# Search for an Immobile Hider on a Stochastic Network

## TL;DR

This paper models a stochastic pursuit-evasion game on a graph where the hider is immobile and edges randomly activate, deriving bounds and strategies for optimal hiding and searching in complex network structures.

## Contribution

It introduces a stochastic game framework for an immobile hider on a graph with random edge activation, extending deterministic strategies to probabilistic settings.

## Key findings

- The game has a well-defined value.
- Bounds for the game's value are established.
- Refined strategies are provided for trees and Eulerian graphs.

## Abstract

Harry hides on an edge of a graph and does not move from there. Sally, starting from a known origin, tries to find him as soon as she can. Harry's goal is to be found as late as possible. At any given time, each edge of the graph is either active or inactive, independently of the other edges, with a known probability of being active. This situation can be modeled as a zero-sum two-person stochastic game. We show that the game has a value and we provide upper and lower bounds for this value. Finally, by generalizing optimal strategies of the deterministic case, we provide more refined results for trees and Eulerian graphs.