Impact of Channel Dynamics on Higher-order Interactions of Oscillators

Emanuel B. S. A. Cambraia; Gabriel Marghoti; Giovana S. Spezzatto; Isabela A. Martins; Sergio R. Lopes; Thiago L. Prado

REVIEW

Impact of Channel Dynamics on Higher-order Interactions of Oscillators

Gabriel Marghoti, Isabela A. Martins, Giovana S. Spezzatto, Emanuel B. S. A. Cambraia, Sergio R. Lopes, Thiago L. Prado

Adaptation and Self-Organizing Systems · Submitted 8 Jul 2026 · 33 references

Pith's verdict: Sound analytical core with a genuinely new decomposition; dynamical findings are preliminary but honestly reported

T0 · AI review

How this review was made: An AI referee (glm-5.2) read the paper on July 9, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 4 major objections · 5 minor · Ask about this paper · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2607.07662 (version 1)

Pith IDI3C6MMIO, Pith's permanent record of this paper and its review

CitationEmanuel B. S. A. Cambraia, Gabriel Marghoti, Giovana S. Spezzatto, Isabela A. Martins, Sergio R. Lopes, Thiago L. Prado. "Impact of Channel Dynamics on Higher-order Interactions of Oscillators." arXiv:2607.07662 (2026). https://pith.science/paper/I3C6MMIO

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

The main thing to know: the analytical derivation in §IV is clean and correct, and the symmetric/antisymmetric decomposition of the environmental tensor (Eqs. 12–15) is a legitimate new result. The claim that the standard (1,1,−2) hyperedge emerges only under B_ijk = B_ikj is mathematically sound and worth publishing. The dynamical findings, however, are preliminary — the most novel one (antisymmetric clustering) is a finite-size effect that vanishes for N≥100, and the paper says so itself. That honesty is a point in its favor, but it limits the practical significance of the dynamical half of the paper. The stress-test note raises a point the paper misses: in the adiabatic limit, the antisymmetric term D_ijk sin(θ_j − θ_k) is independent of the target index i. For uniform D_ijk, this is just a uniform frequency shift — it cannot produce cluster structure on its own. The clustering only emerges through finite-τ memory effects that break this i-independence. This is a structural reason why the phenomenon is inherently non-adiabatic and finite-size, and the authors would strengthen the paper by stating it explicitly rather than leaving it implicit. On the numerical side: N=10 phase diagrams with no error bars, N=30 for the τ-sweeps, and only uniform D_ijk = 1.0 tested. No heterogeneous antisymmetric tensors, no sparse topologies. The authors hypothesize that sparse networks may preserve the clustering, but this is untested. These are real gaps, but they are gaps in the dynamical story, not in the analytical core. The derivation stands on its own. Who benefits: researchers working on higher-order Kuramoto models and anyone interested in when static hypergraph models are valid approximations. The decomposition and the symmetry condition are the contributions that will last; the phase diagrams are exploratory. I'd send this to a serious referee. The analytical section deserves scrutiny from someone who can check the adiabatic elimination and the index-swap argument carefully, and the referee should push the authors to (1) state the i-independence of the antisymmetric adiabatic term explicitly, (2) add error bars or ensemble statistics to the phase diagrams, and (3) at least sketch what heterogeneous D_ijk would do. None of these are reasons to reject; they are reasons to revise.

Referee report

Significance

The analytical derivation in §IV is the paper's principal contribution: it is parameter-free, proceeds by straightforward adiabatic elimination and trigonometric decomposition, and provides a legitimate mechanistic grounding for why the standard (1,1,−2) hyperedge is a special case rather than a universal rule. The decomposition into symmetric and antisymmetric tensor components (Eqs. 12–15) is clean and illuminating. The dynamical findings (bistability for symmetric tensors, clustering for antisymmetric ones) are genuine dynamical predictions rather than fitted results, and the honest acknowledgment that clustering is a finite-size effect in all-to-all networks is commendable. Reproducible code is provided via Zenodo. The paper bridges the higher-order network literature with the physical-transmission-medium perspective in a substantive way.

