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Dynamics on higher-order networks: A review
Soumen Majhi, Matjaz Perc, Dibakar Ghosh
TL;DR
Pairwise networks cannot fully represent group interactions, motivating higher-order network frameworks. This review synthesizes definitions, models, and studies of dynamics on higher-order networks, including synchronization, contagion, cooperation, and consensus. It identifies diverse dynamical effects and open directions involving temporal, multilayer, cluster-synchrony, and chimera phenomena.
Problem
Pairwise network models are limited for systems whose interactions involve groups of three or more constituents across social, biological, and other settings.
Method
The paper reviews higher-order network terminology and synthesizes recent research on synchronization and social dynamics, including contagion, consensus, and evolutionary games.
Results
The reviewed studies show diverse higher-order dynamical effects, including multistable synchronization, discontinuous desynchronization transitions, and higher-order contagion and cooperation dynamics.
Takeaways & Limitations
Higher-order interactions provide a framework for studying dynamics that pairwise network descriptions do not fully capture.
Takeaways & Limitations
Important directions remain open for temporal higher-order networks, interdependent multilayer structures, cluster synchrony, and chimera states.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
Classical pairwise networks are insufficient for representing many group interactions, motivating higher-order network frameworks. The review surveys structural models, analytical tools, and dynamical processes developed for these systems.
- Group interactions occur across neurobiology, social systems, and ecology, but pairwise networks model only dyadic relationships.
- Higher-order structures such as hypernetworks and simplicial complexes represent interactions involving three or more constituents.
- Recent studies report that incorporating higher-order architecture can improve understanding and prediction of network dynamics.
- Research has developed higher-order network models based on simplicial growth, preferential attachment, temporal activity, and configuration mechanisms.
- Analytical and computational tools address higher-order clustering, centrality, robustness, percolation, and dynamical expansion eigenvalues.
- The review covers synchronization, contagion, consensus formation, and evolutionary game dynamics on higher-order structures.
II. BASIC CONCEPTS
Higher-order networks are built from hyperlinks and simplices that encode interactions among more than two nodes. The review focuses on the novel dynamical effects produced by these higher-order interactions.
- A hyperlink can connect any number of nodes, unlike a traditional link that joins only two nodes.
- A hypernetwork H consists of nodes V and hyperlinks E, with E contained in the power set of V.
- A d-simplex is a set of d + 1 fully interacting nodes; examples include nodes, links, triangles, and tetrahedra.
- A simplicial complex is a hypernetwork containing every nonempty subset of each hyperlink.
- The review does not distinguish strictly between hypernetwork and simplicial-complex dynamics, presenting diverse dynamics on higher-order networks generally.
III. SYNCHRONIZATION
Higher-order interactions produce synchronization behaviors that differ from pairwise-network dynamics, including discontinuous transitions, multistability, and synchronization across topological dimensions. The reviewed models combine analytical formalisms and simulations on simplicial complexes and real connectomes.
- Higher-order oscillator interactions can generate infinite multistable synchronized attractors beyond a critical coupling strength.
- Simplicial interactions can produce abrupt transitions to both synchronization and desynchronization, including strong synchrony under repulsive pairwise interactions.
- Contrarian-only oscillator ensembles can achieve collective synchrony when interactions extend beyond dyadic connections.
- Interactions between node and link signals yield explosive topological synchronization with a closed hysteresis loop, tested on simplicial complexes and human and C. elegans connectomes.
- Node-link and node-link-triangle models couple signals across dimensions through order parameters, and simulations test them on configuration and NGF simplicial-complex models.
- The NLT model shows simultaneous discontinuous transitions of all reported order parameters, whereas the NL model permits an independent transition for R_up 1 at zero coupling.
- Higher-order synchronization analysis uses multiorder Laplacians, SBD, and generalized MSF formalisms for heterogeneous hypernetworks and varied coupling functions.
IV. SOCIAL DYNAMICS
Social dynamics research examines how peer-to-peer interactions in social networks influence opinion, culture, language, crowds, hierarchy, cooperation, and spreading. These processes motivate mathematical study of contagion and related dynamics.
