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arxiv: 2511.06766 · v2 · submitted 2025-11-10 · 🌊 nlin.AO

Emergent synchrony in oscillator networks with adaptive arbitrary-order interactions

Dhrubajyoti Biswas , Arpan Banerjee This is my paper

Pith reviewed 2026-05-18 00:17 UTC · model grok-4.3

classification 🌊 nlin.AO
keywords adaptive Kuramoto modelhigher-order interactionshypergraph synchronizationorder parameter dynamicsthermodynamic limitphase transitionsfinite-size fluctuations
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The pith

An adaptive Kuramoto model with hyperedges of arbitrary order yields exact closed equations for the collective order parameter in the thermodynamic limit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates a Kuramoto model in which oscillators couple through adaptive hyperedges that can involve any number of nodes. It obtains closed analytical equations for the synchronization order parameter by taking the limit of infinitely many oscillators. These equations describe the collective evolution and allow prediction of the system's long-term state from the interaction parameters. Numerical simulations of the full system match the analytical trajectories and identify several qualitatively distinct regimes separated by phase transitions. Order-parameter distributions further show that fluctuations in finite-sized networks can shift the final dynamics away from the infinite-limit prediction.

Core claim

By deriving the exact order parameter dynamics in the thermodynamic limit, analytical expressions governing the collective dynamics are obtained. Subsequent numerics confirm the analytical predictions, in addition to capturing qualitatively different dynamical regimes and phase transitions. Further investigations based on order parameter distributions demonstrate how fluctuations, arising due to finite system size, can influence the long-term system dynamics.

What carries the argument

The exact closed dynamical equations for the synchronization order parameter that result from averaging the adaptive arbitrary-order hyperedge interactions in the thermodynamic limit.

Load-bearing premise

The adaptive interactions take a specific functional form that allows the thermodynamic limit to produce exact closed equations for the order parameter without further approximations.

What would settle it

Numerical integration of the full set of oscillator equations for successively larger system sizes should converge to the analytical order-parameter trajectories; sustained mismatch at large but finite sizes would show the closure is not exact.

Figures

Figures reproduced from arXiv: 2511.06766 by Arpan Banerjee, Dhrubajyoti Biswas.

Figure 1
Figure 1. Figure 1: (a): Schematic representation of hyperedges in the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of fd(r) as a function of r ∈ [0, 1] for (a) d = 1 and (b) d = 2, where the colours denote different values of γ. represents the simplest possible functional form of a polyno￾mial adaptation function, which has previously been shown to induce both continuous and explosive phase transitions [81], and has been set up such that it introduces qualitatively differ￾ent adaptation behaviour (i.e., increasing… view at source ↗
Figure 3
Figure 3. Figure 3: (a)-(d): Left: Graphical solution of Eq. (21). The black and magenta curves denotes the LHS and RHS, respectively, whereas the green and red broken vertical lines (in inset) denote the stable and unstable steady-state solutions respectively, as determined from the eigenvalues of the Jacobian of Eqs. (19) and (20); Right: Temporal variation of order parameter r(t). The blue and red curves denote the asynchr… view at source ↗
Figure 4
Figure 4. Figure 4: (a)-(j): Variation of the order parameter [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: r 0 0.25 0.5 0.75 1 0 0.2 0.4 0.6 p ( r ) #10 1 (a) ϵ1 = 0.6 r 0 0.25 0.5 0.75 1 0 0.2 0.4 0.6 p ( r ) #10 1 (b) ϵ1 = 1.3 r 0 0.25 0.5 0.75 1 0 0.2 0.4 0.6 p ( r ) #10 1 (c) ϵ1 = 1.8 r 0 0.25 0.5 0.75 1 0 0.2 0.4 0.6 p ( r ) #10 1 (d) ϵ1 = 2.6 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

Dynamics of complex systems are often driven by interactions that extend beyond pairwise links, underscoring the need to establish a correspondence between interpretable system parameters and emergent phenomena in hypergraph-based networks. The current work formulates an adaptive Kuramoto model that incorporates hyperedges of arbitrary order and explores their effects on synchronization. By deriving the exact order parameter dynamics in the thermodynamic limit, analytical expressions governing the collective dynamics are obtained. Subsequent numerics confirm the analytical predictions, in addition to capturing qualitatively different dynamical regimes and phase transitions. Further investigations based on order parameter distributions demonstrate how fluctuations, arising due to finite system size, can influence the long-term system dynamics. These results provide important insights and can have diverse applications, such as designing optimal surgical procedures for drug-resistant epilepsy and identifying the sources of rumours in a social network.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript formulates an adaptive Kuramoto model on hypergraphs that incorporates interactions of arbitrary order. It claims to derive exact closed equations for the global order parameter in the thermodynamic limit, obtains analytical expressions for collective dynamics, and validates them via numerical simulations that also reveal qualitatively different regimes and phase transitions. Finite-size fluctuations and their influence on long-term behavior are additionally examined, with suggested applications to epilepsy surgery and rumor source identification.

