Fragility and anomalous susceptibility of weakly interacting networks

Percolation is a fundamental concept that brought new understanding on the robustness properties of complex systems. Here we consider percolation on weakly interacting networks, that is, network layers coupled together by much less interlinks than the connections within each layer. For these kinds of structures, both continuous and abrupt phase transition 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 characterised by a remarkable uncertainty in the percolation threshold, which in turns causes an anomalous trend in the observed susceptibility. We develop a simple mathematical model able to describe this phenomenon and to estimate the critical threshold for which the abrupt transition is more likely to occur. Remarkably, finite-size scaling analysis in the abrupt region supports the hypothesis of a genuine first-order phase transition.