Multi-Order Interactions in Complex Systems

• Multi-order interactions are dependencies among groups of more than two entities, modeled using hypergraphs, simplicial complexes, and tensors.
• They influence dynamics such as epidemic spreading, synchronization, and pattern formation by introducing higher-order effects.
• Advanced computational methods and statistical models help infer order-dependent structures, enhancing community detection and system stability analysis.

Multi-order interactions refer to processes, dependencies, or couplings among groups of more than two entities, occurring at varying levels of group cardinality ("order"), and described by distinct mathematical or mechanistic frameworks at each order. This concept generalizes classical pairwise interaction paradigms, enabling the study of systems whose structure and dynamics are fundamentally shaped by higher-order groupings, such as triads, quadruplets, or arbitrary-size collectives. Multi-order interactions arise in physical, biological, ecological, social, and engineered systems whenever the properties or evolution of a system depend on simultaneous relations among multiple units, and their characterization often demands formalism beyond simple graphs—typically hypergraphs, simplicial complexes, multi-way tensors, or context-dependent value functions.

1. Mathematical Foundations and Representations

Formal models of multi-order interactions utilize hypergraphs, simplicial complexes, and associated incidence, adjacency, and Laplacian tensors. A hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$ consists of a set of nodes $\mathcal{V}$ and a set of hyperedges $\mathcal{E}=\bigcup_{o=2}^D\binom{\mathcal{V}{o}}$, where each hyperedge $e$ involves $|e|$ nodes—defining its order. Multi-order structure arises when edge cardinalities vary, generating stratified layers indexed by interaction order. Simplicial complexes further impose inclusion closure on groupings, mapping them to nested sets of $k$-simplices.

Incidence tensors and adjacency tensors formalize the connectivity of arbitrary-order edges. For fixed $k$-uniform interactions, $\mathcal{A}^{(k)}_{i_1\dots i_k}=1$ iff $\{i_1,\dots,i_k\} \in \mathcal{E}$. Multi-order Laplacians generalize the combinatorial Laplacian, yielding operators $L^{(k)}$ whose spectra encode diffusion, synchronization, and spreading properties specific to each order (Lucas et al., 2020, Qu et al., 2021). Practical signal-processing and recovery tasks utilize total-variation regularization across multi-order Laplacians, effectively enforcing smoothness at each scale (Qu et al., 2021).

Order-specific filtering and decomposition methods, as in size-based filtering and layered projections, are critical for empirical analysis, revealing phenomena masked by full aggregation (Landry et al., 2023). In stochastic block models for community detection, multi-order affinity tensors $\mathbf{W}^{(o)}$ govern edge probabilities per order, enabling accurate inference of mesoscale organization (Nakajima et al., 26 Nov 2025).

2. Analytical Dynamics and Equilibrium Phenomena

Multi-order interactions qualitatively alter dynamical processes on networks:

• Epidemic processes: In multi-group SIS models with higher-order contagion over hypergraphs, mean-field ODEs

$$\dot x_i = -\gamma_i\,x_i + \beta_1\,(1-x_i) \sum_{j} a_{ij} x_j + \beta_2\,(1-x_i) \sum_{j,k} b_{ijk} x_j x_k$$

introduce new dynamical domains (Cisneros-Velarde et al., 2020). Notably, with sufficient higher-order rates $\beta_2$, a bistable regime appears: both disease-free and endemic equilibria can be locally stable, with transitions governed by spectral conditions and initial conditions.

• Synchronization in oscillator networks: Higher-order couplings $G_e^{(k)}$ among $k$-body hyperedges are encoded in multi-order Laplacians $$L^{\mathrm{MO}} = \sum_k \gamma_k / \langle K^{(k)} \rangle\, L^{(k)}$$, where $\gamma_k$ is the coupling strength. The stability of synchrony is determined by the Lyapunov exponents $-\Lambda_\alpha^{(\mathrm{MO})}$. All-to-all higher-order interactions broaden the Laplacian spectrum, consistently amplifying the spectral gap and enhancing synchronizability (Lucas et al., 2020, Skardal et al., 2021). Introducing or tuning multi-order weights allows optimal collective dynamics and a trade-off between peak performance and robustness (Skardal et al., 2021).
• Pattern formation: Generalized reaction-diffusion processes on hypergraphs incorporate $d$-body Laplacians $L^{(d)}$ and nonlinear diffusion terms, modifying the onset and structure of Turing patterns. Analytical conditions for instability, involving effective diffusion coefficients from multiple orders, show that higher-order diffusion can both enhance and suppress pattern formation, shifting classical thresholds (Muolo et al., 2022).
• Phase transitions in oscillator systems: Arbitrary asymmetric higher-order Kuramoto interactions contribute to reduced-order equations in terms of "effective orders," defined combinatorially on the coupling coefficients, governing bifurcation and multi-stability beyond what is possible in pairwise models (Costa et al., 10 Jan 2025).

