Dynamics on higher-order networks: A review (2203.06601v1)

Abstract: Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological, and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. We here review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation, and consensus formation. We also outline open challenges and promising directions for future research.

• The paper demonstrates that higher-order interactions fundamentally alter synchronization dynamics, revealing multistability and abrupt transitions.
• It illustrates how considering group interactions refines models of social contagion and consensus in complex systems.
• The review extends evolutionary game and random walk models, enhancing predictions for behaviors in complex systems.

Dynamics on Higher-Order Networks: An Overview

The paper "Dynamics on Higher-Order Networks: A Review" explores the burgeoning field of higher-order networks, which extends traditional network analysis to consider interactions beyond pairwise connections. This review is a comprehensive discourse on the increasingly significant role of higher-order networks in analyzing complex systems across various domains such as social science, biology, and technology.

Key Findings and Contributions

Higher-order networks are described by interactions that involve more than two nodes, thus providing a more nuanced and accurate representation of real-world systems, where relationships are often multi-faceted and complex. This paradigm shift is significant, as traditional networks have been largely based on dyadic interactions, which are insufficient for capturing the full dynamics of systems with group interactions—such as brain networks in neurobiology or social structures in human communities.

The authors Soumen Majhi, Matjaž Perc, and Dibakar Ghosh have structured the paper to cover several key areas of higher-order network dynamics:

1. Synchronization: A rich body of research indicates that higher-order interactions can dramatically alter synchronization dynamics. Traditional models, which focus primarily on pairwise interactions, are expanded to incorporate complex coupling structures, such as hypernetworks and simplicial complexes. These structures give rise to phenomena such as multistable and abrupt transitions in synchronization states, offering explanations for previously misunderstood dynamics in networked systems.
2. Social Dynamics: The analysis of higher-order networks provides insights into social processes such as contagion and consensus formation. For instance, in social contagion models, accounting for group interactions rather than simply individual pair connections can lead to discontinuous transitions between states, illustrating how ideas or behaviors can suddenly take hold or dissipate in a population. These findings have implications for understanding and managing the spread of information or diseases.
3. Evolutionary Game Dynamics: By extending game theory models to higher-order frameworks, the paper highlights how cooperation within groups can be better understood. Group interactions reveal new equilibria and strategic outcomes not evident in pairwise contexts, enhancing our understanding of cooperation and competition in social systems.
4. Random Walks and Diffusion: The review discusses how random walk processes differ in a higher-order network context. These processes, which serve as models for diffusion phenomena, show altered stationary distributions and dynamics, as highlighted in analyses of random walkers on simplicial complexes.

Implications and Future Directions

The implications of this work are extensive. From a theoretical standpoint, higher-order networks challenge existing paradigms and support the development of new mathematical tools and models. Practically, they offer enhanced predictive power for phenomena ranging from the stability of ecosystems to the efficacy of public health interventions.

Future research directions include the exploration of temporal and multilayer higher-order networks, which represent evolving interaction patterns over time. Additionally, understanding the full extent of dynamics such as chimera states and partial synchronization in these complex systems remains an open challenge.

The investigation into higher-order network dynamics represents a critical evolution of network science, providing deeper insights into the interconnected complexities of real-world systems. This paper effectively synthesizes current research efforts in this nascent field, setting a foundation for future explorations and applications.