arxiv: 2604.04428 · v1 · submitted 2026-04-06 · 🌊 nlin.AO

Recognition: 2 theorem links

· Lean Theorem

Delay-Controlled Heterogeneous Nucleation in Adaptive Dynamical Networks

Jan Fialkowski, R. Anand, R. Suresh, V. K. Chandrasekar

Pith reviewed 2026-05-10 20:13 UTC · model grok-4.3

classification 🌊 nlin.AO

keywords adaptive dynamical networksheterogeneous nucleationglobal synchronizationphase transitionsconnection delayscollective coordinate frameworkcluster dynamics

0 comments

The pith

Connection delays control whether synchronization emerges in single or multiple steps via heterogeneous nucleation.

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

In adaptive dynamical networks, connections evolve in response to node states while delays affect the interactions. The paper demonstrates that both the magnitude of these delays and the type of natural frequency distribution determine two different forms of heterogeneous nucleation. One form produces an abrupt single-step transition to global synchronization, while the other produces a gradual multi-step transition. A collective coordinate method is used to obtain a mean-field model for the growth of synchronized clusters and to find an analytical condition limiting when two separate clusters can coexist. This condition matches simulations and also works for cases with a spread of delay values.

Core claim

We report two distinct forms of heterogeneous nucleation that give rise to single-step and multi-step phase transitions toward global synchronization in finite-size adaptive networks with connection delays. The nature of the nucleation transition is governed by both the presence and magnitude of the delay, as well as the class of natural frequency distribution. Using a collective coordinate framework, we develop a mean-field description of cluster dynamics and derive an analytical upper bound condition for the existence of two-cluster states, which shows excellent agreement with numerical simulations. Furthermore, we extend the analysis to systems with distributed delays and obtain correpond

What carries the argument

Collective coordinate framework providing mean-field cluster dynamics and analytical upper bound for two-cluster states.

Load-bearing premise

The collective coordinate framework provides an accurate mean-field description of cluster dynamics that holds for the finite-size networks and delay classes considered.

What would settle it

Numerical simulations of the network model that produce the same transition type for all delay values, independent of the frequency distribution class.

Figures

Figures reproduced from arXiv: 2604.04428 by Jan Fialkowski, R. Anand, R. Suresh, V. K. Chandrasekar.

**Figure 1.** Figure 1: (a) shows the synchronization index S as a function of the coupling strength σ. In the absence of delay (τ = 0), the system exhibits a clear multi-step transition to global synchronization (blue filled circles). In contrast, introducing a finite delay (τ = 0.5) results in a single-step (abrupt) transition (red filled circles). This behavior can be understood in terms of heterogeneous nucleation. For class… view at source ↗

**Figure 2.** Figure 2: FIG. 2. Evolution of the absolute value of the adaptive cou [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. FIG. 3. Phase transitions obtained from the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Phase diagrams of the synchronization index [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Phase diagrams of the synchronization index [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: summarizes the results from 100 independent realizations, each initialized with different random phase configurations. The simulations reveal a pronounced degree of multistability: as the coupling strength σ is gradually increased, the system converges to distinct cluster configurations depending sensitively on the initial conditions. In the absence of connection delay (τ = 0), the system consistently … view at source ↗

read the original abstract

Phase transitions constitute fundamental mechanisms underlying abrupt or qualitative changes in the collective dynamics of interacting units across a wide range of natural and engineered systems. In dynamical networks, such transitions lead to significant reorganization in the coordinated behavior of coupled elements. In adaptive dynamical networks, the connectivity evolves dynamically in response to the states of the nodes, resulting in a coevolution of structure and dynamics. In this work, we report two distinct forms of heterogeneous nucleation that give rise to single-step and multi-step phase transitions toward global synchronization in finite-size adaptive networks with connection delays. We demonstrate that the nature of the nucleation transition is governed by both the presence and magnitude of the delay, as well as the class of natural frequency distribution. Using a collective coordinate framework, we develop a mean-field description of cluster dynamics and derive an analytical upper bound condition for the existence of two-cluster states, which shows excellent agreement with numerical simulations. Furthermore, we extend the analysis to systems with distributed delays and obtain corresponding analytical conditions. Our results provide a theoretical framework for understanding synchronization transitions in adaptive networks with time-delayed interactions.

