Critical Higher-Order Interactions

• Critical higher-order interactions are multi-element dependencies beyond pairwise effects that fundamentally alter system dynamics and phase transitions.
• Mathematical frameworks such as hypergraphs and simplicial complexes capture these interactions, revealing phenomena like explosive transitions and nontrivial scaling behaviors.
• Operational methods, including redundancy metrics and motif-centric inference, effectively isolate critical orders to enhance predictive modeling and control in complex networks.

Critical higher orders of interaction denote those multi-element dependencies in complex systems (beyond pairwise, i.e., k ≥ 3) that are responsible for nontrivial, leading-order, and often genuinely new dynamical behaviors, phase transitions, or structural features. The identification and analysis of such critical orders is essential in network science, statistical physics, and data-driven modeling, as higher-order interactions fundamentally reshape both collective dynamics and the analytical tractability of a system.

1. Defining Critical Higher-Order Interactions

The term “critical higher-order of interaction” refers to either (a) a specific interaction order (e.g., triangles in a simplicial complex, k-uniform hyperedges), or (b) an explicit multi-node motif, whose presence or coupling strength alters qualitative macroscopic properties—such as universality class, nature and sharpness of phase transition, robustness, or information propagation—beyond the regime accessible by simply summing or projecting pairwise (dyadic) effects. Criticality in this context is understood broadly: it may indicate new scaling exponents, multifractality, abrupt onset of order (explosive transitions), or, conversely, the destruction of sharp transitions (Tadic et al., 2024, Schawe et al., 2021, Battiston et al., 2021).

Mathematical frameworks for capturing these higher-order effects typically employ hypergraphs (allowing arbitrary cardinality of interactions) and simplicial complexes (with strong closure under subsimplex inclusion). Key formalisms include generalized Hamiltonians—such as Ising-type models with edge and face (triangle) couplings (Tadic et al., 2024, Robiglio et al., 2024)—and stochastic models where update rules are governed by explicit k-body motifs (Schawe et al., 2021, Guo et al., 11 Jan 2025).

2. Dynamical Consequences and Phase Transition Mechanisms

The introduction of critical higher-order couplings leads to a rich spectrum of dynamical phenomena:

• New universality classes and scaling: In paradigmatic models such as zero-temperature Glauber spin models on simplicial complexes, triangle-embedded couplings (J₂) shift avalanche exponents from mean-field (τ_s ≈ 3/2) to directed percolation/directed sandpile values (τ_s ≈ 4/3), with multifractal temporal statistics and broadened hysteresis (Tadic et al., 2024).
• Destruction or smoothing of sharp phase transitions: In Deffuant bounded-confidence dynamics, increasing the interaction order k beyond ≈4 eliminates the size-dependent peak in susceptibility, leading to a smooth, size-independent crossover rather than a sharp consensus transition (Schawe et al., 2021). This is robust in sparse random or scale-free (BA) hypergraphs but not in regular lattices or spatially embedded hyper-lattices, where a continuous transition with diverging correlation length survives.
• Emergence of discontinuous (explosive) transitions and bistability: Many k-body coupling terms (e.g., in higher-order Kuramoto synchronization or higher-order percolation) fundamentally change the bifurcation structure of the mean-field equations, enabling saddle-node bifurcations and hysteresis cycles impossible in strictly pairwise systems (Battiston et al., 2021, Fariello et al., 2024). Notably, in Ising models constructed with conserved symmetry, three-body interactions produce continuous transitions, while four-body (and higher) interactions produce first-order transitions with discontinuous jumps in the magnetization order parameter (Robiglio et al., 2024).
• Cross-over behavior and multifractality: Tuning the ratio of higher- to lower-order coupling strengths, as in J₂/J₁ for spins or k₂/k₁ for Kuramoto oscillators, generates nontrivial crossovers between universality classes, as evidenced by scaling collapse of avalanche distributions and multifractal spectra—features diagnostic of a genuinely new critical regime (Tadic et al., 2024, Fariello et al., 2024).

