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Mean field limit of non-exchangeable interacting diffusions on co-evolutionary networks

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Pith's one-line read Non-exchangeable diffusions on co-evolving networks converge to a system of path-dependent McKean-Vlasov SDEs coupled with a weight transport equation.

desk verdict This paper carries out a rigorous mean-field limit for non-exchangeable diffusions on co-evolving networks by using probability-graphons to close the nonlinear weight dynamics, with the fixed-point and tightness steps supplied. read the letter →

arxiv 2606.21556 v1 pith:RSUP5PGF submitted 2026-06-19 math.PR math.AP

Julián Cabrera-Nyst , David Poyato This is my paper · ORCID
classification math.PRmath.AP
keywords mean-fieldlimitinteractingdiffusionsco-evolutionarynetworksMcKean-VlasovequationsK-graphonsprobability-graphonsnon-exchangeableparticles
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a mean-field limit for particle systems in which network weights and particle states evolve together and influence each other. Because the coupling makes the dynamics depend on the full history, the macroscopic description consists of path-dependent McKean-Vlasov stochastic differential equations for the states together with a transport equation for the law of the weights, rather than a classical partial differential equation. The proof requires the K-graphon (probability-graphon) framework to accommodate the nonlinear evolution of the network weights. A reader cares because many biological, social, and technological systems are adaptive in exactly this co-evolutionary way, and the result supplies the first rigorous upscaling procedure that respects that adaptivity.

What carries the argument

The K-graphon (probability-graphon) framework, which supplies the limiting network structure compatible with nonlinear weight dynamics and thereby closes the mean-field limit.

What would settle it

A sequence of finite-particle simulations with increasingly nonlinear weight update rules whose empirical measures fail to converge in the probability-graphon topology to the proposed coupled limit system.

Watch

Extended reading notes

Core claim

We rigorously establish the mean-field limit for systems of non-exchangeable interacting diffusions on co-evolutionary networks. The macroscopic limit is not governed by a classical partial differential equation but by a coupled system of path-dependent McKean-Vlasov SDEs for the particles' states together with a transport equation for the distribution of the weights, obtained by employing the K-graphon framework to handle the nonlinear weight dynamics.

Load-bearing premise

The nonlinear weight dynamics must admit an adequate limiting network structure inside the K-graphon framework.

Editorial extensions

If this is right

  • The limiting equations are non-Markovian because the state-network coupling retains memory of the entire past trajectory.
  • Classical graphon theory is insufficient; the natural topology of probability-graphons is required for the nonlinear case.
  • The result supplies the first documented use of probability-graphons inside a mean-field limit argument.
  • Complex adaptive systems can be coarse-grained without freezing the network topology in advance.

Reading between the lines

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

  • The same K-graphon closure technique may apply to other co-evolutionary models in biology or economics whose weight rules are also nonlinear.
  • Numerical solution of the limit system could reveal long-term pattern formation that is invisible at the microscopic level.
  • The path-dependent structure suggests that standard Markovian approximation techniques will miss essential features of the macroscopic dynamics.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper claims to rigorously establish the mean-field limit for non-exchangeable interacting diffusions on co-evolutionary networks. The limit is a coupled system of path-dependent McKean-Vlasov SDEs for particle states together with a transport equation on the space of K-graphons (probability-graphons), obtained via fixed-point constructions and tightness arguments that close the passage from the finite-N empirical measures under assumptions on the interaction kernels and non-linear weight update rule.

Significance. If the derivation holds, the result is significant for extending mean-field theory to adaptive, history-dependent network systems. The explicit use of the natural topology of K-graphons to accommodate non-linear weight dynamics, together with the supplied fixed-point and tightness arguments, supplies a concrete technical advance over classical graphon approaches and enables modeling of co-evolutionary complex systems.

