[Paper Review] Co-Betweenness: A Pairwise Notion of Centrality

Eric D. Kolaczyk, David B. Chua+1 more|ArXiv.org|Sep 21, 2007

Complex Network Analysis Techniques3 references3 citations

TL;DR

This paper introduces co-betweenness, a pairwise centrality measure that quantifies how jointly two vertices control information flow along shortest paths in a network. Extending Brandes' algorithm, it enables efficient computation of pairwise influence, revealing that non-central nodes can play critical relay roles in communication networks.

ABSTRACT

Betweenness centrality is a metric that seeks to quantify a sense of the importance of a vertex in a network graph in terms of its "control" on the distribution of information along geodesic paths throughout that network. This quantity however does not capture how different vertices participate together in such control. In order to allow for the uncovering of finer details in this regard, we introduce here an extension of betweenness centrality to pairs of vertices, which we term co-betweenness, that provides the basis for quantifying various analogous pairwise notions of importance and control. More specifically, we motivate and define a precise notion of co-betweenness, we present an efficient algorithm for its computation, extending the algorithm of Brandes in a natural manner, and we illustrate the utilization of this co-betweenness on a handful of different communication networks. From these real-world examples, we show that the co-betweenness allows one to identify certain vertices which are not the most central vertices but which, nevertheless, act as important actors in the relaying and dispatching of information in the network.

Motivation & Objective

• To address the limitation of traditional betweenness centrality, which evaluates vertices in isolation and fails to capture cooperative roles in information dissemination.
• To develop a formal, computationally efficient measure of pairwise vertex influence in network communication paths.
• To reveal how pairs of vertices collectively control information flow, especially identifying non-central but strategically important relay nodes.
• To extend Brandes' single-vertex betweenness algorithm to compute co-betweenness for all vertex pairs in a network.
• To validate the utility of co-betweenness in real-world communication networks, demonstrating its ability to uncover nuanced structural roles beyond individual centrality.

Proposed method

• Proposes co-betweenness as a measure of the fraction of shortest paths between all vertex pairs (s,t) that pass through both u and v, defined as δ_st(u,v) = σ_st(u,v)/σ_st.
• Derives a recursive decomposition of co-betweenness using dependencies δ_s(u,v) = δ_s(v)/σ_sv × σ_sv(u), enabling efficient computation.
• Adapts Brandes' single-vertex dependency computation to compute pairwise dependencies δ_s(u,v) by propagating contributions from source s through the shortest path tree.
• Employs a three-stage algorithm: (1) BFS from each vertex s to compute shortest path counts and predecessors, (2) compute dependency scores δ_s(v) and δ_s(u,v), (3) aggregate co-betweenness values across all s.
• Uses depth-first traversal to compute contributions along all shortest paths, ensuring each path is processed once with O(σ log n_v) complexity in small-world networks.
• Normalizes co-betweenness to produce correlation-like measures and conditional control metrics, enabling interpretation of joint influence and control dynamics.

Experimental results

Research questions

• RQ1 How can we quantify the joint control of information flow by pairs of vertices in a network, beyond individual vertex centrality?
• RQ2 What is an efficient algorithmic approach to compute co-betweenness for all vertex pairs in a network, extending existing betweenness centrality methods?
• RQ3 Which vertices act as critical relay pairs in real-world communication networks, even if individually not the most central?
• RQ4 How does co-betweenness reveal structural roles that are missed by standard betweenness centrality?
• RQ5 What is the computational complexity of co-betweenness computation, and how does it scale with network size and structure?

Key findings

• Co-betweenness successfully identifies vertex pairs that act as key information relays even when neither vertex has high individual betweenness centrality.
• The algorithm computes co-betweenness in O(n_v n_e + σ log n_v) time, where σ is the total number of shortest paths, making it feasible for large-scale networks.
• Empirical results on communication networks show that co-betweenness reveals cooperative control patterns not detectable via standard betweenness centrality.
• The method enables a standardized measure of correlation between flows over two vertices, providing interpretability beyond raw counts.
• In sparse networks, the algorithm scales as O(n_v^{2+p} log n_v), with p ranging from 0.19 to 0.62 depending on network model, indicating practical efficiency for real-world graphs.
• The recursive dependency formulation allows reuse of Brandes’ framework, ensuring computational efficiency while extending it to pairwise interactions.