Major objections (4)

§V, Fig. 2: The phase diagrams are constructed for N=10 only, with no error bars or statistical uncertainty estimates across the 100 trials mentioned in the text. Given that the central dynamical claims (bistability regimes, partial synchronization boundaries) are drawn from these diagrams, the absence of uncertainty quantification makes it difficult to assess whether the regime boundaries are robust or artifacts of finite-size fluctuations. At minimum, confidence intervals on the order-parameter thresholds used to classify regimes should be reported, or the diagrams should be labeled explicitly as schematic.
§V, Fig. 5 and surrounding text: The most novel dynamical finding — anti-phase clustering driven by antisymmetric tensors with finite τ — is shown to vanish for N≥100. The authors hypothesize that sparse networks may preserve it but provide no test. This is the load-bearing concern for the paper's practical significance. A single sparse-network experiment (even at moderate N with a ring or degree-regular topology) would substantially strengthen or falsify the central claim. Without it, the paper's novel dynamical regime rests on an unverified conjecture.
§IV, Eq. (15): The paper does not explicitly discuss the structural reason for the clustering's fragility. In the adiabatic limit with uniform all-to-all antisymmetric $D_{ijk}$, the antisymmetric term reduces to $Σ_{j,k} D_{ijk} sin(θ_j - θ_k)$, which is independent of the target index $i$. This means the antisymmetric contribution acts as a uniform frequency shift in the adiabatic limit and cannot produce cluster structure — clustering requires finite-τ memory effects that break this $i$-independence. Making this mechanism explicit would clarify why the phenomenon is inherently non-adiabatic and finite-size, and would help frame the sparse-network hypothesis more precisely.
§V: Only uniform tensor entries ($C_{ijk}=1.0$ or $D_{ijk}=1.0$ for all indices) were tested. Since the paper's central analytical point is that symmetry-breaking in $B_{ijk}$ produces qualitatively new dynamics, testing at least one heterogeneous antisymmetric tensor (e.g., random or structured $D_{ijk}$) would demonstrate that the findings are not specific to the uniform case and would strengthen the connection between the analytical and numerical contributions.

Minor objections (5)

§II, Eq. (2): The 1/N normalization on the $K_2$ term, combined with the 1/N in Eq. (1), produces a 1/N² prefactor on the higher-order term (Eq. 7). This is the normalization causing the HOI term to vanish as N→∞ for fixed K₂, which is directly relevant to the finite-size discussion in §V. This connection should be stated explicitly rather than left implicit.
Fig. 3 caption: The 'asymmetric case' (panels c1–c4, where $B_{ijk}$ has unit entries strictly for $j<k$) is introduced only in the figure caption and not in the main text of §V. This case should be described in the body, including how it relates to the symmetric/antisymmetric decomposition of §IV.
§IV.B, Eq. (7): The transition from Eq. (6) to Eq. (7) involves substituting the adiabatic solution into Eq. (1), but the 1/N prefactor from Eq. (1) combined with the 1/N from Eq. (2) yields the 1/N² factor. This compound normalization should be made explicit for the reader.
§V: The O(N³) scaling is cited as the reason larger phase diagrams are computationally unfeasible. A brief comment on whether sparse-tensor representations or subset sampling could mitigate this would be helpful, especially given that the paper itself advocates sparse topologies for future work.
References [4]–[7], [9], [14], [15], [17] include several 2026 arXiv preprints. Where published versions exist, they should be referenced; where only preprints are available, this is fine but the access dates could be noted.

Simulated authors' rebuttal

read the rebuttal and 4 point-by-point responses

We thank the referee for a careful and constructive reading of our manuscript. The referee correctly identifies the analytical derivation in §IV as the principal contribution and raises four major comments concerning: (1) the absence of uncertainty quantification in the N=10 phase diagrams, (2) the untested sparse-network hypothesis for antisymmetric clustering, (3) the need to make explicit the structural mechanism behind clustering fragility, and (4) the restriction to uniform tensor entries. We agree with all four points and will revise the manuscript accordingly. Comments 1, 3, and 4 can be fully addressed in revision. Comment 2 requires new numerical experiments that we will conduct and report, though we are candid about computational constraints on what can be completed within a revision cycle.

Referee: §V, Fig. 2: The phase diagrams are constructed for N=10 only, with no error bars or statistical uncertainty estimates across the 100 trials mentioned in the text. Given that the central dynamical claims (bistability regimes, partial synchronization boundaries) are drawn from these diagrams, the absence of uncertainty quantification makes it difficult to assess whether the regime boundaries are robust or artifacts of finite-size fluctuations. At minimum, confidence intervals on the order-parameter thresholds used to classify regimes should be reported, or the diagrams should be labeled explicitly as schematic.

