k − core percolation on multiplex networks

We generalize the theory of $k\text{\ensuremath{-}}\mathrm{core}$ percolation on complex networks to $\mathbf{k}\text{\ensuremath{-}}\mathrm{core}$ percolation on multiplex networks, where $\mathbf{k}\ensuremath{\equiv}({k}_{1},{k}_{2},...,{k}_{M})$. Multiplex networks can be defined as networks with vertices of one kind but $M$ different types of edges, representing different types of interactions. For such networks, the $\mathbf{k}\text{\ensuremath{-}}\mathrm{core}$ is defined as the largest subgraph in which each vertex has at least ${k}_{i}$ edges of each type, $i=1,2,...,M$. We derive self-consistency equations to obtain the birth points of the $\mathbf{k}\text{\ensuremath{-}}\mathrm{cores}$ and their relative sizes for uncorrelated multiplex networks with an arbitrary degree distribution. To clarify our general results, we consider in detail multiplex networks with edges of two types and solve the equations in the particular case of Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi and scale-free multiplex networks. We find hybrid phase transitions at the emergence points of $\mathbf{k}\text{\ensuremath{-}}\mathrm{cores}$ except the $(1,1)$-core for which the transition is continuous. We apply the $\mathbf{k}\text{\ensuremath{-}}\mathrm{core}$ decomposition algorithm to air-transportation multiplex networks, composed of two layers, and obtain the size of $({k}_{1},{k}_{2})$-cores.