Defence and attack of complex interdependent systems

Abstract

A defender defends and an attacker attacks each of n interdependent targets which can operate or fail and be in 2ⁿ possible states. Interdependence is modelled from each target to each other target. Despite such systems usually being analysed numerically, this paper succeeds in determining analytical solutions, accounting for unit effort costs, target values, and contest intensities. Increasing interdependence from some targets to some other targets induce both players to exert higher efforts into the former targets. For 100% interdependent targets, the attacker encounters a substitution effect. In contrast, for independent targets the defender encounters a substitution effect, defending the essential targets. For similarly advantaged players, increased target contest intensities cause higher efforts and lower expected utilities. Both players may withdraw in both 100% interdependent and independent systems. Applying the model to the US economy, we illustrate how interdependence between petroleum refineries, oil and gas extraction and air transportation impact defence, attack and expected utilities.

Keywords:

• Risk analysis
• survivability
• interdependence
• network
• multi-state system

Acknowledgment

We thank two anonymous referees of this journal for useful comments.