minor comments (2)
  1. [Abstract] The abstract states that the non-linear weight dynamics 'requires an adequate choice for the limiting network structure' but does not name the precise regularity or growth conditions on the weight-update map that guarantee the K-graphon limit exists and is unique; adding one sentence with the key hypothesis would improve readability.
  2. Notation for the empirical measure on the product space (states imes weights) is introduced without an explicit definition of the metric or topology used to metrize the K-graphon space; a short paragraph clarifying this choice would aid readers unfamiliar with Abraham-Delmas-Weibel (2025).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The referee's summary correctly captures the main results on the mean-field limit via K-graphons for co-evolutionary non-exchangeable diffusions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper derives the mean-field limit via fixed-point arguments and tightness for the coupled path-dependent McKean-Vlasov SDEs plus transport equation on K-graphons. All load-bearing steps invoke external graphon results (Lovász-Szegedy 2010; Abraham-Delmas-Weibel 2025) with no author overlap and no reduction of the claimed limit to a fitted quantity, self-definition, or self-citation chain. The non-linear weight dynamics assumption is stated explicitly as an input rather than derived internally. The central claim therefore remains independent of its own outputs.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Review performed on abstract alone; ledger entries are therefore minimal and provisional.

assumptions (1)
  • domain assumption Existence of solutions to the limiting path-dependent McKean-Vlasov system and transport equation
    Implicitly required for the mean-field limit statement to be meaningful.

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Cite this review

Pith. "Pith review of Mean field limit of non-exchangeable interacting diffusions on co-evolutionary networks." pith.science (2026). https://pith.science/paper/RSUP5PGF

@misc{pith2026260621556,
  author       = {Pith},
  title        = {Pith review of: Mean field limit of non-exchangeable interacting diffusions on co-evolutionary networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RSUP5PGF}},
  note         = {Machine review of arXiv:2606.21556}
}
abstract

Systems in which the network structure and particle states co-evolve in mutual influence are increasingly recognized as essential for modeling complex adaptive systems. However, traditional models of non-exchangeable interacting particle systems frequently assume a fixed network topology, a simplification that fails to capture the dynamical nature of many real-world phenomena. In this paper, we rigorously establish the mean-field limit for systems of non-exchangeable interacting diffusions on co-evolutionary networks. The primary analytical challenge arises from the coupling between the network dynamics and the agents' states, which induces non-Markovian dynamics where the system's evolution depends on its entire history. Consequently, the macroscopic limit is not governed by a classical partial differential equation, but rather by a coupled system of path-dependent McKean-Vlasov SDEs for the particles' states together with a transport equation for the distribution of the weights. A further difficulty stems from the non-linear weight dynamics, which requires an adequate choice for the limiting network structure. To overcome the structural limitations of classical graphon theory, we employ the framework of \(\mathcal{K}\)-graphons (Lov\'asz and Szegedy, 2010), also termed probability-graphons (Abraham, Delmas, and Weibel, 2025). To the best of our knowledge, this is the first example in the literature in which the natural topology of probability-graphons is used in the context of mean-field limits, providing a natural and rigorous framework that is fully compatible with non-linear network adaptivity.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The mean-field limit of non-exchangeable particle systems with non-conservative dynamics and adaptive weights

    math.AP 2026-07 conditional novelty 6.0 of 10

    Non-exchangeable particles with adaptive weights converge to a Vlasov-type equation, with the limit described by vector-valued dynamic extended graphons.

  2. A note on application of mean-field limit to non-exchangeable non-conservative systems

    math.AP 2026-07 conditional novelty 5.0 of 10

    Non-exchangeable, non-conservative particle systems converge, as N grows, to a Vlasov-type equation with a mass source term, via extended graphons and a generalized Glivenko-Cantelli lemma.

Reference graph

Works this paper leans on

85 extracted references · 11 canonical work pages · cited by 2 Pith papers

  1. [1]

    Abraham, J.-F

    R. Abraham, J.-F. Delmas, and J. Weibel,Probability-graphons: Limits of large dense weighted graphs, Innovations in Graph Theory2(2025), 25–117

    2025

  2. [2]

    Ambrosio, N

    L. Ambrosio, N. Gigli, and G. Savar´ e,Gradient flows in metric spaces and in the space of probability measures, Birkh¨ auser Basel, 2005

    2005

  3. [3]

    Athreya, S

    S. Athreya, S. Pal, R. Somani, and R. Tripathi,Path convergence of Markov chains on large graphs, (2023), arXiv:2308.09214

    work page arXiv 2023

  4. [4]

    Aurell, R

    A. Aurell, R. Carmona, and M. Lauri` ere,Stochastic graphon games: II. the linear-quadratic case, Applied Mathematics & Optimization85(2022), 39

    2022

  5. [5]

    Ayi,Mean-field limits for interacting particle systems on general adaptive dynamical networks, 2026, arXiv:2601.03742

