Random walk centrality in interconnected multilayer networks

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.

💡 Research Summary

The paper addresses the challenge of measuring node importance in interconnected multilayer networks, where traditional single‑layer centrality metrics fail to capture inter‑layer interactions. Using the rank‑4 adjacency tensor Mᵢαʲβ to encode both intra‑ and inter‑layer edges, the authors formulate a transition tensor Tᵢαʲβ that governs a discrete‑time random walk allowing moves within a layer or switches to the replica of the same node in another layer. The steady‑state of the walk is obtained as the leading eigen‑tensor Πᵢα, which is shown to be proportional to the node‑layer strength sᵢα; aggregating over layers yields the occupation centrality πᵢ.

Building on this foundation, the paper extends random‑walk betweenness and closeness centralities to the multilayer setting. Betweenness is expressed through the cumulative transition tensor G and the probability τ of reaching a destination, resulting in an analytical formula for the expected number of times a node‑layer pair lies on a walk between any source‑target pair. Closeness is defined as the average first‑passage time, computed via the inverse (or power series) of the transition tensor. Both measures naturally incorporate inter‑layer paths, unlike shortest‑path‑based counterparts.

The authors validate their analytical expressions on synthetic multilayer graphs and real‑world datasets (e.g., transportation and social networks), demonstrating exact agreement with Monte‑Carlo simulations. They also show that naïve aggregation of single‑layer centralities can misrepresent node importance, whereas the tensor‑based approach provides a principled, parameter‑free assessment. The work thus offers a rigorous, scalable framework for random‑walk‑based centrality analysis in complex multilayer systems, with implications for epidemic modeling, traffic optimization, and information diffusion studies.