Networks beyond pairwise interactions: structure and dynamics
Federico BattistonGiulia CencettiIacopo IacopiniVito LatoraMaxime LucasAlice PataniaJean-Gabriel YoungGiovanni Petri
Unifies the mathematical frameworks, structural metrics, and dynamical models of higher-order networks to demonstrate how group interactions across hypergraphs and simplicial complexes fundamentally change processes such as diffusion, contagion, and synchronization.
Modern science and industry increasingly depend on understanding complex interconnected systems, ranging from biological and ecological communities to communication, brain, and social networks. For decades, standard network models represented these systems strictly as collections of pairwise links between individual units. However, many real-world interactions occur simultaneously among three or more entities, such as group communication, collaborative teams, or multi-species ecological competition. The article provides a comprehensive overview of the emerging field of networks beyond pairwise interactions, establishing a unified foundation for describing, analyzing, and predicting the structure and behavior of complex group-based systems.
To address this challenge, the article synthesizes structural representations, statistical generative models, and mathematical operators used to model group interactions. It evaluates explicit higher-order frameworks, primarily hypergraphs and simplicial complexes, and contrasts them with traditional graph-based approximations such as bipartite graphs, cliques, and network motifs. The review examines both static equilibrium models—such as higher-order configuration models and stochastic block models—and out-of-equilibrium growth processes that capture dynamic real-world assembly rules.
The findings demonstrate several critical insights for modeling complex systems. First, reducing multi-node interactions to simple pairwise graphs discards essential structural data, causing standard methods to either miss genuine group dynamics or artificially infer connections that do not exist. Second, hypergraphs offer the most flexible and unconstrained representation for arbitrary group sizes, while simplicial complexes provide powerful algebraic topological tools, such as boundary operators and combinatorial Laplacians, to capture multi-scale shapes and higher-order holes in data. Third, higher-order structural constraints dramatically alter system dynamics, shifting diffusive flows, random walk trajectories, and emergent collective behaviors away from what standard graph theory predicts.
These insights carry direct operational and strategic implications for research, technology development, and risk management. Relying on overly simplistic pairwise network models introduces significant blind spots when forecasting systemic risks, epidemic spreading, information cascade dynamics, or multi-agent stability. Adopting higher-order interaction frameworks enables organizations to build more accurate predictive models, optimize collaborative and communication architectures, and prevent costly failures stemming from unobserved group interdependencies.
To capitalize on these findings, decision-makers and technical leaders should update legacy network-analysis pipelines to incorporate hypergraph and simplicial representations where group interactions are present. Before deploying these models at scale, organizations should conduct pilot benchmarking against existing pairwise architectures to measure performance trade-offs, particularly regarding computational overhead. While confidence in the mathematical rigor of these higher-order frameworks is high, analysts must remain cautious regarding data sparsity and the exponential computational complexity associated with large motif and hyperedge enumerations.
- Paper: Hypergraph Neural Networks, Yifan Feng et al. (2018). Provides the foundational hypergraph neural network architecture that formalizes message passing and representation learning over non-pairwise relations.
- Paper: Multilayer networks, Mikko Kivelä et al. (2013). Establishes the generalized mathematical and tensorial representations for complex network structures that move beyond simple single-layer graphs.
- Paper: The structure and dynamics of multilayer networks, S. Boccaletti et al. (2014). Synthesizes the structural metrics and dynamical processes on multi-relational graphs that precede and motivate higher-order modeling frameworks.
- Paper: Epidemic processes in complex networks, Romualdo Pastor-Satorras et al. (2015). Provides the baseline theoretical framework for epidemic spreading and contagion dynamics on standard pairwise complex networks.
- Paper: Temporal Networks, Petter Holme et al. (2011). Surveys the interplay between evolving connectivity patterns and dynamic processes, essential for understanding time-dependent higher-order interactions.
- Paper: Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks, Christopher Morris et al. (2019). Extends standard graph neural message passing to higher-order subgraphs and hyperedges using multidimensional Weisfeiler-Leman invariants.
- Paper: Spatial Networks, Marc Barthelemy (2010). Details the geometric constraints and physical boundaries that govern complex interaction topologies in real-world systems.
- Paper: How Attentive are Graph Attention Networks?, Shaked Brody et al. (2021). Analyzes the expressive limitations of relational attention mechanisms and introduces a dynamically expressive formulation for interconnected network data.
- Paper: Low-dimensional topology of deep neural networks, Junyu Ren et al. (2026). Applies rigorous topological invariant tracking to neural network representations, continuing the investigation of topological mechanics in complex architectures.
- Paper: Storing Infinite Dynamical Attractors in Nonreciprocal Associative Neural Networks, Miguel Aguilera et al. (2026). Extends dynamical systems theory on complex networks to nonreciprocal associative architectures sustaining multiple independent temporal attractors.
