Centrality in Interconnected Multilayer Networks (1311.2906v1)

Abstract: Real-world complex systems exhibit multiple levels of relationships. In many cases, they require to be modeled by interconnected multilayer networks, characterizing interactions on several levels simultaneously. It is of crucial importance in many fields, from economics to biology, from urban planning to social sciences, to identify the most (or the less) influent nodes in a network. However, defining the centrality of actors in an interconnected structure is not trivial. In this paper, we capitalize on the tensorial formalism, recently proposed to characterize and investigate this kind of complex topologies, to show how several centrality measures -- well-known in the case of standard ("monoplex") networks -- can be extended naturally to the realm of interconnected multiplexes. We consider diagnostics widely used in different fields, e.g., computer science, biology, communication and social sciences, to cite only some of them. We show, both theoretically and numerically, that using the weighted monoplex obtained by aggregating the multilayer network leads, in general, to relevant differences in ranking the nodes by their importance.

• The paper extends centrality measures to multilayer networks using tensor formalism, addressing limitations of monoplex models.
• It derives new centrality metrics like random walk, PageRank, and HITS for interconnected layers, validated through numerical experiments.
• The approach enhances understanding of complex systems in fields like biology, communication, and social sciences, guiding strategic interventions.

Insights into Centrality in Interconnected Multilayer Networks

The paper "Centrality in Interconnected Multilayer Networks" by Manlio De Domenico et al., presents a sophisticated extension of centrality measures from conventional monoplex networks to interconnected multilayer networks. This work makes substantial progress in quantifying the importance of nodes in complex systems characterized by multiple interconnected layers. Utilizing tensorial formalism, the paper proposes that traditional centrality measures can naturally extend to this more intricate network framework.

Key Contributions

The paper argues convincingly that monoplex network models often oversimplify the interactions within complex systems, which can lead to significant inaccuracies. By employing interconnected multilayer networks, the complexity inherent in real-world systems—the type often observed in biological, communication, and social sciences—is captured more comprehensively. The authors extend centrality measures traditionally used in monoplex networks, such as eigenvector centrality and betweenness centrality, to the interconnected multilayer network framework using tensors.

Theoretical and Numerical Validation

The theoretical formulation relies heavily on tensor algebra, providing a compact and robust framework to define structures where nodes may exhibit different degrees of importance across layers. The authors advocate for the usage of a 4-order adjacency tensor to encapsulate the full information of multilayer interactions, which effectively generalizes the adjacency matrix used in monoplexes. Importantly, the paper supports theoretical claims with numerical results, demonstrating that aggregation of multilayer networks into monoplexes generally leads to divergent node rankings, which confirms the necessity of this new approach.

Centrality Measures in Multilayer Networks

• Random Walk Centrality: The paper extends the concept of a random walk on networks to multilayer structures, which incorporates the calculation of transition tensors. The authors demonstrate that this measure reflects the probability of occupying nodes more accurately by considering multiple layers.
• PageRank and HITS Centrality: The authors extend PageRank centrality to multilayer networks, illustrating that teleportation might occur across layers, a methodology not previously formalized in literature. Similarly, the well-established HITS algorithm, which distinguishes between hubs and authorities, has been generalized to interconnected networks.
• Centrality Based on Topological Properties: The extension of eigenvector and Katz centrality to interconnected networks is presented, revealing that these measures provide unique insights into node importance when generalizing vector space into tensor space.

Implications and Future Directions

The extension of centrality measures highlights the nuanced importance of nodes in complex multilayer systems. This work has substantial implications for how network dynamics, such as information flow and connectivity resilience, are understood in systems ranging from neuronal networks to socio-economic ecosystems. Practically, this paper advances our ability to identify strategic nodes—whether in disease transmission or communication networks—thus refining intervention strategies.

The research opens pathways for further exploration of dynamic processes over multilayer networks. As tensorial calculations become computationally more feasible, there is potential for increasingly complex and nuanced models to represent layered interactions within systems. Future work could expand upon this framework to include dynamic, temporal layers and to apply these advanced centrality measures to empirical data across diverse fields.

In conclusion, the incorporation of tensorial formalism into network centrality provides a profound improvement over aggregated analysis of monoplex networks. The research enriches our understanding of centrality within the increasingly prevalent multilayer networks, promising substantial enhancements in both theoretical applications and practical implementations across various domains.