Interaction Scaling in Complex Systems

Updated 7 December 2025

Interaction Scaling is a framework that quantifies how variations in interaction strength, range, and frequency drive emergent behaviors, phase transitions, and explainability in physical, computational, and agent systems.
The concept leverages power-law regimes and universality, with applications ranging from quantum metrology and many-body physics to network theory and virtual environments.
Methodologies such as analytic scaling ansatz, curriculum design, parameter fitting, and sparse decoding enable precise predictions and adjustments in models of turbulence, quantum interactions, and explainable AI.

Interaction Scaling denotes the systematic analysis and exploitation of how the strength, range, or frequency of interactions in physical, computational, or agentic systems influences emergent behaviors, criticality, precision, and explainability. Central to this concept are power-law scaling regimes, universality, and phase transitions driven by interaction parameters. Recent advances span quantum metrology, many-body physics, network theory, turbulence, agent architectures, and virtual environments, each elucidating domain-specific interaction-scaling phenomena and their practical implications.

1. Scaling Laws in Physical and Quantum Systems

Interaction scaling critically determines the qualitative and quantitative behavior of both classical and quantum many-body systems. In quantum metrology, the Heisenberg scaling limit (precision $\propto N^{-1}$ ) may be surpassed by engineering $k$ -body interactions, yielding "super-Heisenberg" sensitivity $\delta\theta\propto N^{-k}$ for entangled probes and $N^{-(k-1/2)}$ for unentangled states. Experimental realization with nonlinear optical probes achieves $k=2$ scaling ( $\delta\theta\propto N^{-3/2}$ ) over two orders of magnitude before saturation effects curtail the enhancement (Napolitano et al., 2010).

Ancilla-assisted protocols enable Heisenberg scaling without entanglement by leveraging product-state probes and probe–ancilla coupling, with local measurement on the ancilla sufficing to extract all relevant quantum Fisher information (Fan et al., 23 Jul 2024). Critical to these schemes is optimal selection of initial probe and ancilla states, interaction strengths, and measurement timings, with the precision envelope decaying exponentially under dephasing, which restores the standard scaling in noisy regimes.

In black-hole spacetimes, the Casimir–Polder interaction between two-level atoms exhibits a crossover governed by surface gravity $\kappa$ : at short distances ( $r \ll r_c \sim 1/\kappa$ ), the energy decays thermally as $1/r^2$ , mimicking the Hawking/Unruh effects; at large separations ( $r \gg r_c$ ), the scaling steepens to $1/r^4\, \log r$ , reflecting noninertial spacetime curvature (Menezes et al., 2017). Analogous behavior arises for accelerated atoms in Minkowski backgrounds, emphasizing the geometric origin of interaction scaling.

In extended systems, power-law tails in interaction kernels produce fractional diffusion operators in the scaling limit, while the normalization and range-selection dictate whether the macroscopic limit is standard (Laplacian-driven) or anomalous (fractional) (Bovier et al., 2015).

2. Interaction Range and Universality in Critical Phenomena

Finite-size scaling theories incorporate interaction-scaling exponents to describe phase transitions in spin systems and nonequilibrium models. The extended scaling ansatz takes the form $M(\ell,L) = L^{-\beta/\nu} \mathcal{M}(\ell L^{y_\ell})$ , where $y_\ell$ matches the scaling dimension for a field-like interaction, yielding $M(T_c,\ell)\sim \ell^{1/\delta_\ell}$ with $\delta_\ell$ universally coinciding with the critical exponent for applied fields (e.g., $\delta_\ell\sim5$ in the BEG model) (Özkan et al., 2010).

Models with variable interaction range $\Lambda$ (e.g., block-voter or majority-vote) show that critical amplitudes $M_c(\Lambda)\sim \Lambda^{-X}$ , $\chi_c(\Lambda)\sim \Lambda^{-Y}$ , where the exponents $X$ , $Y$ , $Z$ encode the decay with interaction range, and universal data collapse confirms that scaling functions are independent of both $N$ and $\Lambda$ (Sampaio-Filho et al., 2013). Such frameworks extend canonical Ising or mean-field static exponents with "long-range" interaction-scaling exponents.

In many-body localization, finite-size scaling generalizes to include an interaction-scaling exponent $\mu$ in the critical disorder $W_c(J)\sim J^{\mu}$ , so that the normalized scaling variable is $X = W/J^{\mu}$ and the transition curves for different $J$ collapse to a master function. Empirically, $\mu<1$ (e.g., $\mu=0.28$ ) implies a sub-linear stabilizing effect of interactions against localization (Kudo et al., 2018).

3. Interaction Scaling in Networks and Agents

In real-world interaction networks, degree distributions frequently exhibit scale-free power laws with exponents $1<\gamma<2$ , explained not by vertex-growth preferential attachment but by constant edge addition and positive-feedback reinforcement (Pitman–Yor process), yielding $P(k)\propto k^{-(1+\alpha)}$ and highly connected hubs far denser than standard network models (Crane et al., 2015). This "atypical" scaling is robust across collaboration, communication, and social graphs.

