Betweenness Central Nodes Under Uncertainty: An Absorbing Markov Chain Approach

TL;DR

This paper introduces a novel betweenness centrality measure for stochastic networks using an absorbing Markov chain model, enabling analysis of node importance under uncertainty and robustness to transition kernel perturbations.

Contribution

It develops a new centrality measure based on absorbing Markov chains for networks with uncertain edges, including algorithms and robustness analysis.

Findings

Identifies a small set of dominant nodes in stochastic networks.

Reveals stable and sensitive node rankings under perturbations.

Supports extensions to weighted rewards and constrained candidate sets.

Abstract

We propose a betweenness centrality measure and algorithms for stochastic networks, where edges can fail and weights vary across realizations, making the most central node random. Our approach models the sequence of reported central nodes as an absorbing Markov chain and measures node importance by the share of pre-absorption time spent at each node. This produces a way to study centrality under uncertainty, which can then be estimated with Monte Carlo simulation. We also analyze robustness when the transition kernel is only approximately known, using row-wise perturbations to assess sensitivity and potential ranking changes. The framework further admits extensions to weighted rewards and restricted candidate sets without altering the Markov chain formulation. Experiments on Erd\H{o}s-R\'enyi, Watts-Strogatz, and Les Mis\'erables networks with stochastic edges show that the method identifies a small set of dominant nodes, reveals stable versus sensitive rankings under perturbations, and supports reward-based and structure-constrained variants.