Empirical determination of the optimal attack for fragmentation of modular networks

We perform all possible removals of n nodes from networks of size N, then we identify and measure the largest connected component left in every case. The smallest of these components represents the maximum possible damage (on a network of N vertices), limited to the removal of n nodes, and the set that produces such damage is called the optimal set of size n. We apply the procedure in a series of networks with controlled and varied modularity. Then, we compare the resulting statistics with the effect of removing the same amount of vertices according to state of the art methods of network fragmentation, i.e., High Betweenness Adaptive attack, Collective Influence, and Module-Based Attack. For practical matters we performed mainly attacks of size n=5 on networks of size N=100, because the number of all possible sets (≈108) is at the verge of the computational capability of standard desktops. The results show, in general, that the resilience of networks to attacks has an inverse relationship with modularity, with Qc≈0.73 being the critical value, from which the damage of the optimal attack increases rapidly. Networks are highly vulnerable to targeted attacks when the modularity is greater than the critical value of each heuristic method. On the other hand, for modularities lower than Qc, all the heuristic strategies studied have a similar performance to a random attack.