Giant components in directed multiplex networks

TL;DR

This paper analyzes the complex structure of giant components in directed multiplex networks, extending the classic bow-tie model to multiple edge types and identifying the number and size of these components.

Contribution

It introduces a generalized framework for giant components in directed multiplex networks, including the concept of strongly viable components and their enumeration.

Findings

Total of 9 giant components for 2 types of edges

Number of giant components is 3^m for m edge types

Exact size and emergence point of strongly viable component

Abstract

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.