Authors: The referee is correct. The phase diagrams in Fig. 2 currently lack uncertainty quantification, and this is a genuine gap. We will address this in two ways. First, we will recompute the regime boundaries with confidence intervals on the order parameters (R1 and R2) across the 100 trials, reporting the standard error or bootstrap intervals at each (K1, K2) grid point. Second, we will relabel Fig. 2 explicitly as schematic in the caption, clarifying that the boundaries are illustrative rather than precise transition lines. We note that the figure caption already uses the word 'Schematic,' but the main text does not adequately emphasize this, and no uncertainty estimates are provided anywhere. Both will be corrected. revision: yes
Referee: §V, Fig. 5 and surrounding text: The most novel dynamical finding — anti-phase clustering driven by antisymmetric tensors with finite τ — is shown to vanish for N≥100. The authors hypothesize that sparse networks may preserve it but provide no test. This is the load-bearing concern for the paper's practical significance. A single sparse-network experiment (even at moderate N with a ring or degree-regular topology) would substantially strengthen or falsify the central claim. Without it, the paper's novel dynamical regime rests on an unverified conjecture.

Authors: We agree that this is the most consequential concern. The sparse-network hypothesis is currently unsupported by any numerical evidence, and we will conduct the requested experiment. Specifically, we will run simulations on degree-regular (ring-like) topologies at moderate N (e.g., N=30, 50, 100) with the antisymmetric tensor D_ijk=1.0 and finite τ, measuring R1 and R2 over 100 trials as in the existing Fig. 5 protocol. We will report whether anti-phase clustering persists, is enhanced, or still vanishes. We are candid that the O(N^3) scaling of the higher-order interactions limits how large N can be pushed, but degree-regular topologies at N≤100 are computationally feasible. If the clustering persists on sparse topologies, this substantially strengthens the paper; if it does not, we will report this honestly and reframe the contribution accordingly. In either case, the conjecture will be replaced by evidence. revision: yes
Referee: §IV, Eq. (15): The paper does not explicitly discuss the structural reason for the clustering's fragility. In the adiabatic limit with uniform all-to-all antisymmetric D_ijk, the antisymmetric term reduces to Σ_{j,k} D_ijk sin(θ_j − θ_k), which is independent of the target index i. This means the antisymmetric contribution acts as a uniform frequency shift in the adiabatic limit and cannot produce cluster structure — clustering requires finite-τ memory effects that break this i-independence. Making this mechanism explicit would clarify why the phenomenon is inherently non-adiabatic and finite-size, and would help frame the sparse-network hypothesis more precisely.

Authors: This is an excellent observation that we should have made explicit. The referee's argument is correct: in the adiabatic limit with uniform all-to-all antisymmetric D_ijk, the term Σ_{j,k} D_ijk sin(θ_j − θ_k) is indeed independent of the target index i, reducing to a uniform frequency shift that cannot produce cluster structure. This is precisely why clustering requires finite τ — the memory kernel breaks the i-independence by introducing temporally delayed, non-reciprocal feedback. We will add a paragraph in §IV (after Eq. 15) making this mechanism explicit, and will connect it to the finite-size fragility discussed in §V: in the thermodynamic limit, the mean-field homogenization further suppresses the localized temporal correlations that finite τ introduces. This will also sharpen the motivation for the sparse-network experiments, since sparsity limits the number of interacting channels and may preserve the i-dependence of the delayed feedback. revision: yes
Referee: §V: Only uniform tensor entries (C_ijk=1.0 or D_ijk=1.0 for all indices) were tested. Since the paper's central analytical point is that symmetry-breaking in B_ijk produces qualitatively new dynamics, testing at least one heterogeneous antisymmetric tensor (e.g., random or structured D_ijk) would demonstrate that the findings are not specific to the uniform case and would strengthen the connection between the analytical and numerical contributions.

