Avoidance Markov Metrics and Node Pivotality Ranking

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.