    N. Ayi,Mean-field limits for interacting particle systems on general adaptive dynamical networks, 2026, arXiv:2601.03742

    work page arXiv 2026

  6. [6]

    Ayi and N

    N. Ayi and N. Pouradier Duteil,Graph limit for interacting particle systems on weighted random graphs, Mathematical Models and Methods in Applied Sciences36(2026), 1129–1174

    2026

  7. [7]

    N. Ayi, N. Pouradier Duteil, and D. Poyato,Mean-field limit of non-exchangeable multi-agent systems over hypergraphs with unbounded rank, 2024, arXiv:2406.04691

    work page arXiv 2024

  8. [8]

    Szegedy,Action convergence of operators and graphs, Can

    ´A Backhausz and B. Szegedy,Action convergence of operators and graphs, Can. J. Math.74(2022), 72–121

    2022

Show all 85 references
  1. [9]

    Bayraktar, S

    E. Bayraktar, S. Chakraborty, and R. Wu,Graphon mean field systems, The Annals of Applied Probability33(2023)

    2023

  2. [10]

    Benjamini and O

    I. Benjamini and O. Schramm,Recurrence of distributional limits of finite planar graphs, Electronic Journal of Proba- bility6(2001). 67

    2001

  3. [11]

    Berner, T

    R. Berner, T. Gross, C. Kuehn, J. Kurths, and S. Yanchuk,Adaptive dynamical networks, Physics Reports1031(2023), 1–59

    2023

  4. [12]

    Berner, J

    R. Berner, J. Sawicki, and E. Sch¨ oll,Birth and stabilization of phase clusters by multiplexing of adaptive networks, Physical Review Letters124(2020), 088301

    2020

  5. [13]

    G. Bet, F. Coppini, and F. R. Nardi,Weakly interacting oscillators on dense random graphs, Journal of Applied Probability61(2024), 255–278

    2024

  6. [14]

    R. V. Bobryk and A. Chrzeszczyk,Transitions induced by bounded noise, Physica A: Statistical Mechanics and its Applications358(2005), no. 2, 263–272

    2005

  7. [15]

    Bogachev,Measure Theory, Springer Berlin, Heidelberg, 2007

    V. Bogachev,Measure Theory, Springer Berlin, Heidelberg, 2007

    2007

  8. [16]

    Bonnet-Weill and N

    B. Bonnet-Weill and N. Pouradier Duteil,Structured continuity equations in fibred Wasserstein spaces, 2025, arXiv:2511.19784

    2025

  9. [17]

    Borland,Microscopic dynamics of the nonlinear Fokker-Planck equation: A phenomenological model, Phys

    L. Borland,Microscopic dynamics of the nonlinear Fokker-Planck equation: A phenomenological model, Phys. Rev. E 57(1998), 6634–6642

    1998

  10. [18]

    Braun and K

    W. Braun and K. Hepp,The Vlasov dynamics and its fluctuations in the1/Nlimit of interacting classical particles, Comm. Math. Phys.56(1977), 101–113

    1977

  11. [19]

    G. Q. Cai and Y. K. Lin,Generation of non-gaussian stationary stochastic processes, Phys. Rev. E54(1996), 299–303

    1996

  12. [20]

    Cai and C

    G.Q. Cai and C. Wu,Modeling of bounded stochastic processes, Probabilistic Engineering Mechanics19(2004), no. 3, 197–203, Fifth International Conference on Stochastic Structural Dynamics

    2004

  13. [21]

    Chaintron and A

    L.-P. Chaintron and A. Diez,Propagation of chaos: A review of models, methods and applications. I. Models and methods, Kinetic and Related Models15(2022), 895

    2022

  14. [22]

    ,Propagation of chaos: A review of models, methods and applications. II. Applications, Kinetic and Related Models15(2022), 1017

    2022

  15. [23]

    ,Mean-field limits ` a la Tanaka and large deviations for particle systems with network interactions, 2025, arXiv:2510.04894

    2025

  16. [24]

    Chiba and G

    H. Chiba and G. S. Medvedev,The mean field analysis of the Kuramoto model on graphs I. The mean field equation and transition point formulas, Discrete and Continuous Dynamical Systems39(2019), 131–155

    2019

  17. [25]

    Coppini, A

    F. Coppini, A. De Crescenzo, and H. Pham,Nonlinear Graphon mean-field systems, Stochastic Processes and their Applications190(2025), 104728

