Higher-Order Network Dynamics in Complex Systems

Complex systems across biology, sociology, physics and engineering often feature interactions that extend beyond simple pairwise connections. Higher-order systems capture these multiway relationships, employing mathematical frameworks such as hypergraphs and simplicial complexes to encode group interactions. Such structures reveal nuances invisible to classical network analysis, allowing us to model phenomena where collective behaviour emerges from simultaneous interactions among three or more units. Advances in this field have demonstrated that higher-order organisation can qualitatively alter dynamical processes—including diffusion, synchronization and spreading—leading to novel phase transitions, bistability and topology-driven delays. These insights are crucial for understanding brain dynamics, social contagion, ecological networks and engineered infrastructure, where multiplexed group effects underpin robustness and functional diversity. From theoretical constructs like multiorder Laplacians and simplex renormalisation to practical tools for null model generation, the study of higher-order network dynamics has matured into a multidisciplinary endeavour. This framework enriches our capacity to predict and control emergent behaviour in real-world complex systems.

Research from Nature Portfolio

Recent studies have shown that the choice of higher-order representation can deeply influence collective dynamics. One investigation compared hypergraphs and simplicial complexes in a synchronisation context, revealing that group interactions tend to enhance coherence in hypergraphs yet suppress it in simplicial complexes. This work introduced metrics linking synchronisability to degree heterogeneity and cross-order correlations, with implications for contagion, diffusion and resilience across fields. Complementary research has extended the Master Stability Function formalism to arbitrary simplicial complexes, providing stability criteria for complete synchronisation that generalise classical network results to many-body interactions. Together, these developments underscore the need for precise architectural modelling when assessing dynamical outcomes. Foundational modelling of higher-order contagion further demonstrated discontinuous transitions and bistable regimes in social systems, showing why critical group sizes are required to trigger large-scale adoption. These contributions have solidified the theoretical underpinnings of higher-order dynamics and informed empirical analyses in neuroscience, epidemiology and beyond.

Research from all publishers

An advanced theoretical framework based on simplex path integrals and renormalisation groups has been proposed to capture universality in systems with arbitrary high-order interactions. By defining propagators on simplices and coarse-graining high-order couplings in momentum space, this approach identifies scale-invariant properties across different interaction orders, offering a general methodology to probe emergent topological and statistical features. Meanwhile, a comprehensive review of networks beyond pairwise interactions has consolidated measures, generative models and dynamical analyses for hypergraphs, simplicial complexes and related constructs, charting the field’s evolution and spotlighting applications from ecology to material science. In social science, novel microcanonical null models for directed hypergraphs preserve complex structural constraints, enabling robust hypothesis testing in sociology, epidemiology and economics. These ensembles support efficient sampling algorithms and have been used to uncover political homophily trends, nonlinear contagion effects and dependencies in global trade networks, highlighting the power of tailored null models for higher-order data.

Higher-Order Network Dynamics in Complex Systems publication trend

The graph below shows the total number of articles in higher-order network dynamics in complex systems across all publications each year (not limited to Nature Index journals).

Technical terms

Higher-order interaction: A simultaneous relationship among three or more units, modelled as hyperedges or simplices rather than binary links.

Hypergraph: A generalisation of a network where edges (hyperedges) can connect any number of nodes, capturing group interactions directly.

Simplicial complex: A structured collection of simplices (nodes, edges, triangles, etc.) that encodes nested, multiway relationships with algebraic topology properties.

Synchronisability: A metric of a network’s propensity to achieve coherent dynamics across its units under given coupling schemes.

Renormalisation group: A theoretical tool to study system behaviour under scale transformations, here extended to coarse-grain high-order interactions.

References

1. A simplex path integral and a simplex renormalization group for high-order interactions *. Reports on Progress in Physics (2024).
2. Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes. Nature Communications (2023).
3. Stability of synchronization in simplicial complexes. Nature Communications (2021).
4. Simplicial models of social contagion. Nature Communications (2019).
5. Networks beyond pairwise interactions: Structure and dynamics. Physics Reports (2020).
6. Higher-Order Null Models as a Lens for Social Systems. Physical Review X (2024).
7. Multiorder Laplacian for synchronization in higher-order networks. Physical Review Research (2020).