Authors: The referee is right that testing only uniform tensors leaves open whether the dynamical findings are artifacts of the uniform case. We will add simulations with at least one heterogeneous antisymmetric tensor — specifically, random D_ijk drawn from a uniform distribution on [0, 1] (with D_ikj = -D_ijk to preserve antisymmetry) — at the same parameter values used in Fig. 4 (N=30, K1=0.2, K2=20, varying τ). If the clustering phenomenon persists with heterogeneous tensors, this confirms that the result is not an artifact of uniformity; if it does not, we will report this and discuss what it implies about the robustness of the antisymmetric mechanism. We will also test a structured heterogeneous case (e.g., D_ijk proportional to a distance-dependent kernel) if space permits. This directly connects the analytical generality of the tensor decomposition to the numerical experiments. revision: yes

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Three-body oscillator coupling is just pairwise links with hidden channel memory

The reading

This paper argues that the standard three-body Kuramoto interaction term sin(θ_j + θ_k − 2θ_i), widely used to model higher-order interactions in oscillator networks, is not a fundamental coupling but an artifact that emerges only when one assumes transmission channels respond instantaneously and symmetrically. The authors build a model where each pairwise connection carries a latent transmission variable with finite inertia (a relaxation timescale τ). When that timescale is fast (the adiabatic limit) and the environmental modulation tensor B_ijk is symmetric under swapping j and k, the familiar three-body term drops out. But when either assumption is relaxed — finite channel memory or asymmetric modulation — the interaction takes a more general form, Σ B_ijk sin(θ_j − θ_i) cos(θ_k − θ_i), which does not map onto any static hyperedge. The authors show that symmetric tensors produce bistability and cluster states, while antisymmetric tensors combined with finite channel inertia produce anti-phase cluster synchronization — a dynamical regime with no static-topology counterpart. However, they find that this anti-phase clustering is a finite-size effect that washes out in all-to-all networks of 100 or more oscillators, as mean-field averaging erases the localized non-reciprocal feedback that sustains it. The central claim is that static higher-order topologies are a restrictive special case of a richer, asymmetric, memory-laden transmission dynamics.

What carries the argument

A latent transmission variable u_ij(t) with first-order relaxation timescale τ, modulated by a third-party environmental tensor B_ijk. The variable sits between the phase dynamics and the coupling, so that the effective interaction is a convolution of past phase differences weighted by an exponential kernel. Decomposing B_ijk into symmetric (C_ijk) and antisymmetric (D_ijk) components cleanly separates the standard three-body term from the novel non-reciprocal term.

Extended reading notes

Core claim

The standard (1,1,−2) three-body Kuramoto hyperedge emerges only under the joint conditions of adiabatic channel elimination and strict permutation symmetry B_ijk = B_ikj. When the symmetry is broken, the interaction decomposes into a symmetric part C_ijk that recovers the standard three-body term and an antisymmetric part D_ijk that produces a fundamentally different coupling sin(θ_j − θ_k), which has no static-hyperedge representation. When channel inertia is finite, the transmission variable retains a memory kernel, and the antisymmetric component can drive anti-phase cluster synchronization — but only in small networks, because mean-field homogenization destroys the effect for N ≥ 100 in

Load-bearing premise

The paper's most novel dynamical result — anti-phase cluster synchronization from antisymmetric tensors with finite channel inertia — is shown to vanish for networks of 100 or more oscillators in all-to-all coupling. The authors hypothesize that sparse networks may preserve it, but this is untested.

Editorial extensions

If this is right

Models that assign static (1,1,−2) hyperedges to oscillator networks are implicitly assuming both instantaneous transmission and perfect reciprocity of environmental modulation — assumptions that are physically implausible in neural, social, or directed transport networks.
Antisymmetric interaction tensors, which produce no standard three-body coupling at all in the adiabatic limit, become dynamically consequential once channel memory is included, suggesting that a large class of interactions invisible to static topology analysis may be active in real systems.
The finite-size collapse of anti-phase clustering in all-to-all networks implies that sparse or spatially embedded architectures may be the natural setting where non-reciprocal channel dynamics produce macroscopic effects.
Experimental oscillator systems with controllable transmission delays (e.g., coupled lasers, electronic circuits with band-limited coupling) could test whether antisymmetric modulation and finite channel inertia produce the predicted two-cluster states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If sparse networks do preserve the anti-phase clustering (as the authors hypothesize but do not test), then the interaction between network topology and channel timescale becomes a design parameter: one could tune clustering behavior by adjusting either the sparsity or the transmission delay, not just the coupling strength.
The decomposition into symmetric and antisymmetric tensor components suggests a natural classification scheme for higher-order interactions: symmetric components map to known static topologies, while antisymmetric components represent a qualitatively distinct, dynamically generated interaction class that current hypergraph frameworks cannot represent.
The memory kernel structure implies that the effective interaction at time t depends on the full trajectory of phases, not just their instantaneous values — meaning that systems with the same instantaneous phase configuration but different histories could evolve differently, a property no static topology can capture.

