Line graphs for a multiplex network

It is well known that line graphs offer a good summary of the graphs properties, which make them easier to analyze and highlight the desired properties. We extend the concept of line graph to multiplex networks in order to analyze multi-plexed and multi-layered networked systems. As these structures are very rich, different approaches to this notion are required to capture a variety of situations. Some relationships between these approaches are established. Finally, by means of some simulations, the potential utility of this concept is illustrated.
Complex networks are a valuable tool to study several systems, where nodes represent the system’s constituents and edges account for the interactions between them.1–5 Traditionally, they are employed to model many biological, social, and technological systems such as the World Wide Web, Internet, social networks, protein, metabolic and neural networks, and many others.1,2,6,7 But such systems may include multiple subsystems with many interdependent components in such a way that their components may interact through many different channels (layers). So, the development of this science is providing radical new ways of understanding many different mechanisms and processes. New paradigms, such as interacting or interdependent networks, networks of networks, multiplex networks, multilayer networks, and some others,8–13 have been introduced to provide a solid foundation together with the subsequently developed tools, in order to study multicomponent and multilayer systems in a comprehensive fashion. Consequently, it is necessary to extend some concepts and their applications to this new environment where the interaction of nodes through multiple layers of links cannot be captured by the classical single-layer network representation. In this regard, although some results concerning multiplex networks’ modeling and structure have been recently proposed,8–10,14–22 the study of the line graph associated to a multiplex or a multilayer structure has not yet been addressed. The aim of this paper is to propose different approaches to the definition of the line graph associated to a multiplex network and to illustrate potential applications.