Betweenness Centrality in Large Complex Networks

TL;DR

This paper investigates how betweenness centrality scales in large complex networks, revealing relationships between network structure, loop density, and degree distribution, with specific bounds on the distribution exponents.

Contribution

It provides a theoretical analysis of betweenness centrality in different network topologies, establishing bounds and exact relations for scale-free networks and trees.

Findings

Betweenness centrality follows a power law with an exponent η that varies with loop density.

In scale-free networks, the betweenness distribution exponent δ is bounded below by (γ+1)/2.

For tree networks, δ equals (γ+1)/2, establishing a precise relation.

Abstract

We analyze the betweenness centrality (BC) of nodes in large complex networks. In general, the BC is increasing with connectivity as a power law with an exponent $\eta$. We find that for trees or networks with a small loop density $\eta=2$ while a larger density of loops leads to $\eta<2$. For scale-free networks characterized by an exponent $\gamma$ which describes the connectivity distribution decay, the BC is also distributed according to a power law with a non universal exponent $\delta$. We show that this exponent $\delta$ must satisfy the exact bound $\delta\geq (\gamma+1)/2$. If the scale free network is a tree, then we have the equality $\delta=(\gamma+1)/2$.