Original abstract

Modeling higher-order interactions (HOIs) in nonlinear networks with static topologies is often physically restrictive. We demonstrate that standard 3-body Kuramoto couplings are mathematically equivalent to pairwise connections modulated by latent variables of transmission channels. While standard HOI topologies emerge in the adiabatic limit of these variables, relaxing this constraint reveals that latent channel timescales dictate collective macroscopic states. Specifically, transmission inertia drives bistability for symmetric interaction tensors and anti-phase cluster synchronization for antisymmetric ones. Furthermore, dynamically induced clustering in global topologies emerges as a finite-size effect of the dynamics of the local channels. Ultimately, we show that relying exclusively on static topologies restricts interaction modeling. Integrating latent variables captures the transient inertia and fundamental asymmetry of physical networks, bridging the analytical utility of higher-order functions with the reality of the underlying transmission medium.

Figures

**Figure 1.** FIG. 1. Model reductionism, from physical transmission

**Figure 2.** FIG. 2. Schematic phase diagrams of the synchronization

**Figure 3.** FIG. 3. Snapshots of the phases of a network of

**Figure 4.** FIG. 4. Order parameter distributions for (a) symmetric and

**Figure 5.** FIG. 5. Influence of network size

Figures reproduced from arXiv:2607.07662 by the authors.

References

Works cited (33)

[1]

in favor of a non-reciprocal flow force. Parameter space analysis for larger networks was computationally unfeasible due to theO(N 3) scaling complexity of the higher-order interactions. Evaluating the Kuramoto (R 1) and Daido (R 2, Eq.3 form= 2) order parameters from both fully synchro- nized and uniformly spaced initial conditions under an 5 FIG. 2. Sch...

View on Pith arXiv 2025
[2]

Y. Kuramoto, Chemical turbulence, inChemical Oscilla- tions, Waves, and Turbulence(Springer, 1984) pp. 111– 140

View on Pith 1984
[3]

A. P. Mill´ an, J. J. Torres, and G. Bianconi, Explo- sive higher-order Kuramoto dynamics on simplicial com- plexes, Phys. Rev. Lett.124, 218301 (2020)

View on Pith 2020
[4]

F. Battiston, E. Amico, A. Barrat, G. Bianconi, G. de Ferraz de Arruda, B. Franceschiello, I. Iacopini, S. K´ efi, V. Latora, Y. Moreno, T. Murray, T. P. Peixoto, G. Vaccario, and G. Petri, The physics of higher-order networks in complex systems, Nat. Phys.17, 1093 (2021)

View on Pith 2021
[5]

T. P. Peixoto, L. Peel, T. Gross, and M. De Domenico, Graphs are maximally expressive for higher-order inter- actions, arXiv:2602.16937 (2026)

View on Pith Pith review arXiv 2026
[6]

M. Lucas, L. Gallo, A. Ghavasieh, F. Battiston, and M. De Domenico, Reducibility of higher-order networks from dynamics, Nat. Commun.17, 1551 (2026)

View on Pith 2026
[7]

A. Kirkley, H. Felippe, and F. Battiston, Structural re- ducibility of hypergraphs, Phys. Rev. Lett.135, 247401 (2025)

View on Pith 2025
[8]

J. Llabr´ es, R. Toral, M. San Miguel, and F. V´ azquez, Reducibility of higher-order to pairwise interactions: So- cial impact models on hypergraphs, Phys. Rev. Res.8, 10.1103/wy1x-3px8 (2026)

View on Pith doi:10.1103/wy1x-3px8 2026
[9]

A. D. Letten and D. B. Stouffer, The mechanistic basis for higher-order interactions and non-additivity in com- petitive communities, Ecol. Lett.22, 423 (2019)

View on Pith 2019
[10]

V. Calleja-Solanas, S. Lamata-Ot´ ın, C. G´ omez-Ambrosi, J. G´ omez-Garde˜ nes, and S. Meloni, Higher-order in- teractions in ecology can be hidden in plain sight, arXiv:2605.06301 (2026)

View on Pith Pith review arXiv 2026
[11]

Y. Kuramoto and H. Nakao, On the concept of dynamical reduction: the case of coupled oscillators, Philos. Trans. R. Soc. A377, 20190041 (2019)

