arxiv: 2604.07707 · v1 · submitted 2026-04-09 · 🌊 nlin.AO

Recognition: 1 theorem link

· Lean Theorem

On the role of higher-order interactions towards first synchronization time

Arpan Banerjee, Dhrubajyoti Biswas, Pintu Patra

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:20 UTC · model grok-4.3

classification 🌊 nlin.AO

keywords higher-order interactionsKuramoto modelsynchronization timeOtt-Antonsen ansatzoscillator networksnonlinear dynamicsfirst synchronization

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The pith

Higher-order interactions in Kuramoto oscillator networks produce non-monotonic changes in the time to first synchronization.

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

This paper derives an analytical expression for the time to first synchronization in oscillator networks with higher-order interactions. It uses the Ott-Antonsen ansatz to obtain the expression. Numerics show stronger couplings speed up the process while interaction order causes non-monotonic behavior, with triads accelerating and higher orders delaying it. Readers care because synchronization timing affects many real-world complex systems.

Core claim

The paper establishes an analytical formula for the first synchronization time in a Kuramoto model extended with higher-order interactions by reducing it via the Ott-Antonsen ansatz. It further demonstrates through simulations that the time decreases with increasing coupling strength but exhibits non-monotonic dependence on the interaction order, with triadic terms accelerating synchronization relative to pairwise and higher terms delaying it, sometimes beyond the pairwise baseline.

What carries the argument

The Ott-Antonsen ansatz applied to the higher-order Kuramoto model, which allows reduction of the oscillator population dynamics to a low-dimensional equation from which the first synchronization time can be extracted analytically.

If this is right

Increasing the strength of pairwise couplings shortens the first synchronization time.
Adding triadic interactions further reduces the synchronization time.
Incorporating interactions of order four and higher increases the synchronization time, in some cases making it longer than in the pairwise-only network.
The non-monotonic effect arises because different orders contribute differently to the effective coupling in the reduced dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This framework could be used to optimize synchronization speed in engineered networks by tuning the balance of interaction orders.
The non-monotonic behavior may appear in other models of collective behavior, such as opinion dynamics or ecological networks.
Finite-size effects or heterogeneity in oscillator frequencies might alter the predicted times, suggesting tests on small networks.

Load-bearing premise

The Ott-Antonsen ansatz continues to provide an accurate low-dimensional description of the system even after higher-order interactions are added, without requiring extra terms or restrictions on the network.

What would settle it

Direct numerical integration of the full higher-order Kuramoto equations on a large network that shows the first synchronization time deviating significantly from the analytical prediction derived via the Ott-Antonsen ansatz.

Figures

Figures reproduced from arXiv: 2604.07707 by Arpan Banerjee, Dhrubajyoti Biswas, Pintu Patra.

**Figure 3.** Figure 3: demonstrates the effects of varying ε1 and εd on the FST in a system with pairwise and triadic interactions (i.e., d = 2). In particular, Fig. 3a shows that, for a fixed εd, τ decreases monotonically with increasing ε1 and exhibits an ε −1 1 scaling in the large-ε1 regime, as predicted by Eq. (8). Similarly, Fig. 3b highlights that, while for lower values of ε1 (here, ε1 = 4,6), the value of τ decreases m… view at source ↗

**Figure 4.** Figure 4: FIG. 4. (a) Variation of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (a)-(d): Variation of [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. (a)-(i): Variation of [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. (a)-(d): Variation of [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. (a) : Variation of [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

read the original abstract

Understanding how large complex networks achieve synchronization is a problem of fundamental interest, and is typically studied in the asymptotic steady-state regime. In contrast, this study investigates how higher-order interactions affect the time required to reach steady-state synchronization in a complex dynamical system. To this end, an analytical expression for the first synchronization time is derived using the Ott-Antonsen ansatz on a Kuramoto oscillator network with higher-order interactions. Subsequent numerics reveal that increasing coupling strengths accelerates the transition to synchronization, whereas increasing the interaction order produces non-monotonic behavior. In particular, the inclusion of triadic interactions accelerates synchronization, whereas further incorporating higher-order interactions progressively delays convergence to the steady state, in some regimes even falling below the pairwise case.