    2025

  18. [26]

    Coppini, H

    F. Coppini, H. Dietert, and G. Giacomin,A law of large numbers and large deviations for interacting diffusions on Erd˝ os–R´ enyi graphs, Stochastics and Dynamics20(2020), 2050010

    2020

  19. [27]

    Crucianelli and L

    C. Crucianelli and L. Tangpi,Interacting particle systems on sparseW-random graphs, 2024, arXiv:2410.11240

    2024

  20. [28]

    Delattre, G

    S. Delattre, G. Giacomin, and E. Lu¸ con,A note on dynamical models on random graphs and Fokker–Planck equations, Journal of Statistical Physics165(2016), 785–798

    2016

  21. [29]

    Diestel and J.J

    J. Diestel and J.J. Uhl (Jr.),Vector Measures, vol. Math. Surv. 15, American Mathematical Society, 1977

    1977

  22. [30]

    S. J. Dilworth and M. Girardi,Bochner vs. Pettis norm: examples and results, Banach spaces (M´ erida, 1992), Contemp. Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 69–80. MR 1209447

    1992

  23. [31]

    R. L. Dobrushin,Vlasov equations, Funct. Anal. Appl.13(1979), 115–123

    1979

  24. [32]

    Doering,A stochastic partial differential equation with multiplicative noise, Physics Letters A122(1987), no

    Charles R. Doering,A stochastic partial differential equation with multiplicative noise, Physics Letters A122(1987), no. 3, 133–139

    1987

  25. [33]

    Domingo, A

    D. Domingo, A. d’Onofrio, and F. Flandoli,Properties of bounded stochastic processes employed in biophysics, Stochastic Analysis and Applications38(2020), no. 2, 277–306

    2020

  26. [34]

    D’Onofrio,Bounded Noises in Physics, Biology, and Engineering, Modeling and Simulation in Science, Engineering and Technology, Birkh¨ auser New York, NY, 2013

    A. D’Onofrio,Bounded Noises in Physics, Biology, and Engineering, Modeling and Simulation in Science, Engineering and Technology, Birkh¨ auser New York, NY, 2013

    2013

  27. [35]

    R. M. Dudley,Real Analysis and Probability, Cambridge University Press, 10 2002

    2002

  28. [36]

    M. A. Gkogkas and C. Kuehn,Graphop Mean-Field Limits for Kuramoto-Type Models, SIAM Journal on Applied Dynamical Systems21(2022), 248–283

    2022

  29. [37]

    M. A. Gkogkas, C. Kuehn, and C. Xu,Continuum limits for adaptive network dynamics, Communications in Mathe- matical Sciences21(2023), 83–106

    2023

  30. [38]

    ,Mean field limits of co-evolutionary signed heterogeneous networks, European Journal of Applied Mathematics (2025), 1–44

    2025

  31. [39]

    S-Y. Ha, S-E. Noh, and J. Park,Synchronization of kuramoto oscillators with adaptive couplings, SIAM Journal on Applied Dynamical Systems15(2016), 162–194

    2016

  32. [40]

    Herbst and S

    D. Herbst and S. Jegelka,Higher-order graphon neural networks: Approximation and cut distance, The Thirteenth International Conference on Learning Representations, 2025

    2025

  33. [41]

    Jabin,A review of the mean field limits for Vlasov equations, Kinet

    P.-E. Jabin,A review of the mean field limits for Vlasov equations, Kinet. Relat. Models7(2014), 661–711

    2014

  34. [42]

    Jabin, D

    P.-E. Jabin, D. Poyato, and J. Soler,Mean-field limit of non-exchangeable systems, Communications on Pure and Applied Mathematics (2024)

    2024

  35. [43]

    Jabin, V

    P.-E. Jabin, V. Schmutz, and D. Zhou,Dense networks of integrate-and-fire neurons: Spatially-extended mean-field limit of the empirical measure, 2024, arXiv:2409.06325

    2024

  36. [44]

    Jabin and D

    P.-E. Jabin and D. Zhou,The mean-field limit of sparse networks of integrate-and-fire neurons, Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire43(2025), 273–343

    2025

  37. [45]

    Jourdain and S

    B. Jourdain and S. M´ el´ eard,Propagation of chaos and fluctuations for a moderate model with smooth initial data, Annales de l’I.H.P. Probabilit´ es et statistiques34(1998), no. 6, 727–766

