督聚类的通用方法构建模块的价值。