View on Pith 2019
[12]

I. Le´ on and D. Paz´ o, Phase reduction beyond the first order: The case of the mean-field complex Ginzburg- Landau equation, Phys. Rev. E100, 012211 (2019)

View on Pith 2019
[13]

E. Nijholt, J. L. Ocampo-Espindola, D. Eroglu, I. Z. Kiss, and T. Pereira, Emergent hypernetworks in weakly cou- pled oscillators, Nat. Commun.13, 4849 (2022). 9

View on Pith 2022
[14]

C. Bick, T. B¨ ohle, and C. Kuehn, Higher-order net- work interactions through phase reduction for oscillators with phase-dependent amplitude, J. Nonlinear Sci.34, 10.1007/s00332-024-10053-3 (2024)

View on Pith doi:10.1007/s00332-024-10053-3 2024
[15]

N. Fujii, K. Taga, R. Muolo, B. Rink, and H. Nakao, Emergence of higher-order interactions in systems of coupled Kuramoto oscillators with time delay, arXiv:2512.16193 (2026)

View on Pith Pith review arXiv 2026
[16]

E. Salazar, Composite-operator scaling on triadic hy- pergraphs: Formation transitions in multi-agent ar- chitectures with three-body coupling, arXiv:2604.27038 (2026)

View on Pith Pith review arXiv 2026
[17]

Y. Zhang, P. S. Skardal, F. Battiston, G. Petri, and M. Lucas, Deeper but smaller: Higher-order interactions increase linear stability but shrink basins, Sci. Adv.10, eado8049 (2024)

View on Pith 2024
[18]

C. Kuehn and F. Murphy, Emergent higher-order struc- ture from fast adaptive networks, arXiv:2603.19382 (2026)

View on Pith arXiv 2026
[19]

L. Peel, T. P. Peixoto, and M. De Domenico, Statistical inference links data and theory in network science, Nat. Commun.13, 6794 (2022)

View on Pith 2022
[20]

R. Lambiotte, M. Rosvall, and I. Scholtes, From networks to optimal higher-order models of complex systems, Nat. Phys.15, 313 (2019)

View on Pith 2019
[21]

Y. Zhang, M. Lucas, and F. Battiston, Higher-order in- teractions shape collective dynamics differently in hyper- graphs and simplicial complexes, Nat. Commun.14, 1605 (2023)

View on Pith 2023
[22]

R. Muolo, H. Nakao, and M. Coraggio, When higher- order interactions enhance synchronization: the case of the Kuramoto model, arXiv:2508.10992 (2025)

View on Pith Pith review arXiv 2025
[23]

L. V. Gambuzza, F. Di Patti, L. Gallo, S. Lepri, M. Ro- mance, R. Criado, M. Frasca, V. Latora, and S. Boc- caletti, Stability of synchronization in simplicial com- plexes, Nat. Commun.12, 1255 (2021)

View on Pith 2021
[24]

P. S. Skardal and A. Arenas, Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes, Phys. Rev. Lett.122, 248301 (2019)

View on Pith 2019
[25]

I. Le´ on, R. Muolo, S. Hata, and H. Nakao, Theory of phase reduction from hypergraphs to simplicial com- plexes: A general route to higher-order Kuramoto mod- els, Physica D482, 134858 (2025)

View on Pith 2025
[26]

B.-Y. Ji, C.-Y. Wang, and L. L¨ u, Cluster synchronization on sparse simplicial complexes with higher-order repul- sive coupling, New J. Phys.27, 043012 (2025)

View on Pith 2025
[27]

I. L´ eon, R. Muolo, Y. Zhang, and M. Lucas, Symmetry- based selection rules for higher-order interactions in cou- pled oscillators, arXiv:2606.04904 (2026)

View on Pith Pith review arXiv 2026
[28]

R. Berner, E. Sch¨ oll, and S. Yanchuk, Multiclusters in networks of adaptively coupled phase oscillators, SIAM J. Appl. Dyn. Syst.18, 2227 (2019)

View on Pith 2019
[29]

P. Seliger, S. C. Young, and L. S. Tsimring, Plasticity and learning in a network of coupled phase oscillators, Phys. Rev. E65, 041906 (2002)

View on Pith 2002
[30]

T. Aoki and T. Aoyagi, Co-evolution of phases and con- nection strengths in a network of phase oscillators, Phys. Rev. Lett.102, 034101 (2009)

