How Does the Internal Structure of a Complex System Influence Its Overall Risk? Risk Minimization for Trees

Abstract. The Defender–Attacker problem is often employed as a mathematical framework in risk management. In this problem, the above players with opposite goals allocate limited resources to system elements to minimize or maximize a risk function. It has been well-studied under the assumption of independent system elements. However, in complex systems, elements interact, causing significant differences between the measured and predicted risks. Although models with the interdependence of system elements are regularly considered in the literature, no comprehensive understanding has been formed of how the structure of a complex system influences its overall risk. We address this issue in a series of papers by investigating system structures of increasing complexity. Chains and stars have been analyzed previously; in this paper, the findings are extended to arbitrary trees. We optimize the placement of elements within a tree to minimize risk; derive upper bounds for the relative error of an approximate algorithmic solution of this problem for trees with a few branches and leaves; and explore the dynamics of these bounds when increasing the number of leaves and branches. As demonstrated, the resulting upper bounds do not exceed their counterparts for stars from the previous works.

Keywords: complex systems, risk, system structure, risk management, risk minimization algorithms, the problem of optimal element placement.

PDF (English)

Cite this paper

Shiroky, A.A., Kalashnikov, A.O., How Does the Internal Structure of a Complex System Influence Its Overall Risk? Risk Minimization for Trees. Control Sciences 2, 22–30 (2025).