Enhancing multiplex global efficiency

Abstract

AbstractModeling 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 $$\mathcal {A}\in \mathbb {R}^{N\times N\times L}$$ A ∈ R N × N × L and the parameter $$\gamma \ge 0$$ 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 $$e_\mathcal {A}(\gamma )$$ e A ( γ ) by means of the multiplex path length matrix $$P\in \mathbb {R}^{N\times N}$$ P ∈ R N × 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 $$P^K$$ P K 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 $$e_\mathcal {A}^K(\gamma )$$ e A K ( γ ) for $$e_\mathcal {A}(\gamma )$$ e A ( γ ) , for $$K=1,2,\dots ,N-2$$ K = 1 , 2 , ⋯ , N - 2 . Finally, the sensitivity of $$e_\mathcal {A}^K(\gamma )$$ e A K ( γ ) to changes of the entries of the adjacency tensor $$\mathcal {A}$$ A is investigated to determine edges that should be strengthened to enhance the multiplex global efficiency the most.