Abstract
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, P(0),=,0. Above a critical value of a control parameter, the removal of a tiny fraction Delta 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, Pi ,=,P(1). We obtain asymptotic expressions for the collapse time and relaxation time, above and below the critical point Pi _c, respectively. In the limit Delta rightarrow 0 the total collapse for Pi ,>,Pi _text {c} takes a time T propto 1/(Pi -Pi _text {c}), while there is an exponential relaxation below Pi _text {c} with a relaxation time tau propto 1/[Pi _text {c}-Pi ].
Highlights
Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types of edges
There are two alternative definitions generalizing the notion of cluster in multiplex percolation: the (i) mutually connected cluster, which relies in a global criterion, and the (ii) weakly percolating cluster which is defined through a local, more permissive, r ule[15,16]
The control parameter is the fraction of nodes of degree 1; the critical point does not depend on the rest of the degree distribution
Summary
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. Despite the different characters of these two definitions of cluster, they have many similar effects in the percolation transition Both generalizations of the percolation problem for multilayer networks typically display hybrid phase transitions, with characteristics from both continuous and discontinuous transitions[1,16,18], of the same type as observed in k-core percolation[19]. This jump is followed by a critical square root singularity in the ordered phase, associated with a diverging susceptibility (defined as mean avalanche size induced by the removal of a random node or edge) as in continuous transitions, and diverging relaxation times[24] This hybrid character of the transition is typically found in networks with two or more layers in the case of the stronger definition of cluster (i), while with the weaker definition (ii) at least three layers are required, otherwise the transition is continuous.
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