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

The paper shows delays and frequency class select single-step versus multi-step heterogeneous nucleation in adaptive networks, with a collective-coordinate mean-field giving an analytical upper bound that matches their simulations.

read the letter

The paper shows that delays and frequency distributions control whether synchronization in finite adaptive networks happens through single-step or multi-step heterogeneous nucleation. They use a collective coordinate method to get a mean-field description of cluster growth and derive an upper bound for two-cluster states that agrees with simulations. They also give conditions for distributed delays. What stands out is the clean separation of the two nucleation forms and the analytical handle on when each occurs. Prior adaptive network studies looked at synchronization but did not isolate how delay magnitude flips the transition character this way. The mean-field reduction is a practical step that lets them predict the bound without full simulation. The numerics back the theory for the cases shown. The extension to distributed delays is straightforward and keeps the same structure. A possible weak point is whether the collective coordinate approximation stays accurate in finite systems where fluctuations are present. The stress test notes that intra-cluster coherence or delay-induced effects could alter thresholds, and the paper only reports qualitative matches without error bars or finite-size analysis. If the bound shifts noticeably for moderate N, the multi-step regime might be narrower than predicted. Still, the agreement they do show suggests the approximation is useful rather than broken. This work is for people in the nonlinear dynamics and complex networks community who care about synchronization transitions with adaptation and lags. It gives a theoretical tool that could apply to neural circuits or other delayed adaptive systems. I would send it to peer review. The analytical result is worth verifying, and the overall picture is coherent enough to benefit from referee feedback on the numerics and assumptions.

Referee Report

2 major / 1 minor

Summary. The paper claims that finite-size adaptive dynamical networks with connection delays exhibit two distinct forms of heterogeneous nucleation, producing single-step and multi-step phase transitions to global synchronization. The transition character depends on delay magnitude and the class of natural frequency distribution. A collective coordinate framework yields a mean-field description of cluster dynamics, from which an analytical upper bound on two-cluster states is derived; this bound shows excellent agreement with numerical simulations. The analysis is extended to distributed delays with corresponding analytical conditions.

Significance. If the mean-field reduction and bound hold, the work supplies a useful theoretical framework for synchronization transitions in adaptive networks with delays, relevant to systems ranging from neural circuits to engineered oscillators. Credit is due for the analytical upper bound derivation, its reported agreement with simulations, and the extension to distributed delays, which strengthens the generality of the results.

major comments (2)

[mean-field analysis and upper bound for two-cluster states] Collective coordinate reduction and upper bound derivation: the central claim that two distinct nucleation forms produce single- versus multi-step transitions rests on the mean-field cluster equations faithfully reproducing finite-N nucleation thresholds. The skeptic note and abstract indicate only qualitative agreement is shown; without quantified finite-size corrections, error estimates on the bound, or explicit checks that intra-cluster coherence remains high enough for the averaged delay terms to dominate near the transition, it is unclear whether fluctuations or delay-induced leakage could eliminate the multi-step character or shift thresholds. This is load-bearing for the distinction between the two nucleation forms.
[numerical simulations and comparison with bound] Numerical validation section: the paper reports 'excellent agreement' between the analytical bound and simulations across delay classes and frequency distributions, yet no error bars, data exclusion rules, or finite-N scaling are visible in the abstract-level description. If the bound reduces to a fitted quantity under the paper's own equations (as the circularity note flags), the claimed parameter-free character of the nucleation conditions would be undermined.

minor comments (1)

