Uncertainty in vulnerability of networks under attack.

This study builds conceptual explanations and empirical examinations of the vulnerability response of networks under attack. Two quantities of "vulnerability" and "uncertainty in vulnerability" are defined by scrutinizing the performance loss trajectory of networks experiencing attacks. Both vulnerability and uncertainty in vulnerability quantities are a function of the network topology and size. This is tested on 16 distinct topologies appearing in infrastructure, social, and biological networks with 8 to 26 nodes under two percolation scenarios exemplifying benign and malicious attacks. The findings imply (i) crossing path, tree, and diverging tail are the most vulnerable topologies, (ii) complete and matching pairs are the least vulnerable topologies, (iii) complete grid and complete topologies show the most uncertainty for vulnerability, and (iv) hub-and-spoke and double u exhibit the least uncertainty in vulnerability. The findings also imply that both vulnerability and uncertainty in vulnerability increase with an increase in the size of the network. It is argued that in networks with no undirected cycle and one undirected cycle, the uncertainty in vulnerability is maximal earlier in the percolation process. With an increase in the number of cycles, the uncertainty in vulnerability is accumulated at the end of the percolation process. This emphasizes the role of tailoring preparedness, response, and recovery phases for networks with different topologies when they might experience disruption.