Enhancing multiplex global efficiency

Modeling complex systems that consist of different types of objects leads to multilayer networks, in which vertices are connected by both inter-layer and intra-layer edges. In this paper, we investigate multiplex networks, in which vertices in different layers are identified with each other, and the only inter-layer edges are those that connect a vertex with its copy in other layers. Let the third-order adjacency tensor A∈ RN×N×L and the parameter γ≥ 0 , which is associated with the ease of communication between layers, represent a multiplex network with N vertices and L layers. To measure the ease of communication in a multiplex network, we focus on the average inverse geodesic length, which we refer to as the multiplex global efficiency eA(γ) by means of the multiplex path length matrix P∈ RN×N . This paper generalizes the approach proposed in [15] for single-layer networks. We describe an algorithm based on min-plus matrix multiplication to construct P, as well as variants PK that only take into account multiplex paths made up of at most K intra-layer edges. These matrices are applied to detect redundant edges and to determine non-decreasing lower bounds eAK(γ) for eA(γ) , for K= 1 , 2 , ⋯ , N- 2 . Finally, the sensitivity of eAK(γ) to changes of the entries of the adjacency tensor A is investigated to determine edges that should be strengthened to enhance the multiplex global efficiency the most.