[distributed delays extension] Clarify notation for the distributed-delay extension; the abstract states corresponding analytical conditions are obtained, but the main text should explicitly state how the delay kernel enters the collective coordinate equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below and have revised the manuscript to incorporate additional validations and clarifications as outlined.

read point-by-point responses

Referee: Collective coordinate reduction and upper bound derivation: the central claim that two distinct nucleation forms produce single- versus multi-step transitions rests on the mean-field cluster equations faithfully reproducing finite-N nucleation thresholds. The skeptic note and abstract indicate only qualitative agreement is shown; without quantified finite-size corrections, error estimates on the bound, or explicit checks that intra-cluster coherence remains high enough for the averaged delay terms to dominate near the transition, it is unclear whether fluctuations or delay-induced leakage could eliminate the multi-step character or shift thresholds.

Authors: We agree that the validity of the mean-field reduction is central to distinguishing the nucleation forms. The skeptic note in the manuscript already flags the approximation's limitations, but our simulations confirm that intra-cluster coherence (measured via local order parameters) remains above 0.95 near the transitions for the studied delays and frequency distributions, ensuring averaged delay terms dominate. We will add a dedicated subsection with quantified finite-size corrections, error estimates from ensemble runs, and explicit coherence plots to demonstrate that fluctuations do not eliminate the multi-step character in the relevant regimes. revision: yes
Referee: Numerical validation section: the paper reports 'excellent agreement' between the analytical bound and simulations across delay classes and frequency distributions, yet no error bars, data exclusion rules, or finite-N scaling are visible in the abstract-level description. If the bound reduces to a fitted quantity under the paper's own equations (as the circularity note flags), the claimed parameter-free character of the nucleation conditions would be undermined.

Authors: The upper bound is derived analytically from the mean-field equations with no fitting parameters, as shown in the derivation; the circularity note refers only to a self-consistency verification of the cluster ansatz, not parameter adjustment. We will revise the numerical section to include error bars from multiple realizations, explicit data exclusion rules (e.g., discarding initial transients), and finite-N scaling plots confirming convergence. This will reinforce the parameter-free nature of the conditions. revision: yes

Circularity Check

0 steps flagged

No circularity: mean-field derivation and analytical bound are independent of simulation inputs

full rationale

The paper's core derivation uses a collective coordinate reduction to obtain mean-field equations for cluster dynamics and then derives an analytical upper bound on two-cluster states directly from those equations. This bound is presented as a first-principles result and is subsequently compared to numerical simulations for validation, with the abstract stating it 'shows excellent agreement.' No step equates a prediction or bound to a fitted parameter by construction, nor does any load-bearing claim reduce to a self-citation chain or ansatz smuggled via prior work. The frequency distribution and delay classes enter as explicit inputs to the mean-field model rather than being redefined from outputs. Finite-size effects and fluctuation concerns raised in the skeptic note affect predictive accuracy but do not create circularity in the derivation itself, which remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; ledger incomplete. No explicit free parameters, axioms, or invented entities are stated. The collective coordinate mean-field description is invoked without listed assumptions or fitted values.

pith-pipeline@v0.9.0 · 5495 in / 977 out tokens · 37680 ms · 2026-05-10T20:13:29.831343+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 unclear
Using a collective coordinate framework, we develop a mean-field description of cluster dynamics and derive an analytical upper bound condition for the existence of two-cluster states (Eqs. 9–24, Sec. IV)

Reference graph

Works this paper leans on

57 extracted references · 1 canonical work pages

[1]

Wiesenfeld, P

K. Wiesenfeld, P. Colet, and S. H. Strogatz, Synchroniza- tion transitions in a disordered josephson series array, Physical review letters76, 404 (1996)

1996
[2]

A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K. Showalter, Dynamical quorum sensing and synchro- nization in large populations of chemical oscillators, sci- ence323, 614 (2009). 10

2009
[3]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, Journal of theoretical biology16, 15 (1967)

1967
[4]

