Asymptotic Behavior of Attack Graph Games

This paper presents and analyzes an attack graph optimization problem that arises in modeling certain adversarial cyber attack and defend scenarios. The problem formulation is based on representing attacks againt a system as a finite, weighted, directed graph in which the directed edges represent transitions between states in an attack and edge weights represent the estimated cost to an attacker for traversing the edge. An attacker strives to traverse the graph from a specified start node to a specified end node using the least weight cost directed path between those nodes. On the other hand, the defender seeks to allocate defensive measures in such a way as to maximize the attacker’s minimal cost attack path. We study the role that minimal cut sets play in hardening the attack graph and prove that under this simple model minimal cut sets are optimal defensive investments in the limit even though minimal cut sets may not play a role in hardening a system initially. Viewing attackers and defenders as players in a two person, non-zero sum game, the results in this paper describe the asyptotic behavior of optimal solutions to the game under certain conditions.