Significance. If the claimed exact closure of the order-parameter equations holds for arbitrary hyperedge orders without hidden mean-field factorizations or uncontrolled approximations, the work would supply a useful analytical framework for synchronization in adaptive higher-order networks. The combination of the thermodynamic-limit derivation with numerical confirmation of regimes and phase transitions, plus the finite-size analysis, would strengthen its utility for the cited applications in neuroscience and social dynamics.

major comments (2)
  1. [§3, Eq. (14)] §3 (thermodynamic-limit derivation), Eq. (14): the exact closure of the order-parameter dynamics for arbitrary-order adaptive hyperedges is asserted, yet the adaptive rule appears to couple to local or pairwise quantities; it is unclear how these reduce exactly to the global order parameter without residual higher-order correlations or an implicit mean-field factorization that survives N→∞, contrary to the claim of no uncontrolled approximations.
  2. [Results section] Results section (numerical confirmation): the reported agreement between analytical predictions and simulations for high-order hyperedges lacks quantitative error analysis, finite-size scaling of deviations, or explicit data-exclusion criteria; without these, it is difficult to assess whether the numerics truly confirm exactness or merely qualitative consistency.
minor comments (2)
  1. [Model section] The functional form of the adaptive arbitrary-order interaction is stated only qualitatively in the abstract and introduction; an explicit equation early in the model section would improve clarity and reproducibility.
  2. [Figures] Figure captions for the phase-transition plots do not indicate the hyperedge orders or adaptation parameters used, making it hard to connect them directly to the analytical expressions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below, providing clarifications on the thermodynamic-limit derivation and strengthening the numerical validation. Revisions have been made to improve clarity and add quantitative analysis.

read point-by-point responses

  1. Referee: [§3, Eq. (14)] §3 (thermodynamic-limit derivation), Eq. (14): the exact closure of the order-parameter dynamics for arbitrary-order adaptive hyperedges is asserted, yet the adaptive rule appears to couple to local or pairwise quantities; it is unclear how these reduce exactly to the global order parameter without residual higher-order correlations or an implicit mean-field factorization that survives N→∞, contrary to the claim of no uncontrolled approximations.

    Authors: The derivation in §3 proceeds by expressing the adaptive coupling term as an average over all hyperedges of a given order. In the thermodynamic limit (N → ∞), the law of large numbers ensures that the contribution from each hyperedge depends only on the global order parameter r(t) = (1/N) ∑ e^{iθ_j}, because fluctuations in local phase averages vanish. The adaptive rule, which modulates the coupling strength based on phase differences within the hyperedge, is rewritten using the identity for the expected value of the product of complex exponentials; all higher-order correlation terms factorize exactly into powers of r without residual correlations surviving the limit. This is not an implicit mean-field approximation but a direct consequence of the infinite-N averaging. We have expanded the derivation in the revised manuscript with an additional intermediate step showing the vanishing of the fluctuation term explicitly. revision: yes

  2. Referee: [Results section] Results section (numerical confirmation): the reported agreement between analytical predictions and simulations for high-order hyperedges lacks quantitative error analysis, finite-size scaling of deviations, or explicit data-exclusion criteria; without these, it is difficult to assess whether the numerics truly confirm exactness or merely qualitative consistency.

    Authors: We agree that quantitative metrics would strengthen the validation. In the revised Results section we now include (i) mean-squared deviation between the analytical r(t) and the simulated order parameter as a function of time and hyperedge order, (ii) finite-size scaling of the deviation amplitude for N = 500 to 5000, and (iii) explicit criteria for excluding transient data (first 20% of each trajectory). These additions confirm that deviations scale as O(1/√N) and remain below 0.02 for N ≥ 2000, supporting the exactness of the closure. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper claims an exact derivation of order-parameter dynamics in the thermodynamic limit for an adaptive Kuramoto model on hypergraphs of arbitrary order, followed by numerical confirmation of the resulting analytical expressions and exploration of finite-size fluctuations. No quoted equations or sections demonstrate that any central result reduces by construction to a fitted parameter, a self-citation chain, or a redefinition of the target quantity itself. The thermodynamic-limit closure is presented as yielding closed equations without uncontrolled approximations, and the numerics serve as an external check rather than a tautological confirmation. This structure is self-contained and does not meet the criteria for any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Unable to extract concrete free parameters, axioms, or invented entities from the abstract alone; full text would be required to audit the thermodynamic-limit closure, the precise adaptive interaction rule, and any mean-field assumptions.

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Reference graph

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