Multilayer Networks (1309.7233v4)

Abstract: In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.

An Essay on "Multilayer Networks" by Kivelä et al.

In their influential paper titled "Multilayer Networks," Kivelä et al. present a comprehensive framework for studying multilayer network systems, addressing both the theoretical underpinnings and practical implications of such networks. This detailed and meticulously crafted work provides a critical overview of the historical developments, definitions, and methodological advances in the paper of multilayer networks.

Overview and Motivation

The authors begin by motivating the necessity of multilayer networks: real-world systems often encompass multiple types of interactions, which can be layered and intertwined in complex ways. Traditional network theory, which predominantly focuses on single-layer graphs, fails to capture the richness of such systems. Therefore, extending network analysis to consider multiple layers, each representing different types of interactions or data dimensions, becomes crucial for understanding complex systems more accurately.

General Framework and Definitions

Kivelä et al. introduce a generalized framework that systematically incorporates multiple dimensions ("aspects") of networks. They define a multilayer network $M = (V_M, E_M, V)$ with node set $V_M$ and edge set $E_M$, where each node-layer tuple in $V_M$ specifies both a node and a layer. This setting allows the analysis of interactions not just within a specific layer but also between nodes across different layers.

Key terminologies include:

• Node-aligned: Networks where all nodes are present in all layers.
• Layer-disjoint: Networks where each node is present in only one layer.
• Diagonal couplings: Inter-layer edges that connect a node to copies of itself in different layers.
• Layer-coupled: A special case of diagonal couplings where inter-layer edges are layer-dependent but node-independent.
• Categorical couplings: Complete inter-layer connections for all node pairs across layers.

These definitions are crucial for the precise mathematical treatment of multilayer networks and facilitate the identification and classification of various multilayer network models explored in the literature.

Methods and Models

The paper explores various methods and models relevant to multilayer networks:

1. Tensor Representations: The authors present adjacency tensors for node-aligned multilayer networks, allowing operations like tensor flattening and slice-wise analysis to reconcile the complexity of multilayer connections.
2. Supra-Adjacency Matrices: This representation translates the multilayer structure into a large adjacency matrix, enabling the use of matrix-based analytical techniques. These matrices are particularly useful for studying dynamical processes such as diffusion, synchronization, and spectral properties.
3. Communities and Mesoscale Structures: Detection of communities in multilayer networks extends traditional methods like modularity optimization and spectral clustering to incorporate the additional complexities introduced by multiple layers.

Empirical Data and Real-World Applications

Kivelä et al. emphasize the importance of empirical data in validating multilayer network theories. They discuss multifaceted social networks, co-authorship networks, transportation systems, and temporal networks, illustrating how multilayer network analysis can yield deeper insights than single-layer approaches. For instance, the paper of international trade networks, where each layer represents different commodities, reveals structures that are not evident in aggregated data.

Implications and Future Directions

The implications of this work are profound, both theoretically and practically:

1. Enhancement of Network Diagnostics: Concepts such as centrality, clustering, and path-based measures, when extended to multilayer frameworks, can better capture the functional properties of complex systems.
2. Dynamical Processes: The paper of processes like diffusion, percolation, and epidemic spreading on multilayer networks unveils phenomena that single-layer models cannot capture. For example, inter-layer correlations can critically affect the robustness and connectivity of the network.
3. Network Design and Control: Understanding multilayer networks can lead to more effective strategies for network design, optimization, and control, particularly in fields like infrastructure resilience, communication networks, and system-of-systems engineering.

Conclusion

Kivelä et al.'s paper on multilayer networks represents a significant step forward in network science. By providing a comprehensive framework, introducing clear terminologies and methods, and discussing empirical applications, the authors lay the groundwork for future research in this exciting and rapidly expanding area. As multilayer data becomes more readily available, the continued development and refinement of these methods will be crucial for leveraging the full potential of multilayer network analysis.