Title: Empirical determination of the optimum attack for fragmentation of modular networks,
Abstract: 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.