Large-agent environments require systematic interaction scaling in the Generation–Execution–Feedback (GEF) loop. Axes of environment scale include state-space and action-space cardinalities and task horizons, along with qualitative dimensions of realism, diversity, feedback density, and robustness—each controlled by explicit scaling methodologies such as curricula, co-evolution, automated evaluation, or interactive execution. Benchmarks are now structured to probe compositional depth, parallel tool invocation, and dynamic curricula (Huang et al., 12 Nov 2025).

For web and tool-using agents, interaction scaling is implemented as horizon-scheduled test-time interaction (TTI): training under a curriculum of increasing interaction steps yields agents that adaptively balance exploration and exploitation, outperforming static reasoning-token scaling (chain-of-thought, CoT) under partial observability (Shen et al., 9 Jun 2025). Empirical gains are most pronounced when per-episode horizons are flexibly scheduled, with both success rate and sample efficiency improved versus fixed-length rollouts.

In visual reasoning, over-turn masking and multi-turn data generation enable deep interaction scaling—despite training-time constraints, agents generalize to trajectories spanning tens of reasoning steps, with accuracy continually improving at higher turn limits. Removal of any of the core scaling components substantially degrades multi-turn performance (Lai et al., 9 Sep 2025).

4. Scaling in Virtual Environments and Human–Scene Interaction

In virtual and telepresence environments, object–user scale mismatches reveal a "plausibility paradox," where physical realism (e.g., Newton–Euler laws scaled down by $\alpha$ or $s$ ) conflicts with user expectations. At small scales ( $s\ll1$ ), "true" physics leads to perceived unnatural dynamics; most subjects rate "movie physics" as more plausible, even when strictly inaccurate (Pouke et al., 2019, Pouke et al., 2021). Quantitative forced-choice and Likert studies consistently favor human-scale interactions at small $s$ and divergent, context-collapsed blending at large $s$ .

Design guidelines now recommend blended kinematics, pseudo-haptic cues, and adjustable "authenticity sliders" for interaction scaling in VR/telepresence, especially in collaborative or multiscale settings.

Scaling up human–scene interaction datasets (e.g., TRUMANS) incorporates extensive contact-informed motion and geometry augmentation, enabling autoregressive diffusion models to synthesize arbitrary-length HSI sequences that generalize zero-shot across diverse scenes with near-human fidelity. Robustness is maintained via per-vertex contact annotation and temporal smoothing, with model performance validated by both quantitative metrics and blinded human studies (Jiang et al., 13 Mar 2024).

5. Interaction Scaling in Computational Methods and Explainability

In computational chemistry, scaling procedures transform cost-inefficient correlated interaction energies (e.g., CCSD(T)) into practical approximations via proportionality against cheaper methods (e.g., MP2), with fitted scaling coefficients nearly invariant under distance and transferable across classes of noncovalent interactions. These scaling corrections reduce errors by an order of magnitude and propagate through basis-set extrapolations with minimal computational overhead (Fabiano et al., 2015).

In interpretable machine learning, scaling algorithms for post-hoc feature interaction attributions (SPEX) leverage sparse Boolean Fourier transforms and BCH channel decoding to extract high-order interactions at LLM scales ( $n\sim1000$ ) with sublinear sample complexity. The method recovers key interaction sets that dominate model outputs, outperforming marginal indices and aligning with human annotations in multi-hop QA and VQA scenarios (Kang et al., 19 Feb 2025). The underlying scaling assumes natural sparsity and low interaction degree in real data; violations thereof may compromise faithful recovery.

6. Scaling Transformations, Nonlocality, and Symmetry

Group-theoretic approaches show that scaling transformations (dilatations) furnish both operator/coordinate and spinor/unitary representations, culminating in mixed vector/chiral interaction vertices invariant under combined coordinate, momentum, and spinor scaling. Nonlocal interaction Lagrangians generally vary under scaling—analogous to running effects in renormalization; only for massless fields, fixed-point form factors, and chiral vertices does exact scale invariance (and associated conserved Noether currents) emerge. An additional spin–angular-momentum term ("scalum") appears to close the conservation law, with experimental detection feasible via polarized, spin-dependent electron–hadron scattering (Wanng, 2012).

7. Scaling of Interaction and Coherent Structures in Turbulence

In high-Reynolds-number wall turbulence, resolvent-mode analysis exposes geometric self-similarity: modal amplifications scale as $\lambda^3$ and nonlinear triadic coupling coefficients as $\lambda^{-5/2}$ , with the composite interaction scaling as $\lambda^{7/2}$ for scaled hierarchy factor $\lambda$ . This structure enables reduced-order modeling based on a single reference triad, analytic generation of all log-layer couplings, and physical insight into self-sustaining turbulence cascades—limited to regions where logarithmic scaling remains valid (Sharma et al., 2016).

Interaction Scaling thus organizes and quantifies how fundamental or engineered changes to interaction strength, range, or multiplicity drive phase boundaries, universality classes, precision, explainability, and emergent behavior across a wide range of physical, computational, and human-agent systems. Standard methodologies combine analytic scaling ansatz, curriculum and horizon design, parameter-fitting, sparse decoding, and post-hoc adjustment or blending, with practical guidelines tailored to domain and context.