Arenas, A

A. Arenas, A. D´ ıaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Synchronization in complex networks, Physics reports469, 93 (2008)

2008
[5]

H. E. Stanley, Phase transitions and critical phenomena, Moscow, Mir (1971)

1971
[6]

M. E. Newman, The structure and function of complex networks, SIAM review45, 167 (2003)

2003
[7]

Boccaletti, A

S. Boccaletti, A. N. Pisarchik, C. I. Del Genio, and A. Amann, Synchronization: from coupled systems to complex networks, (2018)

2018
[8]

Sch¨ oll, Nonequilibrium phase transitions in semicon- ductors: self-organization induced by generation and re- combination processes, (2012)

E. Sch¨ oll, Nonequilibrium phase transitions in semicon- ductors: self-organization induced by generation and re- combination processes, (2012)

2012
[9]

C. Guo, J. Wang, J. Li, Z. Wang, and S. Tang, Kinetic pathways and mechanisms of two-step nucleation in crys- tallization, The journal of physical chemistry letters7, 5008 (2016)

2016
[10]

L. V. Gambuzza, M. Frasca, F. Sorrentino, L. M. Pecora, and S. Boccaletti, Controlling symmetries and clustered dynamics of complex networks, IEEE Transactions on Network Science and Engineering8, 282 (2020)

2020
[11]

C. Xu, S. Boccaletti, Z. Zheng, and S. Guan, Univer- sal phase transitions to synchronization in kuramoto- like models with heterogeneous coupling, New Journal of Physics21, 113018 (2019)

2019
[12]

F. A. Rodrigues, T. K. D. Peron, P. Ji, and J. Kurths, The kuramoto model in complex networks, Physics Re- ports610, 1 (2016)

2016
[13]

Menara, G

T. Menara, G. Baggio, D. S. Bassett, and F. Pasqualetti, Stability conditions for cluster synchronization in net- works of heterogeneous kuramoto oscillators, IEEE Transactions on Control of Network Systems7, 302 (2019)

2019
[14]

Frolov, V

N. Frolov, V. Maksimenko, S. Majhi, S. Rakshit, D. Ghosh, and A. Hramov, Chimera-like behavior in a heterogeneous kuramoto model: The interplay between attractive and repulsive coupling, Chaos: An Interdisci- plinary Journal of Nonlinear Science30(2020)

2020
[15]

Lu¸ con and C

E. Lu¸ con and C. Poquet, Long time dynamics and disorder-induced traveling waves in the stochastic ku- ramoto model, (2017)

2017
[16]

L. F. Abbott and S. B. Nelson, Synaptic plasticity: tam- ing the beast, Nature neuroscience3, 1178 (2000)

2000
[17]

Berner, T

R. Berner, T. Gross, C. Kuehn, J. Kurths, and S. Yanchuk, Adaptive dynamical networks, Physics Re- ports1031, 1 (2023)

2023
[18]

Taylor, E

D. Taylor, E. Ott, and J. G. Restrepo, Spontaneous syn- chronization of coupled oscillator systems with frequency adaptation, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics81, 046214 (2010)

2010
[19]

Horstmeyer and C

L. Horstmeyer and C. Kuehn, Adaptive voter model on simplicial complexes, Physical Review E101, 022305 (2020)

2020
[20]

Markram, J

H. Markram, J. L¨ ubke, M. Frotscher, and B. Sakmann, Regulation of synaptic efficacy by coincidence of postsy- naptic aps and epsps, Science275, 213 (1997)

1997
[21]

Jain and S

S. Jain and S. Krishna, A model for the emergence of cooperation, interdependence, and structure in evolving networks, Proceedings of the National Academy of Sci- ences98, 543 (2001)

2001
[22]

Berner, S

R. Berner, S. Vock, E. Sch¨ oll, and S. Yanchuk, Desyn- chronization transitions in adaptive networks, Physical Review Letters126, 028301 (2021)

2021
[23]

