  Network dismantling aims at breaking a network into disconnected components,
and attacking vertices that intersect with many loops has proven to be a most
efficient strategy. But the existing loop-focusing methods treat the short
loops within densely connected local clusters (e.g., cliques) as equally
important as the long loops connecting different clusters. Here we propose for
highly clustered artificial and real-world networks a bipartite factor-graph
formulation that retains all the long loops while simplifies the local dense
clusters as individual factor nodes. We develop a mean-field theory for the
associated long-loop feedback vertex set problem and apply its message-passing
equation as a solver for network dismantling. The proposed factor-graph loop
algorithm outperforms the current state-of-the-art graph loop algorithms by a
considerable margin on various real networks. Further improvement in
dismantling performance is achievable by optimizing the choice of the local
dense clusters.
