Enhancing Controllability Robustness of -Snapback Networks through Redirecting Edges

Figure 4

Density of driver-nodes (), as a function of the proportion of the removed nodes (). Initially, when no node has been removed (), both networks A and C require only 1 driver-node, while network B requires 2. When , meaning that 3 out of 4 nodes are removed, each network has only one node remaining, so the required number of driver nodes is also 1 for each network. The ordinal ranks are calculated and marked at each value, where represents the rank of network (, or ) at step of an attack, . After averaging over all 4 points, network A has an average rank , network B has , and network C has . Here, represents the number of winnings. Network A wins the first rank for 4 values, network B wins only once when , and network C wins 3 times.