Random walk centrality in interconnected multilayer networks

TL;DR

This work addresses centrality in interconnected multilayer networks by adopting a tensorial representation with a rank-4 adjacency tensor $M^{i\\alpha}_{j\\beta}$ and a multilayer random-walk transition tensor $T^{i\\alpha}_{j\\beta}$. It extends two classical random-walk centralities—occupation-based centrality and random-walk-based betweenness and closeness—to the multilayer setting, deriving analytical expressions for the steady-state occupancy $\\Pi_{i\\alpha}$, absorbing-transition based mean crossing times, and MFPT-based closeness measures, along with a PageRank variant $R^{i\\alpha}_{j\\beta}$. The authors demonstrate exact agreement between the analytical results and simulations across various multilayer topologies, and discuss that aggregating centrality over layers preserves occupation centrality but can misrepresent other measures. The framework provides a principled, scalable approach for evaluating node influence in complex systems (e.g., social, economic, and transportation networks) where multiple relation types and inter-layer dynamics coexist.

Abstract

Real-world complex systems exhibit multiple levels of relationships. In many cases they require to be modeled as interconnected multilayer networks, characterizing interactions of several types simultaneously. It is of crucial importance in many fields, from economics to biology and from urban planning to social sciences, to identify the most (or the less) influential nodes in a network using centrality measures. However, defining the centrality of actors in interconnected complex networks is not trivial. In this paper, we rely on the tensorial formalism recently proposed to characterize and investigate this kind of complex topologies, and extend two well known random walk centrality measures, the random walk betweenness and closeness centrality, to interconnected multilayer networks. For each of the measures we provide analytical expressions that completely agree with numerically results.

Figures (4)

• Figure 1: A, an interconnected multilayer network representing the same actors exhibiting different relationships on different levels. The cost to move from one layer to the other is represented by dotted vertical lines. B, edge-colored graph representing the same actors with the same relations in A with two different types of interactions (solid and dashed edges). In this case the representation does not allow modeling the cost to move between layers. C, classical approach of representing the different types of relations using an aggregated network. The network represents the same actor and relation in A and B but disregarding the type of relation.
• Figure 2: Schematic of a walk (dotted trajectories) between two individuals $s$ and $t$ using a multilayer network. A walker can jump between nodes within the same layer, or it might switch to another layer. This illustration evidences how multilayer structure allows a walker to move between nodes that belong to different (disconnected) components on a given layer (L1).
• Figure 3: Comparison of the random walk betweenness centrality obtained by simulation and by our analytical approach for different multilayer network topologies. Each multilayer network is composed of two layers with 1000 nodes per layer. A, results on a multilayer network with two Erdős-Rényi networks as layers. B, results on a multilayer network with one Erdős-Rényi network and one Barabàsi-Albert network as layers. C, results on a multilayer network with two Barabàsi-Albert networks as layers.
• Figure 4: Comparison of the random walk closeness centrality obtained by simulation and by our analytical approach for different multilayer network topologies. Each multilayer network is composed of two layers with 1000 nodes per layer. A, results on a multilayer network with two Erdős-Rényi networks as layers. B, results on a multilayer network with one Erdős-Rényi network and one Barabàsi-Albert network as layers. C, results on a multilayer network with two Barabàsi-Albert networks as layers.