Giant components in directed multiplex networks

We describe the complex global structure of giant components in directed multiplex networks that 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 nine 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.

Figure 1

Structure of giant components in (a) ordinary directed networks and (b) multiplex directed networks.

Figure 2

Schematic representation of the self-consistency equations for the probabilities (a) $x_A$ and (b) $y_A$, with $m=2$. The solid black and dashed blue lines with infinity symbols at one of their ends represent probabilities $x_A,y_A$ and $x_B,y_B$ respectively. The right-hand sides of these equations indicate the full sets of configurations contributing to the corresponding probabilities. For the sake of clarity, we do not show the edges leading to finite components.

Figure 3

Schematic representation of a vertex belonging to the giant strongly viable component.

Figure 4

Schematic representation of the probability that a vertex belongs to the giant viable components (a) II, (b) OO, (c) IO, (d) OI, (e) IS, (f) OS, (g) SI, and (h) SO in the case of directed multiplex networks with edges of two colors $A$ and $B$. For the analytical representation, see Eqs. (4)–(11).

Figure 5

Relative size of the giant strongly viable component SS of directed multiplex Erdős-Rényi networks with two types of edges vs mean in- and out-degrees $c_A=c_B\equiv c$ of the networks. The W curve shows the dependence of the giant weakly viable component on $c$. This component emerges at $c=2.4554.../2$.

Figure 6

Lower limit estimation of the sizes of different giant viable components compared with the size of the giant strongly component.

Figure 7

Value of the discontinuity of the giant strongly viable component vs the ratio $c_A/c_B$ of the mean in- and out-degrees of networks $A$ and $B$.

Figure 8

Line on the plane $(c_A,c_B)$, which separates the phase with the giant strongly viable component from the normal phase.

Figure 9

Schematic representation of the probability that an edge of type $A$ belongs to the giant strongly viable component.