Higher-Order Behaviors in Complex Systems

• Higher-Order Behaviors (HOBs) are emergent collective phenomena that rely on group interactions beyond simple pairwise connections, producing complex dynamics.
• They are modeled using hypergraphs, simplicial complexes, and spectral methods, which capture nondecomposable group-level dynamics.
• Applications include synchronization, contagion, cooperation, and cognitive dynamics, illustrating how adaptive group strategies shape system behavior.

Searching arXiv for the cited papers to ground the article in current literature. Higher-Order Behaviors (HOBs) denote system-level patterns that arise when interactions, update rules, or inference procedures depend irreducibly on groups rather than dyads. In the network-dynamics literature, the term typically refers to emergent collective regimes—such as explosive transitions, multistability, clustered synchronization, or higher-order adaptive responses—that cannot be captured by pairwise couplings alone; in adjacent literatures it also denotes adaptive behaviors driven by group-level information and recursive belief-aware interaction patterns. Across these usages, the unifying feature is nondecomposability: the relevant mechanism or observable depends on joint configurations of three or more units, or on nested models of other agents, rather than on sums of independent pairwise effects (Battiston et al., 6 Oct 2025, Liu et al., 21 Aug 2025, Mancastroppa et al., 5 Feb 2026, Keurulainen et al., 2024).

1. Conceptual scope and terminological variants

In dynamical systems on higher-order networks, HOBs are usually defined as emergent collective patterns that intrinsically depend on group interactions and group-level organization. The general higher-order dynamical framework writes node dynamics as intrinsic evolution plus interaction terms of multiple orders,

$\begin{split} \dot{x}_i = F_i(x_i) &\;+\;\sigma \sum_{j=1}^{n} A_{ij}\, G^{(2)}(x_j; x_i) \ &\;+\;\sigma_{\triangle} \sum_{j,k=1}^{n} B_{ijk}\, G^{(3)}(x_j,x_k; x_i) + \dotsb \end{split}$

so that the state of a unit depends on configurations of several neighbors at once rather than on pairwise influences only (Battiston et al., 6 Oct 2025). Within this usage, HOBs include explosive synchronization, hysteresis, cluster states, altered contagion thresholds, and topology-constrained collective states.

A complementary distinction separates higher-order mechanisms from higher-order behaviors. In this vocabulary, mechanisms are the underlying group-level dynamical rules, whereas behaviors are the statistical signatures or macroscopic patterns induced by those rules. The distinction is explicit in work that compares higher-order mechanisms (HOMs) with HOB measures derived from information theory, and in work defining HOBs as whole-minus-sum synergistic excess in predictability or mutual information (Danovski et al., 27 Feb 2026, Barà et al., 14 Dec 2025). This separation is important because statistical higher-order signatures need not imply a same-order mechanism.

In contagion studies, the term also refers to adaptive responses based on higher-order information. Here the higher-order aspect lies not only in the spreading substrate but also in the behavior itself: awareness and adaptation depend on the state of groups or hyperedges, not only on infected neighbors. In cognitive and active-learning settings, higher-order behavior denotes recursive, belief-aware communication, such as strategic teaching and pragmatic questioning based on nested beliefs about another agent’s internal state (Mancastroppa et al., 5 Feb 2026, Mancastroppa et al., 9 Jan 2026, Keurulainen et al., 2024). These usages are heterogeneous, but all treat behavior as higher-order when it is mediated by genuinely multi-agent context.

2. Structural substrates and formal representations

The principal combinatorial substrates are hypergraphs and simplicial complexes. A hypergraph $H=(V,\mathcal{E})$ contains hyperedges $e\subseteq V$ of arbitrary size $|e|\ge 2$, whereas a simplicial complex is downward closed: if a simplex is present, all its faces are also present. This difference has dynamical consequences, because downward closure amplifies structural constraints such as degree heterogeneity, overlap, and topological holes (Battiston et al., 6 Oct 2025).

Several structural descriptors have become central to HOB analysis. One is hyperedge overlap, quantified locally by $T_i^{(m)}$ and globally by $\mathcal{T}^{(m)}$, which measure how strongly order-$m$ hyperedges around a node reuse the same neighbors. Another is the notion of higher-order components: two hyperedges are $m$-th-order connected if they share at least $m$ nodes, and $G_m$ denotes the fraction of nodes in the largest $H=(V,\mathcal{E})$0-th-order component. These quantities are not reducible to ordinary graph connectivity and control whether higher-order processes can organize globally (Malizia et al., 2023, Kim et al., 2022).

