A Note on Computing Betweenness Centrality from the 2-core

TL;DR

This paper introduces a recursive method to compute betweenness centrality from a graph's 2-core and analyzes how removing degree-one nodes affects estimation accuracy and computational efficiency.

Contribution

It provides a novel recursive formula for betweenness centrality based on the 2-core and analyzes the impact of degree-one node removal on estimation accuracy and runtime.

Findings

Removing degree-1 nodes reduces sample complexity.

The recursive formula simplifies betweenness centrality computation.

Empirical results show decreased runtime with node removal.

Abstract

A central task in network analysis is to identify important nodes in a graph. Betweenness centrality (BC) is a popular centrality measure that captures the significance of nodes based on the number of shortest paths each node intersects with. In this note, we derive a recursive formula to compute the betweenness centralities of a graph from the betweenness centralities of its 2-core.Furthermore, we analyze mathematically the significant impact of removing degree-one nodes on the estimation of betweenness centrality within the context of the popular pivot sampling scheme for Single-Source Shortest Path (SSSP) computations, as described in the Brandes-Pich approach and implemented in widely used software such as NetworkX. We demonstrate both theoretically and empirically that removing degree-1 nodes can reduce the sample complexity needed to achieve better accuracy, thereby decreasing the overall runtime.