3. Game-Theoretic and Statistical Interaction Decomposition

In value-function or neural network contexts, multi-order interactions are defined via generalizations of Shapley value and marginal contributions. For $n$ features, the $m$th-order component of the interaction between $i$ and $j$ is given by

$$I^{(m)}(i,j) = \mathbb{E}_{S \subset N \setminus \{i,j\}, |S|=m}[ \Delta_{ij} v(S) ]$$

with $\Delta_{ij}v(S)$ capturing joint effects not explainable by separate marginal contributions (Zhang et al., 2020). Fundamental properties such as linearity, efficiency, recursive relationships, and symmetry hold. For computational modeling, multi-order interaction tensors can be realized efficiently with rank-1 PARAFAC factorization in MOI layers in neural architectures, enabling explicit control over interaction orders with linear complexity (Lee et al., 2021, Li et al., 2022).

4. Empirical Analysis and Structure Discovery

Multi-order interactions manifest stratified, scale-dependent patterns across empirical systems. Filtering by edge size uncovers distinct roles for individuals, variations in assortativity, centrality, and community structure that are invisible in aggregate projections (Landry et al., 2023). Structural metrics (effective information, degree assortativity, betweenness centrality) reveal that intermediate group sizes can maximize network assortativity and information uniqueness, while community organization may shift dramatically as one filters by interaction order.

Recent work demonstrates that optimal stochastic block model partitions often require multiple distinct order-dependent affinity matrices, reflecting fundamentally different mesoscale mechanisms at different group sizes—a property observed in real contact, co-citation, and collaboration hypergraphs (Nakajima et al., 26 Nov 2025).

5. Impact on Dynamics, Stability, and Biodiversity

Multi-order interactions contribute nontrivially to the stability and diversity of ecological, physical, and chemical systems:

• Ecology: In competitive Lotka–Volterra-type models, augmenting pairwise with higher-order (e.g., triple-wise) interaction terms yields nontrivial stabilization effects; in symmetric (identical) communities, any nonzero higher-order fraction suffices to stabilize intransitive networks, but in heterogeneous or structured populations, a finite threshold must be exceeded, and sometimes no stabilizing solution exists (Duran-Sala et al., 15 Jan 2025, Vandermeer et al., 2023). In complex community models mixing intransitive cycles and trait-mediated predator-prey effects, multi-order interactions can spawn limit cycles, multi-stability, and deterministic chaos.
• Multiphase liquids and biomolecular condensates: Composition-dependent higher-order terms (e.g., binary-cubic and ternary couplings in free-energy expansions) can both promote and oppose phase separation, destabilize or stabilize extra phases, and render linear-stability analysis inadequate; only full equilibrium minimization reveals the rich phase diagram topology induced by higher-order physical interactions (Luo et al., 11 Mar 2024).
• Quantum and condensed matter systems: Multi-spin $n$-local couplings can be robustly distinguished from lower-order effects via dynamic (Rabi-type) detection methods, with sensitivity scaling optimally in $n$ (Bergamaschi et al., 2021). In itinerant magnets, frustrated exchange and higher-order multi-site interactions stabilize unexpected collinear multi-Q states, as observed in spin-polarized STM experiments and modeled analytically (Gutzeit et al., 2022).

6. Computational, Algorithmic, and Modeling Considerations

Algorithmic frameworks for multi-order interaction inference and modeling exploit convex decomposition, greedy extraction, and context-dependent factorization:

• Efficient recovery of time-ordered multibody interactions from Markov-chain data leverages parsimonious representation via nested interaction types, yielding substantial reductions in model complexity, and validating the robustness of complexity measures against statistical noise (Alvarez-Rodriguez et al., 2021).
• In neural-system design, explicit multi-order context modules (e.g., MOI and gated aggregation layers) increase the expressive capacity without incurring quadratic cost, and optimize discriminatory performance particularly in middle-order synergies (Lee et al., 2021, Li et al., 2022).

7. Open Questions and Outlook

Central challenges in multi-order interaction research include principled parameterization and inference of interaction order-dependence in empirical systems, the development of null models and statistical tests for order-stratified subhypergraphs, and extension of multi-order Laplacian theory and dynamics to temporal, multiplex, and data-rich settings. Understanding the dynamical and structural consequences of multi-order stratification—especially in questions of stability, phase behavior, and information processing—remains a critical frontier (Battiston et al., 2020, Nakajima et al., 26 Nov 2025, Landry et al., 2023). A plausible implication is that robust modeling and control of real-world complex systems will require explicit attention to the stratification and generative mechanisms of multi-order interactions.