- Peer-to-peer interactions in social networks influence opinion, cultural, language, crowd, hierarchy, cooperation, and spreading dynamics.
- Contagion effects motivate mathematical investigation of social-process dynamics.
A. Contagion processes
Higher-order contagion models combine dyadic and group interactions, revealing nonlinear infection growth, discontinuous outbreaks, bistability, and dependence on temporal structure and initial conditions.
- Simplicial and temporal contagion: Simplicial contagion combines pairwise and higher-order contacts, producing discontinuous endemic transitions and coexistence of healthy and endemic states.Temporal extensions show that identical infectious seeds can yield different outcomes depending on network timing.
- Structural effects: Higher-order structural differences can change epidemic prevalence even when networks share clustering and degree-distribution profiles.A new metric measures order-four structures to distinguish these higher-order differences.
- Exposure heterogeneity: Heterogeneous exposure periods and minimal infective doses produce a nonlinear infection-risk relation and discontinuous outbreaks with bistable healthy and outbreak states.The model analyzes heterogeneity in environments and individual participation over time.
- Nonlinear infection kernels: ν > 1 yields superexponential prevalence growth, whereas ν ≤ 1 produces regular exponential growth until saturation.The infection kernel scales as θ_m(ρ) ∝ ρ^ν.
- Thresholds and bistability: The absorbing healthy state is unstable above the invasion threshold β_u and globally stable below the persistence threshold β_s.Transitions are continuous when β_s = β_u and discontinuous with bistability when β_s < β_u.
- Competing epidemic processes: Competing SIS dynamics on 1- and 2-simplices can switch epidemic dominance with triadic infection strength and initial infection seeds.Weak triadic strength favors absolute epidemic dominance, while stronger triadic strength can produce alternative dominance depending on the initial seed.
B. Consensus formation
Consensus on higher-order networks depends on nonlinear interaction functions and temporal connectivity. These ingredients can shift consensus away from the average state, alter convergence rates, and generate multistability.
- Nonlinear consensus: Multibody interactions affect consensus only with nonlinear interaction functions, which can shift the final state away from the system average.The shift depends on the underlying network and initial configuration.
- Temporal consensus: Temporal higher-order consensus is modeled through time-indexed adjacency tensors, with τ specifying the duration between successive network structures.The tensors represent changing 3-hypergraph connectivity.
- Clustered configurations: In two connected clusters, the fraction p of oriented hyperedges controls the asymmetric configuration used to study consensus values and convergence.The setup contains two ten-node clusters connected by 20 randomly placed 3-links.
- Adaptive opinion dynamics: Peer pressure accelerates transitions to single-opinion and two-opinion states in adaptive higher-order voter models.The model can also exhibit multiple time scales, with 2-simplices vanishing before active links are exhausted.
- Temporal effects: Asymmetric configurations converge faster, and decreasing τ drives the consensus value toward that of the aggregated dynamics.Figure 4 compares first-mover A, first-mover B, and aggregated-dynamics scenarios.
- Multistability: Higher-order interactions can produce multistability in steady states.This provides a distinct dynamical outcome beyond a single consensus state.
C. Evolutionary game dynamics
Higher-order evolutionary games differ from pairwise games when payoffs depend nonlinearly on multiple neighbors’ strategies. The reviewed studies show that group size, structural heterogeneity, and simplicial interactions can substantially alter cooperation, reciprocity, and honesty.
- Higher-order payoffs: Nonlinear payoff dependence on multiple cooperative neighbors makes higher-order interactions dynamically distinct from sums of pairwise games.General multiplayer games capture this dependence by making a player’s payoff a function of the strategies of all neighbors.
- Cooperation in hypernetworks: Larger group sizes can help preserve cooperation in networks whose dyadic interactions alone do not capture group effects.The reviewed hypernetwork work preserves the dyadic projection while examining how higher-order structure affects cooperation.
