Multilayer Networks in a Nutshell (1804.03488v1)

Abstract: Complex systems are characterized by many interacting units that give rise to emergent behavior. A particularly advantageous way to study these systems is through the analysis of the networks that encode the interactions among the system's constituents. During the last two decades, network science has provided many insights in natural, social, biological and technological systems. However, real systems are more often than not interconnected, with many interdependencies that are not properly captured by single layer networks. To account for this source of complexity, a more general framework, in which different networks evolve or interact with each other, is needed. These are known as multilayer networks. Here we provide an overview of the basic methodology used to describe multilayer systems as well as of some representative dynamical processes that take place on top of them. We round off the review with a summary of several applications in diverse fields of science.

• The paper presents a detailed mathematical framework that extends single-layer analysis to capture interdependent interactions.
• It employs advanced techniques like tensor calculus to model both intra-layer and inter-layer connections and dynamic processes.
• Practical applications demonstrate the framework’s effectiveness in decoding complex behaviors in social, biological, and technological systems.

The paper "Multilayer Networks in a Nutshell" provides a comprehensive overview of multilayer networks, an advanced framework for analyzing complex systems. These systems, found in diverse domains such as natural, social, biological, and technological areas, are characterized by numerous interactions leading to emergent behaviors. Traditional network science has contributed significantly to understanding these systems; however, it often falls short when dealing with interconnected and interdependent networks.

Key Concepts and Methodologies

1. Definition and Structure:
   • Multilayer networks consist of multiple interconnected layers, where each layer represents a distinct type of interaction or relationship.
   • These layers can interact with one another, providing a more nuanced representation of complex systems.
2. Mathematical Formalism:
   • The paper outlines the mathematical framework for multilayer networks, including nodes, edges, and their extensions across layers.
   • Methods for analyzing multilayer networks involve advanced matrix representations and tensor calculus, capturing both intra-layer and inter-layer connections.
3. Dynamical Processes:
   • It discusses various dynamical processes that occur on multilayer networks, such as diffusion, synchronization, and spreading processes.
   • These processes are significantly affected by the multilayer structure, influencing system dynamics compared to single-layer networks.

Applications

The paper highlights several applications of multilayer networks across different scientific fields, including:

• Social Networks: Understanding overlapping social circles and intersocial-platform interactions.
• Biological Systems: Modeling complex interactions in biological networks, such as gene regulation and protein interactions.
• Technological and Infrastructural Systems: Assessing the robustness and vulnerability of power grids or transportation systems, which are naturally multilayered due to their intertwined services.

Implications and Conclusion

The work underscores the importance of multilayer networks in capturing the complexity and interdependencies of real-world systems. By advancing beyond traditional single-layer analyses, multilayer network theory provides richer and more accurate insights, fostering a deeper understanding of the emergent behaviors in complex systems.

This overview serves as a foundational reference for both newcomers and seasoned researchers interested in exploring multilayer networks, offering methodological insights and illustrating their far-reaching implications across various disciplines.