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We describe the critical behavior of weak multiplex percolation, ageneralization of percolation to multiplex or interdependent networks. A nodecan determine its active or inactive status simply by referencing neighboringnodes. This is not the case for the more commonly studied generalization ofpercolation to multiplex networks, the mutually connected clusters, whichrequires an interconnecting path within each layer between any two vertices inthe giant mutually connected component. We study the emergence of a giantconnected component of active nodes under the weak percolation rule, findingseveral non-typical phenomena. In two layers, the giant component emerges witha continuos phase transition, but with quadratic growth above the criticalthreshold. In three or more layers, a discontinuous hybrid transition occurs,similar to that found in the giant mutually connected component. In networkswith asymptotically powerlaw degree distributions, defined by the decayexponent $\gamma$, the discontinuity vanishes but at $\gamma=1.5$ in threelayers, more generally at $\gamma = 1+ 1/(M-1)$ in $M$ layers.

Exotic Critical Behavior of Weak Multiplex Percolation