Scaling of random walk betweenness in networks

TL;DR

This paper investigates how random walk betweenness centrality scales in different network models, revealing model-dependent scaling behaviors and highlighting the usefulness of a normalized measure that simplifies analysis.

Contribution

It introduces a normalized random walk betweenness measure and demonstrates its simpler scaling behavior across network models, including preferential attachment graphs.

Findings

Scaling collapse with no adjustable parameters is achieved.

Normalized random betweenness simplifies the scaling analysis.

Probability of passing through the root node tends to one in large preferential attachment graphs.

Abstract

The betweenness centrality of graphs using random walk paths instead of geodesics is studied. A scaling collapse with no adjustable parameters is obtained as the graph size $N$ is varied; the scaling curve depends on the graph model. A normalized random betweenness, that counts each walk passing through a node only once, is also defined. It is argued to be more useful and seen to have simpler scaling behavior. In particular, the probability for a random walk on a preferential attachment graph to pass through the root node is found to tend to unity as $N\rightarrow\infty.$