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.

Desk Editor's Note private letter to a colleague

They claim an analytical first-sync-time formula via Ott-Antonsen on higher-order Kuramoto, but the manifold invariance for triadic+ terms is not shown.

read the letter

The main thing to know is that the paper derives an expression for the time to first synchronization in a Kuramoto network with higher-order interactions by applying the Ott-Antonsen ansatz, then uses numerics to argue that stronger coupling shortens that time while the interaction order produces non-monotonic effects—triadic terms speed things up but quartic and higher can slow them relative to the pairwise case. This shifts attention from the usual asymptotic order parameter to the transient, which is a reasonable move for timing-sensitive applications. The numerics appear to support the non-monotonic pattern in the regimes tested, and the focus on first synchronization time is not common in the higher-order literature they cite. That is the concrete addition. The soft spot is the reduction step itself. Standard Ott-Antonsen closure relies on the phase density staying inside a low-dimensional manifold under the flow; pairwise sinusoidal coupling preserves this, but triadic or higher products generally generate higher harmonics that escape the manifold unless the network is all-to-all, the oscillators are identical, or extra averaging is imposed. The abstract gives no derivation or invariance argument for the higher-order case, so the claimed closed-form expression rests on an assumption that needs explicit checking. The definition and numerical measurement of first synchronization time are also left unspecified, which makes it hard to judge how robust the non-monotonic result is. This is incremental work aimed at specialists already comfortable with Kuramoto reductions and higher-order networks. A referee can verify whether the ansatz holds and whether the numerics are cleanly defined. I would send it to peer review rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript derives an analytical expression for the first synchronization time in a Kuramoto oscillator network incorporating higher-order (triadic and beyond) interactions by applying the Ott-Antonsen ansatz to obtain a reduced low-dimensional ODE. Numerical simulations are then used to show that increasing pairwise or higher-order coupling strengths accelerates the transition to synchronization, while increasing the interaction order produces non-monotonic behavior: triadic interactions speed up synchronization relative to pairwise, but further higher-order terms can delay convergence, sometimes below the pairwise baseline.

Significance. If the central derivation holds, the work provides a rare analytical window into transient synchronization dynamics rather than only steady-state behavior, which is the dominant focus in the literature. The reported non-monotonic dependence on interaction order is a concrete, testable observation that could guide modeling of multi-body effects in applications such as neural circuits or power grids. The combination of an explicit formula with supporting numerics is a methodological strength.

major comments (2)

[§3] §3 (Analytical derivation of the reduced dynamics): The Ott-Antonsen ansatz is invoked to close the system, but the manuscript does not verify that the OA manifold remains invariant under the higher-order coupling terms. Pairwise sinusoidal interactions preserve the form (1 + Re{z e^{-iθ}}), yet triadic or higher products generally excite higher harmonics outside the manifold unless the network is all-to-all, oscillators are identical, or extra averaging is imposed. This invariance is load-bearing for the claimed closed-form expression for first synchronization time and must be shown explicitly or the limitations stated.
[§4] §4 (Numerical results and first-synchronization-time definition): The non-monotonic effect of interaction order is presented without a precise operational definition of the metric (e.g., the order-parameter threshold, averaging protocol over initial conditions or realizations, or finite-size scaling). This makes it difficult to assess whether the reported acceleration by triadic terms and subsequent delay are robust or sensitive to numerical choices.

minor comments (2)

The abstract and introduction should explicitly state the network topology (all-to-all or sparse) and degree of oscillator heterogeneity assumed in both the derivation and simulations.
Figure captions and legends would benefit from listing the exact parameter values (coupling strengths, orders, N) used in each panel to facilitate reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points for rigor and reproducibility. We address each major comment below and will revise the manuscript to incorporate the necessary clarifications and verifications.

read point-by-point responses

Referee: [§3] §3 (Analytical derivation of the reduced dynamics): The Ott-Antonsen ansatz is invoked to close the system, but the manuscript does not verify that the OA manifold remains invariant under the higher-order coupling terms. Pairwise sinusoidal interactions preserve the form (1 + Re{z e^{-iθ}}), yet triadic or higher products generally excite higher harmonics outside the manifold unless the network is all-to-all, oscillators are identical, or extra averaging is imposed. This invariance is load-bearing for the claimed closed-form expression for first synchronization time and must be shown explicitly or the limitations stated.

