Hidden transition in multiplex networks.

Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, [Formula: see text]. Above a critical value of a control parameter, the removal of a tiny fraction [Formula: see text] of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discontinuity is not accompanied by any singularity of the giant component, in contrast to the discontinuous hybrid transition which usually appears in such problems. The control parameter is the fraction of nodes in each layer with a single connection, [Formula: see text]. We obtain asymptotic expressions for the collapse time and relaxation time, above and below the critical point [Formula: see text], respectively. In the limit [Formula: see text] the total collapse for [Formula: see text] takes a time [Formula: see text], while there is an exponential relaxation below [Formula: see text] with a relaxation time [Formula: see text].