Avoidance Markov Metrics and Node Pivotality Ranking

TL;DR

This paper introduces avoidance Markov metrics for flexible modeling of random walks that avoid certain nodes, enabling more nuanced pivotality ranking of nodes in network reachability beyond classical methods.

Contribution

It develops new avoidance Markov metrics and theories to assess node pivotality in network reachability, enhancing traditional models with greater flexibility and detail.

Findings

Metrics effectively rank nodes based on their pivotality in reachability.

Synthetic and real-world examples demonstrate the metrics' practical utility.

The approach provides deeper insights into node importance in network connectivity.

Abstract

We introduce the avoidance Markov metrics and theories which provide more flexibility in the design of random walk and impose new conditions on the walk to avoid (or transit) a specific node (or a set of nodes) before the stopping criteria. These theories help with applications that cannot be modeled by classical Markov chains and require more flexibility and intricacy in their modeling. Specifically, we use them for the pivotality ranking of the nodes in a network reachabilities. More often than not, it is not sufficient simply to know whether a source node $s$ can reach a target node $t$ in the network and additional information associated with reachability, such as how long or how many possible ways node $s$ may take to reach node $t$, is required. In this paper, we analyze the pivotality of the nodes which capture how pivotal a role that a node $k$ or a subset of nodes $S$ may play in the reachability from node $s$ to node $t$ in a given network. Through some synthetic and real-world network examples, we show that these metrics build a powerful ranking tool for the nodes based on their pivotality in the reachability.