E. Estrada, New Journal of Physics 21 (2019) 05004

We give a formal definition of a multiplex network and using its supra-adjacency matrix representation we construct the multiplex communicability matrix. Then we prove that the communicability function naturally induces an embedding of the multiplexes in a hyperspherical Euclidean space. We then study (i) intra-layer, (ii) inter-layer, and (iii) inter-layer self-communicability distance and angles in multiplex networks. Using these multiplex metrics we study a social multiplex related to an office politics and the multiplex of synaptic interactions between neurons in the worm C. elegans. We find that the average communicability angles in these multiplexes exhibits a minimum for certain value of the interlayer coupling strength. We provide an explanation for this phenomenon which emerges from the multiplexity of these systems and related it to other important phenomena like the synchronizability of these systems. Finally, we define and study communicability shortest paths in the multiplexes. We show how the communicability shortest paths avoid the most central nodes in the multiplexes in terms of their degree and betweenness, which is a main difference with (topological) shortest paths. We explain this behavior in terms of a diffusive model in which the 'information' not only diffuses between the nodes but it is also processed internally on the entities of the complex system. Finally, we give some new ideas on how to extend the current work and represent complex systems as 'multiplex hypergraphs' and 'multi-simplicial complexes'.

We give a formal definition of a multiplex network and using its supra-adjacency matrix representation we construct the multiplex communicability matrix. Then we prove that the communicability function naturally induces an embedding of the multiplexes in a hyperspherical Euclidean space. We then study (i) intra-layer, (ii) inter-layer, and (iii) inter-layer self-communicability distance and angles in multiplex networks. Using these multiplex metrics we study a social multiplex related to an office politics and the multiplex of synaptic interactions between neurons in the worm C. elegans. We find that the average communicability angles in these multiplexes exhibits a minimum for certain value of the interlayer coupling strength. We provide an explanation for this phenomenon which emerges from the multiplexity of these systems and related it to other important phenomena like the synchronizability of these systems. Finally, we define and study communicability shortest paths in the multiplexes. We show how the communicability shortest paths avoid the most central nodes in the multiplexes in terms of their degree and betweenness, which is a main difference with (topological) shortest paths. We explain this behavior in terms of a diffusive model in which the 'information' not only diffuses between the nodes but it is also processed internally on the entities of the complex system. Finally, we give some new ideas on how to extend the current work and represent complex systems as 'multiplex hypergraphs' and 'multi-simplicial complexes'.