Fialkowski, S

J. Fialkowski, S. Yanchuk, I. M. Sokolov, E. Sch¨ oll, G. A. Gottwald, and R. Berner, Heterogeneous nucleation in finite-size adaptive dynamical networks, Physical review letters130, 067402 (2023)

2023
[24]

Aoki and T

T. Aoki and T. Aoyagi, Co-evolution of phases and con- nection strengths in a network of phase oscillators, Phys- ical review letters102, 034101 (2009)

2009
[25]

Aoki and T

T. Aoki and T. Aoyagi, Self-organized network of phase oscillators coupled by activity-dependent interactions, Physical Review E84, 066109 (2011)

2011
[26]

Kasatkin, S

D. Kasatkin, S. Yanchuk, E. Sch¨ oll, and V. Nekorkin, Self-organized emergence of multilayer structure and chimera states in dynamical networks with adaptive cou- plings, Physical Review E96, 062211 (2017)

2017
[27]

Kasatkin and V

D. Kasatkin and V. Nekorkin, Synchronization of chimera states in a multiplex system of phase oscilla- tors with adaptive couplings, Chaos: An Interdisciplinary Journal of Nonlinear Science28(2018)

2018
[28]

Berner, J

R. Berner, J. Fialkowski, D. Kasatkin, V. Nekorkin, S. Yanchuk, and E. Sch¨ oll, Hierarchical frequency clus- ters in adaptive networks of phase oscillators, Chaos: An Interdisciplinary Journal of Nonlinear Science29(2019)

2019
[29]

Berner, J

R. Berner, J. Sawicki, and E. Sch¨ oll, Birth and stabiliza- tion of phase clusters by multiplexing of adaptive net- works, Physical review letters124, 088301 (2020)

2020
[30]

Gerstner, A neuronal learning rule for sub-millisecond temporal coding, Nature395, 37 (1998)

W. Gerstner, A neuronal learning rule for sub-millisecond temporal coding, Nature395, 37 (1998)

1998
[31]

Caporale and Y

N. Caporale and Y. Dan, Spike timing–dependent plas- ticity: a hebbian learning rule, Annu. Rev. Neurosci.31, 25 (2008)

2008
[32]

Bi and M.-m

G.-q. Bi and M.-m. Poo, Synaptic modification by corre- lated activity: Hebb’s postulate revisited, Annual review of neuroscience24, 139 (2001)

2001
[33]

Meisel and T

C. Meisel and T. Gross, Adaptive self-organization in a realistic neural network model, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics80, 061917 (2009)

2009
[34]

L¨ ucken, O

L. L¨ ucken, O. V. Popovych, P. A. Tass, and S. Yanchuk, Noise-enhanced coupling between two oscillators with long-term plasticity, Physical Review E93, 032210 (2016)

2016
[35]

Donald, The organization of behavior, New York 1952 Donald The Organization of Behaviour 1952 (1949)

H. Donald, The organization of behavior, New York 1952 Donald The Organization of Behaviour 1952 (1949)

1952
[36]

L¨ owel and W

S. L¨ owel and W. Singer, Selection of intrinsic horizontal connections in the visual cortex by correlated neuronal activity, Science255, 209 (1992)

1992
[37]

L. I. Zhang, H. W. Tao, C. E. Holt, W. A. Harris, and M.-m. Poo, A critical window for cooperation and com- petition among developing retinotectal synapses, Nature 395, 37 (1998)

1998
[38]

Berner, S

R. Berner, S. Yanchuk, and E. Sch¨ oll, What adaptive neuronal networks teach us about power grids, Physical Review E103, 042315 (2021)

2021
[39]

Anand, V.K.Chandrasekar, and R

R. Anand, V.K.Chandrasekar, and R. Suresh, Emergence of solitary and chimera states in adaptive pendulum net- works under diverse learning rules, Chaos, Solitons & Fractals207, 118040 (2026)

2026
[40]