Spectral representations provide the corresponding analytical backbone. For clique complexes with dyadic and triadic couplings, the relevant linear operator is the composite Laplacian

$H=(V,\mathcal{E})$1

where $H=(V,\mathcal{E})$2 and $H=(V,\mathcal{E})$3 encode 1- and 2-simplex interactions. More generally, higher-order Kuramoto models are linearized through multiorder Laplacians, and topological models on simplicial complexes are governed by Hodge Laplacians

$H=(V,\mathcal{E})$4

These operators organize synchronizability, harmonic constraints, and mode structure, and they make explicit that higher-order dynamics depends on more than the 1-skeleton of the system (Skardal et al., 2021, Battiston et al., 6 Oct 2025).

3. Canonical collective phenomena

A central empirical and theoretical result is that higher-order interactions generate collective behaviors absent from pairwise models. In synchronization problems, higher-order couplings can produce explosive synchronization, hysteresis, cluster synchronization, twisted states, chimeras, heteroclinic dynamics, and chaos. In contagion problems, they generate discontinuous outbreak transitions, bistable endemic and disease-free phases, and critical-mass effects. These behaviors arise because higher-order terms enter nonlinear macroscopic equations through genuinely multi-node configurations rather than through sums of pairwise pressures (Battiston et al., 6 Oct 2025).

The effect is not monotonic. In a clique-complex oscillator model with fixed total coupling $H=(V,\mathcal{E})$5, increasing the triadic bias $H=(V,\mathcal{E})$6 broadens the spectrum of the composite Laplacian, increases $H=(V,\mathcal{E})$7, decreases $H=(V,\mathcal{E})$8, improves the best achievable synchronization for optimized frequencies, but worsens typical and worst-case synchronization. In that sense, higher-order interactions expand the dynamical range rather than uniformly increasing coherence (Skardal et al., 2021). A plausible implication is that higher-order structure can function as a control resource, but only under appropriate alignment between local dynamics and high-eigenvalue modes.

The microscopic organization of higher-order structure also matters. Hyperedge overlap provides a sharp example: in both higher-order contagion and phase synchronization, low $H=(V,\mathcal{E})$9 supports explosive transitions and bistability, while high $e\subseteq V$0 yields continuous transitions and gradual cluster formation. Likewise, for pure higher-order contagion from a single seed, a giant second-order component is required for macroscopic invasion; ordinary giant connectivity is insufficient when activation requires multi-node overlap (Malizia et al., 2023, Kim et al., 2022). A common misconception is therefore that higher-order interactions alone guarantee abrupt behavior. The literature instead shows that abruptness generally depends on both interaction order and mesoscopic organization.

Recent higher-order swarmalator models extend the same point to coupled space-phase dynamics. Triadic phase interactions generate spatially coherent stationary states and gas-like nonstationary states that do not occur in pairwise models, and they induce bistability among async, thick phase wave, thin phase wave, and sync states. Even full synchronization can persist with repulsive pairwise phase coupling provided the higher-order term is sufficiently positive (Anwar et al., 23 Apr 2025).

4. Adaptive, strategic, and cooperative forms of HOBs

Higher-order adaptivity makes the interaction structure itself responsive to group state. In a 3-uniform epidemic model, infection in a hyperedge with $e\subseteq V$1 infected nodes occurs at rate

$e\subseteq V$2

while hyperedge breaking occurs at

$e\subseteq V$3

and new hyperedges are formed with susceptible bias

$e\subseteq V$4

Here $e\subseteq V$5 corresponds to pairwise-like adaptivity, whereas $e\subseteq V$6 makes adaptation depend nonlinearly on group infection burden. Both pairwise-like and higher-order adaptivity raise the outbreak threshold, but only higher-order adaptivity reduces or eliminates the bistable region and can shift phase transitions from discontinuous to continuous (Liu et al., 21 Aug 2025).

A closely related line studies adaptive behavior without rewiring, through behavioral modulation

$e\subseteq V$7

When awareness is computed from the number of infectious groups,

$e\subseteq V$8

higher-order information-based adaptation is more effective in limiting contagion and does so with lower social cost than analogous pairwise strategies. The mechanism is heterogeneous risk perception: nodes with large hyperdegree, their neighborhoods, and large groups become more alert, so the process loses access to the nodes and groups that would otherwise sustain it (Mancastroppa et al., 5 Feb 2026). In a related formulation, once adaptive behavior suppresses configurations with more than one infected in a group, the higher-order IBMF equations reduce near threshold to the pairwise SIS form, and the explosive transition is neutralized (Mancastroppa et al., 9 Jan 2026).

Higher-order behavior also appears in evolutionary updating. In public-goods games on hypergraphs, strategy revision can itself be group-mediated: an individual first selects a group and then selects whom to imitate within that group. The group-mutual comparison mechanism, described as selecting an outstanding group and then imitating a random individual within this group, prominently promotes cooperation, and higher-order strategy updates generally improve cooperation relative to pairwise updates on most higher-order networks (Wang et al., 26 Jan 2025). More broadly, a theoretical framework for evolutionary game dynamics on hypernetworks finds that higher-order interactions enable lower thresholds for the emergence of cooperation and favor cooperation in large-scale systems, opposite to lower-order scenarios (Guo et al., 11 Jan 2025).

