# Absorbing Random Walks Interpolating Between Centrality Measures on Complex Networks

## TL;DR

This paper introduces a novel method using absorbing Markov chains to interpolate between shortest-path and non-restrictive centrality measures, revealing complex node importance dynamics in real networks.

## Contribution

It develops a continuous interpolation framework for betweenness and closeness centralities, connecting traditional and information-based measures through Markov chain modeling.

## Key findings

- Node centrality rankings change non-monotonically with the interpolation parameter.
- Nodes near community boundaries exhibit complex betweenness behaviors.
- The method uncovers nuanced importance shifts in real network structures.

## Abstract

Centrality, which quantifies the "importance" of individual nodes, is among the most essential concepts in modern network theory. As there are many ways in which a node can be important, many different centrality measures are in use. Here, we concentrate on versions of the common betweenness and it closeness centralities. The former measures the fraction of paths between pairs of nodes that go through a given node, while the latter measures an average inverse distance between a particular node and all other nodes. Both centralities only consider shortest paths (i.e., geodesics) between pairs of nodes. Here we develop a method, based on absorbing Markov chains, that enables us to continuously interpolate both of these centrality measures away from the geodesic limit and toward a limit where no restriction is placed on the length of the paths the walkers can explore. At this second limit, the interpolated betweenness and closeness centralities reduce, respectively, to the well-known it current betweenness and resistance closeness (information) centralities. The method is tested numerically on four real networks, revealing complex changes in node centrality rankings with respect to the value of the interpolation parameter. Non-monotonic betweenness behaviors are found to characterize nodes that lie close to inter-community boundaries in the studied networks.