Yadav, J

A. Yadav, J. Fialkowski, R. Berner, V.K.Chandrasekar, and D.V.Senthilkumar, Disparity-driven heterogeneous nucleation in finite-size adaptive networks, Physical Re- view E109, L052301 (2024). 11

2024
[41]

Yadav, Q

A. Yadav, Q. S. Rahaman, V.K.Chandrasekar, and D.V.Senthilkumar, Heterogeneous nucleation in a mul- tiplex adaptive network, Physical Review E111, 054318 (2025)

2025
[42]

Senthamizhan, R

R. Senthamizhan, R. Gopal, and V.K.Chandrasekar, Swarmalators with frequency-weighted interactions, arXiv preprint arXiv:2510.05663 (2025)

work page arXiv 2025
[43]

C. Xu, Y. Sun, J. Gao, T. Qiu, Z. Zheng, and S. Guan, Synchronization of phase oscillators with frequency- weighted coupling, Scientific reports6, 21926 (2016)

2016
[44]

Timms and L

L. Timms and L. Q. English, Synchronization in phase- coupled kuramoto oscillator networks with axonal delay and synaptic plasticity, Physical Review E89, 032906 (2014)

2014
[45]

Thamizharasan, V.K.Chandrasekar, M

S. Thamizharasan, V.K.Chandrasekar, M. Senthilvelan, and D.V.Senthilkumar, Dynamics of adaptive network with connection delay and external stimulus, Chaos, Soli- tons & Fractals199, 116747 (2025)

2025
[46]

Thamizharasan, V.K.Chandrasekar, M

S. Thamizharasan, V.K.Chandrasekar, M. Senthilve- lan, and D.V.Senthilkumar, Stimulus-induced dynamical states in an adaptive network with symmetric adapta- tion, Physical Review E110, 034217 (2024)

2024
[47]

M. S. Yeung and S. H. Strogatz, Time delay in the ku- ramoto model of coupled oscillators, Physical review let- ters82, 648 (1999)

1999
[48]

Wu and M

H. Wu and M. Dhamala, Dynamics of kuramoto oscilla- tors with time-delayed positive and negative couplings, Physical Review E98, 032221 (2018)

2018
[49]

C. J. Stam, Modern network science of neurological dis- orders, Nature Reviews Neuroscience15, 683 (2014)

2014
[50]

V. V. Klinshov and A. A. Zlobin, Kuramoto model with delay: The role of the frequency distribution, Mathemat- ics11, 2325 (2023)

2023
[51]

Montbri´ o, D

E. Montbri´ o, D. Paz´ o, and J. Schmidt, Time delay in the kuramoto model with bimodal frequency distribu- tion, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics74, 056201 (2006)

2006
[52]

P. S. Skardal, Low-dimensional dynamics of the kuramoto model with rational frequency distributions, Physical Re- view E98, 022207 (2018)

2018
[53]

L. D. Smith and G. A. Gottwald, Model reduction for the collective dynamics of globally coupled oscillators: From finite networks to the thermodynamic limit, Chaos: An Interdisciplinary Journal of Nonlinear Science30(2020)

2020
[54]

L. D. Smith and G. A. Gottwald, Chaos in networks of coupled oscillators with multimodal natural frequency distributions, Chaos: An Interdisciplinary Journal of Nonlinear Science29(2019)

2019
[55]

D. H. Zanette, Propagating structures in globally coupled systems with time delays, Physical Review E62, 3167 (2000)

2000
[56]

P. S. Skardal, D. Taylor, and J. G. Restrepo, Com- plex macroscopic behavior in systems of phase oscillators with adaptive coupling, Physica D: Nonlinear Phenom- ena267, 27 (2014)

2014
[57]

Zhang, S

X. Zhang, S. Boccaletti, S. Guan, and Z. Liu, Explo- sive synchronization in adaptive and multilayer networks, Physical review letters114, 038701 (2015)

2015