View on Pith 2009
[31]

H. Daido, Multibranch entrainment and scaling in large populations of coupled oscillators, Phys. Rev. Lett.77, 1406 (1996)

View on Pith 1996
[32]

J. A. Acebr´ on, L. L. Bonilla, C. J. P´ erez Vicente, F. Ri- tort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys.77, 137 (2005)

View on Pith 2005
[33]

G. Marghoti, Channeldynamicsonhigherorderinteraction, https://doi.org/10.5281/zenodo.21220174(2026)

View on Pith doi:10.5281/zenodo.21220174(2026 2026

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

in favor of a non-reciprocal flow force. Parameter space analysis for larger networks was computationally unfeasible due to theO(N 3) scaling complexity of the higher-order interactions. Evaluating the Kuramoto (R 1) and Daido (R 2, Eq.3 form= 2) order parameters from both fully synchro- nized and uniformly spaced initial conditions under an 5 FIG. 2. Sch...

View on Pith arXiv 2025

[2] [2]

Y. Kuramoto, Chemical turbulence, inChemical Oscilla- tions, Waves, and Turbulence(Springer, 1984) pp. 111– 140

View on Pith 1984

[3] [3]

A. P. Mill´ an, J. J. Torres, and G. Bianconi, Explo- sive higher-order Kuramoto dynamics on simplicial com- plexes, Phys. Rev. Lett.124, 218301 (2020)

View on Pith 2020

[4] [4]

F. Battiston, E. Amico, A. Barrat, G. Bianconi, G. de Ferraz de Arruda, B. Franceschiello, I. Iacopini, S. K´ efi, V. Latora, Y. Moreno, T. Murray, T. P. Peixoto, G. Vaccario, and G. Petri, The physics of higher-order networks in complex systems, Nat. Phys.17, 1093 (2021)

View on Pith 2021

[5] [5]

T. P. Peixoto, L. Peel, T. Gross, and M. De Domenico, Graphs are maximally expressive for higher-order inter- actions, arXiv:2602.16937 (2026)

View on Pith Pith review arXiv 2026

[6] [6]

M. Lucas, L. Gallo, A. Ghavasieh, F. Battiston, and M. De Domenico, Reducibility of higher-order networks from dynamics, Nat. Commun.17, 1551 (2026)

View on Pith 2026

[7] [7]

A. Kirkley, H. Felippe, and F. Battiston, Structural re- ducibility of hypergraphs, Phys. Rev. Lett.135, 247401 (2025)

View on Pith 2025

[8] [8]

J. Llabr´ es, R. Toral, M. San Miguel, and F. V´ azquez, Reducibility of higher-order to pairwise interactions: So- cial impact models on hypergraphs, Phys. Rev. Res.8, 10.1103/wy1x-3px8 (2026)

View on Pith doi:10.1103/wy1x-3px8 2026

[9] [9]

A. D. Letten and D. B. Stouffer, The mechanistic basis for higher-order interactions and non-additivity in com- petitive communities, Ecol. Lett.22, 423 (2019)

View on Pith 2019

[10] [10]

V. Calleja-Solanas, S. Lamata-Ot´ ın, C. G´ omez-Ambrosi, J. G´ omez-Garde˜ nes, and S. Meloni, Higher-order in- teractions in ecology can be hidden in plain sight, arXiv:2605.06301 (2026)

View on Pith Pith review arXiv 2026

[11] [11]

Y. Kuramoto and H. Nakao, On the concept of dynamical reduction: the case of coupled oscillators, Philos. Trans. R. Soc. A377, 20190041 (2019)

View on Pith 2019

[12] [12]

I. Le´ on and D. Paz´ o, Phase reduction beyond the first order: The case of the mean-field complex Ginzburg- Landau equation, Phys. Rev. E100, 012211 (2019)

View on Pith 2019

[13] [13]

E. Nijholt, J. L. Ocampo-Espindola, D. Eroglu, I. Z. Kiss, and T. Pereira, Emergent hypernetworks in weakly cou- pled oscillators, Nat. Commun.13, 4849 (2022). 9

View on Pith 2022

[14] [14]

C. Bick, T. B¨ ohle, and C. Kuehn, Higher-order net- work interactions through phase reduction for oscillators with phase-dependent amplitude, J. Nonlinear Sci.34, 10.1007/s00332-024-10053-3 (2024)

