Fragility and anomalous susceptibility of weakly interacting networks.

Percolation is a fundamental concept that has brought new understanding of the robustness properties of complex systems. Here we consider percolation on weakly interacting networks, that is, network layers coupled together by much fewer interlinks than the connections within each layer. For these kinds of structures, both continuous and abrupt phase transitions are observed in the size of the giant component. The continuous (second-order) transition corresponds to the formation of a giant cluster inside one layer and has a well-defined percolation threshold. The abrupt transition instead corresponds to the merger of coexisting giant clusters among different layers and is characterized by a remarkable uncertainty in the percolation threshold, which in turns causes an anomalous behavior of the observed susceptibility. We develop a simple mathematical model able to describe this phenomenon, using a susceptibility measure that defines the range where the abrupt transition is more likely to occur. Finite-size scaling analysis in the abrupt region supports the hypothesis of a genuine first-order phase transition.