Empirical determination of the optimum attack for fragmentation of modular networks

All possible removals of $n=5$ nodes from networks of size $N=100$ are performed in order to find the optimal set of nodes which fragments the original network into the smallest largest connected component. The resulting attacks are ordered according to the size of the largest connected component and compared with the state of the art methods of network attacks. We chose attacks of size $5$ on relative small networks of size $100$ because the number of all possible attacks, ${100}\choose{5}$ $\approx 10^8$, is at the verge of the possible to compute with the available standard computers. Besides, we applied the procedure in a series of networks with controlled and varied modularity, comparing the resulting statistics with the effect of removing the same amount of vertices, according to the known most efficient disruption strategies, i.e., High Betweenness Adaptive attack (HBA), Collective Index attack (CI), and Modular Based Attack (MBA). Results show that modularity has an inverse relation with robustness, with $Q_c \approx 0.7$ being the critical value. For modularities lower than $Q_c$, all heuristic method gives mostly the same results than with random attacks, while for bigger $Q$, networks are less robust and highly vulnerable to malicious attacks.