- Cooperation in hypernetworks: Increasing connection order can cause higher reciprocity through a mechanism that replaces some first-order 3-cliques with second-order triangles.The analysis combines mean-field treatment for homogeneous interactions with invasion analysis for heterogeneous structures.
- Public-goods games: Public-goods dynamics on uniform hypernetworks without hyperdegree correlations is consistent with replicator dynamics in the well-mixed regime.Introducing heterogeneity in interaction order and hyperdegree changes the evolutionary game dynamics.
- Signaling games: Honesty can persist under higher-order signaling interactions even when individuals face temptation to lie or lying benefits the receiver at a cost to the sender.This contrasts with the stated dyadic-only instance in which the reviewed result is not observed.
- Simplicial games: In a simplicial framework, the frequency of cooperation depends on the density and game type of three-body interactions, with cooperation decreasing when Game2 is Snowdrift or Prisoner’s Dilemma at T2 ≥1.Game1 and Game3 are held identical, while Game2 varies across Harmony, Stag Hunt, Snowdrift, and Prisoner’s Dilemma regimes.
V. RANDOM WALK AND DIFFUSION
Higher-order random walks extend diffusion beyond pairwise networks by biasing motion toward hyperlinks of different sizes and linking stationary behavior to higher-order structure. These dynamics support community detection and reveal how bias and geometry shape diffusion-related organization.
- Higher-order random walks: Random walks on simplicial complexes form Markov chains whose stationary distributions are related to harmonics of the Hodge Laplacian.The reviewed framework extends random-walk dynamics beyond pairwise interactions.
- Size-biased walks: A size-bias parameter σ makes hypernetwork random walks favor hyperlinks of high or low size, thereby producing different network projections.Large positive and negative values of σ govern dynamics through large and small hyperlinks, respectively.
- Community detection: Markov stability is generalized to hypernetworks to identify communities as a function of the random-walk time horizon and size bias.The method assumes a partition of nodes into non-overlapping communities and evaluates its quality through Markov stability.
- Community detection: In the 16-node hierarchical hypernetwork, large positive σ yields 16 singleton communities or two communities of eight, while negative σ reveals intermediate and ultimately size-two communities.The number of communities changes across Markov time and σ, exposing the hierarchy encoded by hyperlink sizes.
- Community detection: The Simpson diversity index identifies community sizes: 1/Y = 2, 4, and 8 correspond to two groups of eight, four groups of four, and eight groups, respectively.For uniformly distributed communities, Y is approximately 1/Q, where Q is the number of groups.
- Diffusion and geometry: Higher-order Laplacians can have order-dependent finite spectral dimensions that influence return-time probabilities in diffusion processes.Related work also connects network geometry in NGF and STC models to diffusion dynamics.
VI. SUMMARY AND FUTURE PROSPECTS
The review surveys how higher-order interactions shape diverse dynamical processes beyond dyadic networks and identifies several underdeveloped directions for future research.
- Summary: Higher-order network architecture can strongly influence dynamical processes, motivating analysis beyond predominantly pairwise models.The review examines how diverse phenomena are affected when higher-order connections are included.
- Summary: The review covers dynamical processes on networks beyond dyadic interactions, including synchronization, spreading, cooperation, consensus, random walks, and diffusion.Random-walk and diffusion dynamics are specifically treated as effects of higher-order interactions.
- Future prospects: Temporal higher-order networks remain an open direction spanning structural intricacies and dynamics on time-varying higher-order structures.The review identifies this as a promising area for further study.
- Future prospects: Interdependent multilayer or multiplex frameworks with higher-order interactions are proposed as another promising research direction.The review specifically highlights inspection of these combined structures.
- Future prospects: Cluster synchrony requires further analysis, while chimera states in simplicial networks remain largely unexplored despite resemblance to neuronal developments.The review also suggests studying swarmalator collective behavior and adaptive higher-order systems.
- Future prospects: Adaptive higher-order systems require more attentive study because adaptivity increases the complexity of their dynamical scenarios.This is identified as a further research need alongside swarmalator systems with higher-order interactions.