Enhancing multiplex global efficiency

• Published in: Numerical algorithms Vol. 96; no. 1; pp. 397 - 416
• Main Authors: Noschese, Silvia, Reichel, Lothar
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2024; Springer Nature B.V
• Subjects: Algebra; Algorithms; Apexes; Communication; Complex systems; Computer Science; Efficiency; Graph theory; Lower bounds; Monolayers; Multilayers; Multiplexing; Networks; Numeric Computing; Numerical Analysis; Original Paper; Public transportation; Roads & highways; Tensors; Theory of Computation; Travel; 05C50; Network analysis; 15A16; 65F15; Perron root; Multiplex path length matrix; Multiplex network; Global efficiency
• ISSN: 1017-1398; 1572-9265; 1572-9265
• DOI: 10.1007/s11075-023-01651-5

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 ∈ R N × 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 e A ( γ ) by means of the multiplex path length matrix 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 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 A K ( γ ) for e A ( γ ) , for K = 1 , 2 , ⋯ , N - 2 . Finally, the sensitivity of e A K ( 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.