Giant components in directed multiplex networks

We describe the complex global structure of giant components in directed multiplex networks which generalizes the well-known bow-tie structure, generic for ordinary directed networks. By definition, a directed multiplex network contains vertices of one type and directed edges of $m$ different types. In directed multiplex networks, we distinguish a set of different giant components based on the existence of directed paths of different types between their vertices, such that for each type of edges, the paths run entirely through only edges of that type. If, in particular, $m=2$, we define a strongly viable component as a set of vertices, in which for each type of edges, each two vertices are interconnected by at least two directed paths in both directions, running through the edges of only this type. We show that in this case, a directed multiplex network contains, in total, $9$ different giant components including the strongly viable component. In general, the total number of giant components is $3^m$. For uncorrelated directed multiplex networks, we obtain exactly the size and the emergence point of the strongly viable component and estimate the sizes of other giant components.