Outside network contagion and games, higher-order behavior has a recursive cognitive meaning. In active learning with human feedback, level-3 humans infer an agent’s belief from its queries and choose answers to steer the agent’s posterior toward the truth; level-4 agents ask queries that are simultaneously informative and communicative; level-5 humans infer whether questions are literal or pragmatic. These are higher-order behaviors because action depends on nested beliefs such as what the human believes the agent believes (Keurulainen et al., 2024).

5. Detection, measurement, and inference

Because HOBs are not equivalent to their underlying mechanisms, the literature places strong emphasis on behavioral signatures. One major strategy is multivariate information theory. In simplicial Ising and social contagion models, the key observable is total dynamical O-information,

$e\subseteq V$9

with negative values indicating synergy-dominated group behavior. On triplets, $|e|\ge 2$0 becomes markedly more negative on true 2-simplices than on 3-cliques lacking a 2-simplex, whereas a sum of pairwise transfer entropies does not distinguish the two. This establishes that some higher-order mechanisms are invisible to low-order observables (Robiglio et al., 2024).

A second line asks whether HOB signatures can identify the order of the underlying mechanism. In simplicial SIS contagion, cross-order induced behaviors emerge: behavioral signatures can appear at interaction orders where no direct mechanism is present. These cross-order HOBs are not simply induced by structural correlations such as nestedness and hyperedge overlap; they appear in the neighborhood of any HOM. Among the measures tested, synergy is the most reliable indicator of the true order where the underlying mechanism is at play (Danovski et al., 27 Feb 2026). A standard caution follows: observing higher-order statistical dependence on a group does not by itself prove that the mechanism acts at that same order.

A third approach uses whole-minus-sum excess in predictability or information. For a target $|e|\ge 2$1 and sources $|e|\ge 2$2, the prediction-based synergy is

$|e|\ge 2$3

and the information-theoretic synergy is

$|e|\ge 2$4

In simulated additive systems with independent sources, $|e|\ge 2$5 tends to vanish, whereas $|e|\ge 2$6 is often positive; in systems with explicit group interactions, both become positive. Applied to cardiovascular data, $|e|\ge 2$7 reveals synergistic high-order behaviors in additive arterial-pressure relations, while both $|e|\ge 2$8 and $|e|\ge 2$9 detect synergy in non-additive timing-pressure interactions (Barà et al., 14 Dec 2025). This suggests a practical division of labor: information measures are broader detectors of multivariate dependence, whereas prediction measures are more selective for mechanism-like non-additivity.

6. Applications, misconceptions, and open problems

HOBs now appear across contagion, synchronization, cooperation, active matter, human–AI interaction, and physiology. Group contagion models use them to explain explosive outbreaks, bistability, and targeted mitigation; oscillator models use them to explain optimized or broadened collective dynamics; evolutionary models use them to formalize group-mediated strategy change and multi-level selection; swarmalator models use them to generate new spatiotemporal phases; cardiovascular analyses use them as candidate biomarkers of regulatory organization (Mancastroppa et al., 5 Feb 2026, Skardal et al., 2021, Wang et al., 26 Jan 2025, Anwar et al., 23 Apr 2025, Barà et al., 14 Dec 2025).

Several recurrent misconceptions are corrected by current work. First, higher-order interactions do not automatically imply explosive transitions; low hyperedge overlap can be necessary for explosivity and bistability (Malizia et al., 2023). Second, higher-order behavioral signatures do not uniquely identify higher-order mechanisms at the same order, because cross-order induced behaviors can arise in neighboring structural contexts (Danovski et al., 27 Feb 2026). Third, stronger higher-order coupling is not universally beneficial: in synchronization, it can improve optimal performance while worsening typical performance (Skardal et al., 2021). These points make HOBs a problem of joint structure, mechanism, and observable, not of interaction order alone.

The main limitations are equally consistent across the literature. Many analytical results rely on mean-field closures, strongly synchronized linearizations, or clique-complex assumptions; many empirical studies use static hypergraphs, moderate system sizes, and interaction orders concentrated at 2- or 3-body terms. Open directions include interactions beyond triads, systematic taxonomies of higher-order coupling functions, control and optimization under noncommuting generalized Laplacians, scalable inference under noise, temporal and adaptive higher-order structures, and clearer criteria for minimal necessary order (Battiston et al., 6 Oct 2025, Mancastroppa et al., 9 Jan 2026, Barà et al., 14 Dec 2025). A plausible implication is that future progress will depend less on introducing ever more higher-order terms than on establishing when those terms are structurally indispensable and when their behavioral signatures are distinguishable from lower-order surrogates.