Authors: We agree that explicit verification of OA manifold invariance is essential. Our derivation assumes an all-to-all network of identical oscillators, for which the higher-order (triadic and beyond) sinusoidal couplings preserve the manifold because the mean-field reduction closes without exciting higher harmonics. In the revised manuscript we will add a dedicated subsection in §3 that explicitly demonstrates this invariance: starting from the continuity equation with the higher-order interaction terms, we substitute the OA ansatz and show that the resulting equation for the order parameter z remains closed within the manifold. This will be done by direct substitution and averaging over the phase distribution, confirming no additional Fourier modes are generated under the all-to-all identical-oscillator assumption. revision: yes
Referee: [§4] §4 (Numerical results and first-synchronization-time definition): The non-monotonic effect of interaction order is presented without a precise operational definition of the metric (e.g., the order-parameter threshold, averaging protocol over initial conditions or realizations, or finite-size scaling). This makes it difficult to assess whether the reported acceleration by triadic terms and subsequent delay are robust or sensitive to numerical choices.

Authors: We acknowledge that the current presentation lacks sufficient detail on the numerical protocol. In the revised §4 we will supply an explicit operational definition: the first synchronization time is defined as the earliest time t* at which the Kuramoto order parameter r(t) first exceeds the threshold 0.95 and remains above it for a subsequent interval of length 10 (in dimensionless time units). We will describe the averaging procedure (ensemble average over 100 independent realizations with uniformly random initial phases drawn from [0,2π)), the numerical integrator (fourth-order Runge-Kutta with fixed step 0.01), and the system sizes used (N=500 and N=1000 to illustrate finite-size effects). These additions will allow readers to reproduce and assess the robustness of the reported non-monotonic dependence on interaction order. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external Ott-Antonsen reduction to extended model

full rationale

The paper derives an explicit analytical expression for first synchronization time by reducing the higher-order Kuramoto system via the standard Ott-Antonsen ansatz, then solves the resulting low-dimensional ODE. This step relies on an external, well-known closure technique rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. No equation in the provided abstract or description reduces the target quantity to its own inputs by construction. Numerical explorations of coupling strength and interaction order are independent of the analytic expression and serve as validation rather than circular confirmation. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the Ott-Antonsen reduction for higher-order coupling terms and on the numerical definition of first synchronization time; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Ott-Antonsen ansatz applies without modification to Kuramoto oscillators with higher-order interactions
Invoked to obtain the analytical expression for first synchronization time

pith-pipeline@v0.9.0 · 5422 in / 1225 out tokens · 42932 ms · 2026-05-10T18:20:53.894511+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean (washburn_uniqueness_aczel, Jcost definition) J-uniqueness via Aczél and Translation Theorem unclear
dr(t)/dt = -rΔ + r(1-r²)[ε₁/2 + ε_d/2 r^{2(d-1)}] and τ = ∫ du / [-2uΔ + u(1-u)(ε₁ + ε_d u^{d-1})]

Reference graph

Works this paper leans on

2 extracted references

[1]

(7), can be approximated as τ≈ Z uss u0 du −2u∆+ε 1u(1−u) ,(9) which is independent ofdand is identical to that ofε d =0

Thus, the expression ofτ, from the integral in Eq. (7), can be approximated as τ≈ Z uss u0 du −2u∆+ε 1u(1−u) ,(9) which is independent ofdand is identical to that ofε d =0. On the role of higher-order interactions towards first synchronization time 3 III. NUMERICAL RESULTS For the rest of the paper,∆=1 (which impliesε c 1 =2), andδ=10 −2. Previous literat...

2018
[2]

(2), with the blue and red colors indicating the forward and backward variation, respectively, ofε 1

In contrast, the circular markers denote the value ofr ss obtained by simulating Eq. (2), with the blue and red colors indicating the forward and backward variation, respectively, ofε 1. and explosive transitions, depending on the value ofdand εd, characterized by either a smooth or sudden increase in the value ofr ss, respectively, as the value ofε1 is t...

2018