Networks beyond pairwise interactions: structure and dynamics (2006.01764v1)

Abstract: The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a great variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, in face-to-face human communication, chemical reactions and ecological systems, interactions can occur in groups of three or more nodes and cannot be simply described just in terms of simple dyads. Until recently, little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can greatly enhance our modeling capacities and help us to understand and predict their emerging dynamical behaviors. Here, we present a complete overview of the emerging field of networks beyond pairwise interactions. We first discuss the methods to represent higher-order interactions and give a unified presentation of the different frameworks used to describe higher-order systems, highlighting the links between the existing concepts and representations. We review the measures designed to characterize the structure of these systems and the models proposed in the literature to generate synthetic structures, such as random and growing simplicial complexes, bipartite graphs and hypergraphs. We introduce and discuss the rapidly growing research on higher-order dynamical systems and on dynamical topology. We focus on novel emergent phenomena characterizing landmark dynamical processes, such as diffusion, spreading, synchronization and games, when extended beyond pairwise interactions. We elucidate the relations between higher-order topology and dynamical properties, and conclude with a summary of empirical applications, providing an outlook on current modeling and conceptual frontiers.

• The paper introduces frameworks for modeling complex systems using hypergraphs and simplicial complexes to capture higher-order interactions.
• It extends traditional network measures to include generalized degrees, adjacency tensors, and higher-order clustering, enhancing structural analysis.
• The study demonstrates altered dynamics such as higher-order diffusion, synchronization, and evolutionary game dynamics, with significant empirical applications.

Networks beyond Pairwise Interactions: Structure and Dynamics

The paper "Networks beyond pairwise interactions: structure and dynamics" presents a comprehensive overview of complex systems modeled via higher-order interactions. It introduces new frameworks for representing these systems, extends traditional network measures to higher-order structures, and explores implications for dynamics such as diffusion, synchronization, and evolutionary games.

Representation of Higher-Order Interactions

The representation of higher-order interactions takes multiple forms, including hypergraphs and simplicial complexes.

• Hypergraphs describe interactions where hyperedges can connect any number of nodes. These structures are inherently flexible but can be complex to handle analytically.
• Simplicial Complexes are more structured, representing interactions such that all sub-interactions within a group are also included. This leads to a rich algebraic framework suitable for modeling complex dependencies.

Measures of Structure in Higher-Order Systems

The extension of network measures to higher-order systems involves the definition of new concepts such as:

• Generalized Degree: For nodes in higher-order networks, considering the number of $d$-dimensional simplices they participate in.
• Adjacency Tensors: Generalizing adjacency matrices to higher dimensions to capture multi-node interactions.
• Centrality Measures: Including higher-order degree centrality and path-based centralities extended to hypergraphs and simplicial complexes.
• Clustering Coefficient: Extended to account for the likelihood of nodes forming higher-order simplices rather than just triangles.

Dynamics in Higher-Order Systems

Higher-Order Diffusion

Diffusion processes on higher-order systems, where substances can move beyond pairwise interactions, involve novel formulations of the Laplacian:

• Combinatorial Laplacian: Defined at each order $k$ to model diffusion on $k$-dimensional simplices.
• Higher-Order Random Walks: Modifying classical random walks to account for transitions across higher-dimensional simplices in a simplicial complex.

Synchronization

The synchronization of coupled dynamical systems in higher-order settings introduces new regimes:

• Kuramoto Model Extensions: Address multi-body interactions, leading to phenomena like abrupt desynchronization and multistability.
• Higher-Order Coupling: The introduction of coupling beyond pairwise interactions into models such as complex geometrical networks and simplicial complexes highlights the critical impact of higher-order interactions on global synchronization dynamics.

Evolutionary Games

Higher-order interactions also revamp evolutionary game theory:

• Public Goods Game on Hypergraphs: Enhanced cooperation potential when considering the true group structure rather than its pairwise projection.
• Coordination Games: Higher-order structures can lead to distinct stable states and evolutionary dynamics distinct from pairwise settings.

Empirical Applications

Empirical applications of higher-order frameworks span multiple domains:

Social Networks

Studies on affiliation networks and scientific collaborations show that:

• Node Structure Vector: Used to measure centrality in affiliation networks, revealed that high-dimensional nodes play crucial roles in maintaining social cohesion.
• Simplicial Closure: In scientific collaborations, authors participating in multiple group papers exhibit significantly high simplicial closure, indicating robust collaborative structures.

Neuroscience

Higher-order methods in neuroscience reveal that:

• Topological Data Analysis: Persistent homology techniques elucidate the latent topological structures encoded in neuronal spike trains and functional brain networks.
• Functional Topology: Changes in the topological structure of brain functional networks are linked to states of consciousness and neurological disorders.

Conclusion and Future Directions

The paper underscores the significance of incorporating higher-order interactions into the modeling of complex systems. This approach not only enriches the structural understanding but also unveils new dynamic behaviors not observed in traditional pairwise models. Future research could expand the theoretical framework, improve computational methods for inference, and apply these high-dimensional models across more empirical domains, particularly in uncovering hidden interdependencies in massive datasets from genomics, neuroscience, and social systems.