Optimizing Robustness against Cascading Failures under Max-Load Targeted Attack

Motivated by reliability of networks in critical infrastructures, we consider optimal robustness of a class of flow networks against a \textit{targeted} attack, namely max-load targeted attack, that triggers cascading failures due to removal of largest load carrying portion of lines. The setup involves a network of $N$ lines with initial loads $L_1, \ldots, L_N$, drawn from independent and identical uniform distribution, and free-spaces or redundancies $S_1, \ldots, S_N$ to be allocated. In the failure propagation mechanism, a line fails initially due to attack and later due to overloading. The load that was carried at the moment of failing gets redistributed equally among all remaining lines in the system. We analyze robustness of this network against the max-load targeted attack that removes the largest load carrying $p$-fraction of the lines from the system. The system designer allocates $S_i$ as a stochastic function of the load in each line. Assuming an average available resource budget, we show that allocating all lines the free-spaces equally among nodes is optimal under some regulatory assumptions. We provide numerical results verifying that equal free-space allocation to all lines perform optimally in more general targeted attack scenarios.