Identifying optimal targets of network attack by belief propagation

For a network formed by nodes and undirected links between pairs of nodes, the network optimal attack problem aims at deleting a minimum number of target nodes to break the network down into many small components. This problem is intrinsically related to the feedback vertex set problem that was successfully tackled by spin-glass theory and an associated belief propagation-guided decimation (BPD) algorithm [Zhou, Eur. Phys. J. B 86, 455 (2013)]. In the present work we apply the BPD algorithm (which has approximately linear time complexity) to the network optimal attack problem and demonstrate that it has much better performance than a recently proposed collective information algorithm [Morone and Makse, Nature 524, 65 (2015)] for different types of random networks and real-world network instances. The BPD-guided attack scheme often induces an abrupt collapse of the whole network, which may make it very difficult to defend.