View on Pith doi:10.1007/s00332-024-10053-3 2024

[15] [15]

N. Fujii, K. Taga, R. Muolo, B. Rink, and H. Nakao, Emergence of higher-order interactions in systems of coupled Kuramoto oscillators with time delay, arXiv:2512.16193 (2026)

View on Pith Pith review arXiv 2026

[16] [16]

E. Salazar, Composite-operator scaling on triadic hy- pergraphs: Formation transitions in multi-agent ar- chitectures with three-body coupling, arXiv:2604.27038 (2026)

View on Pith Pith review arXiv 2026

[17] [17]

Y. Zhang, P. S. Skardal, F. Battiston, G. Petri, and M. Lucas, Deeper but smaller: Higher-order interactions increase linear stability but shrink basins, Sci. Adv.10, eado8049 (2024)

View on Pith 2024

[18] [18]

C. Kuehn and F. Murphy, Emergent higher-order struc- ture from fast adaptive networks, arXiv:2603.19382 (2026)

View on Pith arXiv 2026

[19] [19]

L. Peel, T. P. Peixoto, and M. De Domenico, Statistical inference links data and theory in network science, Nat. Commun.13, 6794 (2022)

View on Pith 2022

[20] [20]

R. Lambiotte, M. Rosvall, and I. Scholtes, From networks to optimal higher-order models of complex systems, Nat. Phys.15, 313 (2019)

View on Pith 2019

[21] [21]

Y. Zhang, M. Lucas, and F. Battiston, Higher-order in- teractions shape collective dynamics differently in hyper- graphs and simplicial complexes, Nat. Commun.14, 1605 (2023)

View on Pith 2023

[22] [22]

R. Muolo, H. Nakao, and M. Coraggio, When higher- order interactions enhance synchronization: the case of the Kuramoto model, arXiv:2508.10992 (2025)

View on Pith Pith review arXiv 2025

[23] [23]

L. V. Gambuzza, F. Di Patti, L. Gallo, S. Lepri, M. Ro- mance, R. Criado, M. Frasca, V. Latora, and S. Boc- caletti, Stability of synchronization in simplicial com- plexes, Nat. Commun.12, 1255 (2021)

View on Pith 2021

[24] [24]

P. S. Skardal and A. Arenas, Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes, Phys. Rev. Lett.122, 248301 (2019)

View on Pith 2019

[25] [25]

I. Le´ on, R. Muolo, S. Hata, and H. Nakao, Theory of phase reduction from hypergraphs to simplicial com- plexes: A general route to higher-order Kuramoto mod- els, Physica D482, 134858 (2025)

View on Pith 2025

[26] [26]

B.-Y. Ji, C.-Y. Wang, and L. L¨ u, Cluster synchronization on sparse simplicial complexes with higher-order repul- sive coupling, New J. Phys.27, 043012 (2025)

View on Pith 2025

[27] [27]

I. L´ eon, R. Muolo, Y. Zhang, and M. Lucas, Symmetry- based selection rules for higher-order interactions in cou- pled oscillators, arXiv:2606.04904 (2026)

View on Pith Pith review arXiv 2026

[28] [28]

R. Berner, E. Sch¨ oll, and S. Yanchuk, Multiclusters in networks of adaptively coupled phase oscillators, SIAM J. Appl. Dyn. Syst.18, 2227 (2019)

View on Pith 2019

[29] [29]

P. Seliger, S. C. Young, and L. S. Tsimring, Plasticity and learning in a network of coupled phase oscillators, Phys. Rev. E65, 041906 (2002)

View on Pith 2002

[30] [30]

T. Aoki and T. Aoyagi, Co-evolution of phases and con- nection strengths in a network of phase oscillators, Phys. Rev. Lett.102, 034101 (2009)

View on Pith 2009

[31] [31]

H. Daido, Multibranch entrainment and scaling in large populations of coupled oscillators, Phys. Rev. Lett.77, 1406 (1996)

View on Pith 1996

[32] [32]

J. A. Acebr´ on, L. L. Bonilla, C. J. P´ erez Vicente, F. Ri- tort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys.77, 137 (2005)

View on Pith 2005

[33] [33]

G. Marghoti, Channeldynamicsonhigherorderinteraction, https://doi.org/10.5281/zenodo.21220174(2026)

View on Pith doi:10.5281/zenodo.21220174(2026 2026