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A new edge betweenness measure using a game theoretical approach: an application to hierarchical community detection

Daniel Gómez, Javier Castro, Inmaculada Gutiérrez, Rosa Espínola

TL;DR

This work addresses hierarchical community detection by formalizing the Hierarchical Clustering Network Problem (HCNP) and proposing a divisive algorithm that uses a node-weighted shortest-path edge betweenness grounded in cooperative game theory. Node importance is captured with the Shapley value of a linear modularity game, producing a node-game SP betweenness that better discriminates edges for hierarchical splitting, and a fast quadratic-time variant is introduced for scalability. Empirical results on real networks and synthetic benchmarks show the method often yields more homogeneous, modular partitions and competitive or superior performance compared to established hierarchical clustering algorithms. The approach advances hierarchical clustering in networks by integrating principled power measures into edge betweenness and providing scalable implementations with rigorous evaluation criteria.

Abstract

In this paper we formally define the hierarchical clustering network problem (HCNP) as the problem to find a good hierarchical partition of a network. This new problem focuses on the dynamic process of the clustering rather than on the final picture of the clustering process. To address it, we introduce a new ierarchical clustering algorithm in networks, based on a new shortest path betweenness measure. To calculate it, the communication between each pair of nodes is weighed by he importance of the nodes that establish this communication. The weights or importance associated to each pair of nodes are calculated as the Shapley value of a game, named as the linear modularity game. This new measure, (the node-game shortest path betweenness measure), is used to obtain a hierarchical partition of the network by eliminating the link with the highest value. To evaluate the performance of our algorithm, we introduce several criteria that allow us to compare different dendrograms of a network from two point of view: modularity and homogeneity. Finally, we propose a faster algorithm based on a simplification of the node-game shortest path betweenness measure, whose order is quadratic on sparse networks. This fast version is competitive from a computational point of view with other hierarchical fast algorithms, and, in general, it provides better results.

A new edge betweenness measure using a game theoretical approach: an application to hierarchical community detection

TL;DR

This work addresses hierarchical community detection by formalizing the Hierarchical Clustering Network Problem (HCNP) and proposing a divisive algorithm that uses a node-weighted shortest-path edge betweenness grounded in cooperative game theory. Node importance is captured with the Shapley value of a linear modularity game, producing a node-game SP betweenness that better discriminates edges for hierarchical splitting, and a fast quadratic-time variant is introduced for scalability. Empirical results on real networks and synthetic benchmarks show the method often yields more homogeneous, modular partitions and competitive or superior performance compared to established hierarchical clustering algorithms. The approach advances hierarchical clustering in networks by integrating principled power measures into edge betweenness and providing scalable implementations with rigorous evaluation criteria.

Abstract

In this paper we formally define the hierarchical clustering network problem (HCNP) as the problem to find a good hierarchical partition of a network. This new problem focuses on the dynamic process of the clustering rather than on the final picture of the clustering process. To address it, we introduce a new ierarchical clustering algorithm in networks, based on a new shortest path betweenness measure. To calculate it, the communication between each pair of nodes is weighed by he importance of the nodes that establish this communication. The weights or importance associated to each pair of nodes are calculated as the Shapley value of a game, named as the linear modularity game. This new measure, (the node-game shortest path betweenness measure), is used to obtain a hierarchical partition of the network by eliminating the link with the highest value. To evaluate the performance of our algorithm, we introduce several criteria that allow us to compare different dendrograms of a network from two point of view: modularity and homogeneity. Finally, we propose a faster algorithm based on a simplification of the node-game shortest path betweenness measure, whose order is quadratic on sparse networks. This fast version is competitive from a computational point of view with other hierarchical fast algorithms, and, in general, it provides better results.
Paper Structure (13 sections, 3 theorems, 17 equations, 4 figures)

This paper contains 13 sections, 3 theorems, 17 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cooperative Game Theory
  4. Hierarchical Clustering for Graphs
  5. Modularity Function in Networks
  6. The SP Edge Betweenness Measure
  7. A New SP Edge Betweenness Measure Based on Game Theory
  8. Computational Complexity
  9. Reducing the Computational Complexity of the Node-Game Betweeness Measure
  10. Computational Results in Real Well-Known Examples
  11. Computational Results in Random Networks
  12. Discussion
  13. Final Remarks and Conclusions

Key Result

Proposition 1

Let $(N,v_{mod})$ denote the linear modularity game, let $i$ denote a node in $N$ and let $\Psi$ denote a semivalue with $\sum_{S\in S^k}{P(S)}= \sum_{S\in S^{(l_i-k)}}{P(S)}$. Then,

Figures (4)

  • Figure S1: Graph $G=(N,E)$.
  • Figure S2: Graph $G=(N,E)$ with 7 nodes, and the considered partition, $P=\{\{1,2,3,4\},\{5,6,7\}\}.$
  • Figure S3: Graph $G=(N,E)$ with adjacency matrix $A$. Toy example to illustrate $S^k_i$ calculation.
  • Figure S4: Chain $Ch=(N,E)$ of $12$ nodes.

Theorems & Definitions (17)

  • Definition 1: Shapley value Shapley
  • Definition 2: Semivalue Dubey1981
  • Definition 3: Feasible partition
  • Definition 4: Finer partition
  • Definition 5: Hierarchical partition
  • Example 1
  • Example 2
  • Definition 6: SP betweenness measure GirvanNewman:2002
  • Example 3
  • Example 4
  • ...and 7 more