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 $\mathcal{A}\in\R^{N\times N\times L}$ and the parameter $\gamma\geq 0$, 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 $e_\mathcal{A}(\gamma)$ by means of the multiplex path length matrix $P\in\R^{N\times N}$. This paper generalizes the approach proposed in \cite{NR23} for single-layer networks. We describe an algorithm based on min-plus matrix multiplication to construct $P$, as well as variants $P^K$ 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 $e_\mathcal{A}^K(\gamma)$ for $e_\mathcal{A}(\gamma)$, for $K=1,2,\dots,N-2$. Finally, the sensitivity of $e_\mathcal{A}^K(\gamma)$ to changes of the entries of the adjacency tensor $\mathcal{A}$ is investigated to determine edges that should be strengthened to enhance the multiplex global efficiency the most.

翻译：由不同类型对象组成的复杂系统建模可导出多层网络，其中顶点通过层间边与层内边相连。本文研究多重网络——各层中的顶点彼此对应，唯一层间边仅连接某顶点及其在其他层的副本。设三阶邻接张量$\mathcal{A}\in\R^{N\times N\times L}$与参数$\gamma\geq 0$（与层间通信难易程度相关）共同表征含$N$个顶点、$L$层的多重网络。为度量多重网络中的通信效率，我们聚焦于平均逆测地长度——借助多重路径长度矩阵$P\in\R^{N\times N}$将其定义为多重全局效率$e_\mathcal{A}(\gamma)$。本文推广了文献\cite{NR23}针对单层网络提出的方法。我们描述了一种基于最小-加矩阵乘法构建$P$的算法，以及仅考虑最多包含$K$条层内边的多重路径的变体$P^K$。这些矩阵被用于检测冗余边，并确定$e_\mathcal{A}(\gamma)$的非递减下界$e_\mathcal{A}^K(\gamma)$（$K=1,2,\dots,N-2$）。最后，研究$e_\mathcal{A}^K(\gamma)$对邻接张量$\mathcal{A}$元素变化的敏感性，以识别需加强的边，从而最大程度提升多重全局效率。