The use of game theory to measure the vulnerability of stochastic networks

Abstract

Conventional approaches to network reliability analysis are based on either connectivity or capacity. This paper proposes an alternative method which seeks to identify those links or nodes whose failure would impair network performance the most. It is assumed that all links have two costs, a normal cost and a failed cost, both of which can be traffic-dependent. A 2-player, noncooperative, zero-sum game is envisaged between a router, seeking a least-cost path, and a virtual network tester, seeking to maximize trip-cost by failing 1 link. At the mixed strategy Nash equilibrium, link-use probabilities are optimal for the router, and link-failure probabilities are optimal for the tester. Finding the equilibrium involves solving a maximin programming problem. When link costs are fixed (not traffic-dependent), the maximin problem can be recast as a linear programming problem. Two forms of the linear programming problem are presented, one requiring path enumeration, and the other not. The interpretation of the primal and dual variables is elucidated by two propositions. Where link costs are traffic-dependent (e.g., where queuing is a feature), the mixed strategy Nash equilibrium can be found by the VISA (method of successive averages). A numerical example illustrates the approach on a stochastic network with queuing. While the example relates to single commodity e.g., where there are multiple origins and destinations.