Multi-Relational Structural Entropy

TL;DR

The paper addresses the limitation of Structural Entropy (SE) in handling heterogeneous relation types by introducing Multi-relational Structural Entropy (MrSE). It builds on a random surfing interpretation to define MrSE on a multi-relational graph with an adjacency tensor and dual stationary distributions, and proposes a greedy 2D MrSE minimization to reveal communities. Empirical results on synthetic and real-world networks show that MrSE provides sharper decoding of multi-relational structure and boosts performance in tasks like multi-relational node clustering and social event detection. This yields a principled, scalable tool for analyzing and interpreting complex networks with heterogeneous relations.

Abstract

Structural Entropy (SE) measures the structural information contained in a graph. Minimizing or maximizing SE helps to reveal or obscure the intrinsic structural patterns underlying graphs in an interpretable manner, finding applications in various tasks driven by networked data. However, SE ignores the heterogeneity inherent in the graph relations, which is ubiquitous in modern networks. In this work, we extend SE to consider heterogeneous relations and propose the first metric for multi-relational graph structural information, namely, Multi-relational Structural Entropy (MrSE). To this end, we first cast SE through the novel lens of the stationary distribution from random surfing, which readily extends to multi-relational networks by considering the choices of both nodes and relation types simultaneously at each step. The resulting MrSE is then optimized by a new greedy algorithm to reveal the essential structures within a multi-relational network. Experimental results highlight that the proposed MrSE offers a more insightful interpretation of the structure of multi-relational graphs compared to SE. Additionally, it enhances the performance of two tasks that involve real-world multi-relational graphs, including node clustering and social event detection.

Figures (5)

• Figure 1: Decode the essential structures of a multi-relational graph with MrSE (ours) and SE. (a) is a multi-relational graph $G'$. (b) shows the essential structures of $G'$, decoded with MrSE. (c) is a single-relational graph $G$ reduced from $G'$. (d) shows the essential structures of $G$, decoded with SE.
• Figure 2: The 1D and 2D SE and RSSE of single-relational graphs with varying sizes (a) and sparsities (b).
• Figure 3: The $\Delta$SE, $\Delta$RSSE, and $\Delta$MrSE of multi-relational graphs with varying sizes (a), the total number of relations (b), and sparsities (c).
• Figure 4: Examples of single-relational graph, encoding tree, and 2D SE minimization. (a) is a single-relational graph $G$. (b) is the encoding tree of height 1, which encodes the 1st-order structures, i.e., nodes, in $G$. (c) - (e) demonstrate how 2D SE minimization detects the 2nd-order structures, i.e., communities, in $G$. Initially, each node in $G$ is assigned to its own cluster. $\mathcal{P}$ in (c) shows the initial clusters. Following the vanilla greedy 2D SE minimization algorithm li2016structural, at each step, any two clusters that would reduce SE the most are merged. Eventually, the optimal encoding tree of height 2, as shown in (e), is associated with the minimum possible SE value, and encodes the communities, in $G$. $\mathcal{P}$ in (e) shows the detected communities.
• Figure 5: Examples of multi-relational graph, encoding tree, and 2D MrSE minimization. (a) is a multi-relational graph $G'$. (b) is the adjacency tensor of $G'$. (c) is the encoding tree of height 1, which encodes the 1st-order structures, i.e., nodes, in $G'$. (d) - (f) demonstrate how 2D MrSE minimization detects the 2nd-order structures, i.e., communities, in $G'$. Initially, each node in $G'$ is assigned to its own cluster. $\mathcal{P}$ in (d) shows the initial clusters. Following our proposed 2D MrSE minimization algorithm (Algorithm \ref{['algorithm:2D_MrSE']}), at each step, any two clusters that would reduce MrSE the most are merged. Eventually, the optimal encoding tree of height 2, as shown in (f), is associated with the minimum possible MrSE value, and encodes the communities, in $G'$. $\mathcal{P}$ in (f) shows the detected communities.

Theorems & Definitions (8)

• Definition 2.1
• Definition 2.2
• Proposition 3.1
• proof
• Definition 3.2
• Proposition 3.3
• proof
• Definition 3.4