3. Information-Theoretic and Inference Frameworks for Critical Orders

Criticality can also be defined through structural and inferential frameworks:

• Redundancy and compressibility: The structural reducibility approach defines per-order scores $R_s$ measuring the information saved by optimally compressing or removing each hyperedge order, using conditional entropy relative to higher layers. Critical orders are those for which $R_s$ exceeds a chosen threshold (e.g., >0.5), i.e., the layer encodes substantial non-redundant information not present at any higher order. The method finds an optimal representative set $\mathcal R^*$, ensuring that all essential higher-order structure is preserved while removing maximal redundancy (Kirkley et al., 5 Jan 2026).
• Motif-centric inference and model selection: Fully generative nonparametric models, such as the subgraph-configuration model (SGCM), decompose observed networks into minimized-length representations in terms of edges and explicit motifs. “Critical” motifs are those whose inclusion yields the greatest decrease in description length and are operationally those required for an accurate generative fit. In real data, critical motifs systematically include higher-order cliques, cycles, and specialized subgraphs covering a large proportion of edges or links (e.g., 85–97%) (Wegner et al., 2024).
• Statistical detection and hypothesis testing: For multivariate data, the Streitberg interaction measure and its associated kernel permutation test detects which interaction orders are statistically significant. The computationally feasible range typically caps at d≤5, but the method can isolate “critical” orders in neuroimaging and synthetic benchmarks, with significant over-representation in domain-specific functional modules (Liu et al., 2023).

4. Quantifying and Isolating Critical Orders in Practice

Operational determination of critical orders depends on empirical context:

• Stepwise reduction and performance metrics: In link prediction tasks, the so-called n-reduced graph approach systematically decomposes all k>n hyperedges into their n-node subsets and measures the incremental gain (e.g., in AUC) as a function of n. The “critical order” is the lowest n above which further inclusion of higher orders yields diminishing return; this can be as low as 3 (triadic) in social or email networks but as high as 7–8 in complex biochemical networks (Bian et al., 2024).
• Centrality and dynamical impact: Hyperedge-level Laplacians and associated centrality measures (diffusion Fréchet, degree, betweenness, closeness) allow ranking higher-order interactions by their impact on diffusion, epidemic thresholds, or network robustness. SIR simulations on perturbed networks validate that removing top-ranked hyperedges produces maximal reduction in spreading, confirming their critical dynamical role (Aktas et al., 2021).
• Percolation and robustness analysis: In multiplex hypergraphs with multiple layers of different interaction orders, percolation thresholds, nature of the transition (continuous vs. hybrid), and vulnerability to node/hyperedge removal depend sensitively on the distribution and correlation of higher-order structures. For instance, positive interlayer degree correlations strongly lower critical points, and interdependent percolation processes (requiring activation in all layers) lead invariably to hybrid, discontinuous transitions (Sun et al., 2021).

5. Theoretical and Physical Significance

Fundamentally, the existence of critical higher orders is a symptom of the inadequacy of pairwise approximations to capture essential features of real complex systems:

• Non-reducibility and emergence: Layer-wise redundancy scores show that, depending on system architecture, most or nearly all information at a given order may be strictly non-redundant. In practice, only a minimal set of orders suffice to capture structural or dynamical complexity, and these orders correspond to the “critical” scaffold of collective behavior (Kirkley et al., 5 Jan 2026).
• Phase structure and critical regimes: The explicit presence of higher-order interactions is both necessary and sufficient to produce observed phenomena such as explosive synchronization, first-order transitions, multifractal scaling, and the smoothing or destruction of sharp consensus transitions (Robiglio et al., 2024, Bian et al., 2024, Schawe et al., 2021, Battiston et al., 2021).
• Domain dependence: The critical order varies systematically with network type: small social groups lack substantial higher-order motifs beyond triangles, whereas biochemical or multi-class systems possess essential higher-order structure up to order 7–8 (Bian et al., 2024, Wegner et al., 2024).

6. Open Challenges and Future Directions

Several unsolved problems persist in the quantitative theory and application of critical higher orders:

• Mathematical characterization of universality: While many examples show that higher-order terms change the universality class, a comprehensive theory linking topological/hypergraph invariants with phase structure remains incomplete (Battiston et al., 2021).
• Efficient algorithms for high orders: Both inferential (SGCM, kernel permutation) and structural reducibility methods are computationally limited for large d or large numbers of layers. Improved heuristics, sampling, or model reduction strategies are required to scale these analyses to very large or dense hypergraphs (Liu et al., 2023, Kirkley et al., 5 Jan 2026).
• Causal inference and data-driven validation: Determining the causal, not just structural, necessity of a critical order requires advanced experimental designs and statistical interventions, especially in empirical domains (e.g., neuroscience, systems biology) where underlying generative processes are only partially observed (Battiston et al., 2021, Liu et al., 2023).

Advances in these directions will solidify the role and operational characterization of critical higher orders of interaction, enabling predictive modeling, robust control, and principled reduction of complex systems across scientific disciplines.