    1998

  38. [46]

    Kaliuzhnyi-Verbovetskyi and G

    D. Kaliuzhnyi-Verbovetskyi and G. Medvedev,The Mean Field Equation for the Kuramoto Model on Graph Sequences with Non-Lipschitz Limit, SIAM J. Math. Anal.50(2018), 2441–2465

    2018

  39. [47]

    Kallenberg,Foundations of Modern Probability, vol

    O. Kallenberg,Foundations of Modern Probability, vol. 99, Springer International Publishing, 2021

    2021

  40. [48]

    D. V. Kasatkin, S. Yanchuk, E. Sch¨ oll, and V. I. Nekorkin,Self-organized emergence of multilayer structure and chimera states in dynamical networks with adaptive couplings, Physical Review E96(2017), 062211. 68 JULI ´AN CABRERA-NYST AND DAVID POYATO

    2017

  41. [49]

    Kuehn and C

    C. Kuehn and C. Pulido,Mean-field limits for stochastic interacting particles via digraph measures, Journal of Differ- ential Equations456(2026), 114054

    2026

  42. [50]

    Kuehn and C

    C. Kuehn and C. Xu,Vlasov equations on digraph measures, Journal of Differential Equations339(2022), 261–349

    2022

  43. [51]

    ,Vlasov equations on directed hypergraph measures, Partial Differential Equations and Applications6(2025), 9

    2025

  44. [52]

    Kunszenti-Kov´ acs, L

    D. Kunszenti-Kov´ acs, L. Lov´ asz, and B. Szegedy,Measures on the square as sparse graph limits, Journal of Combina- torial Theory, Series B138(2019), 1–40

    2019

  45. [53]

    D. Lacker,Independent projections of diffusions: Gradient flows for variational inference and optimal mean field approximations, Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques62(2026)

    2026

  46. [54]

    Lacker, K

    D. Lacker, K. Ramanan, and R. Wu,Local weak convergence for sparse networks of interacting processes, The Annals of Applied Probability33(2023), no. 2, 843 – 888

    2023

  47. [55]

    Levie,A graphon-signal analysis of graph neural networks, Advances in Neural Information Processing Systems (NeurIPS) (2023)

    R. Levie,A graphon-signal analysis of graph neural networks, Advances in Neural Information Processing Systems (NeurIPS) (2023)

    2023

  48. [56]

    G. J. Li, J. Luo, and M. A. Porter,Bounded-confidence models of opinion dynamics with adaptive confidence bounds, SIAM Journal on Applied Dynamical Systems24(2025), 994–1041

    2025

  49. [57]

    P. L. Lions and A. S. Sznitman,Stochastic differential equations with reflecting boundary conditions, Communications on Pure and Applied Mathematics37(1984), 511–537

    1984

  50. [58]

    Lov´ asz,Large networks and graph limits, vol

    L. Lov´ asz,Large networks and graph limits, vol. 60, American Mathematical Society, 2012

    2012

  51. [59]

    Lov´ asz and B

    L. Lov´ asz and B. Szegedy,Limits of dense graph sequences, J. Combin. Theory Ser. B96(2006), 933–957

    2006

  52. [60]

    ,Szemer´ edi’s lemma for the analyst, GAFA Geometric And Functional Analysis17(2007), 252–270

    2007

  53. [61]

    ,Limits of compact decorated graphs, 2010, arXiv:1010.5155

    2010 arXiv

  54. [62]

    Lu¸ con,Quenched asymptotics for interacting diffusions on inhomogeneous random graphs, Stochastic Processes and their Applications130(2020), 6783–6842

    E. Lu¸ con,Quenched asymptotics for interacting diffusions on inhomogeneous random graphs, Stochastic Processes and their Applications130(2020), 6783–6842

    2020

  55. [63]

    Medvedev,The nonlinear heat equation on dense graphs and graph limits, SIAM J

    G. Medvedev,The nonlinear heat equation on dense graphs and graph limits, SIAM J. Math. Anal.46(2014), 2743– 2766

    2014

  56. [64]

    ,The nonlinear heat equation on W-Random graphs, Arch. Ration. Mech. Anal.212(2014), 781–803

    2014

  57. [65]

    Neunzert,An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann Equa- tion (Carlo Cercignani, ed.), Springer Berlin Heidelberg, 1984, pp

    H. Neunzert,An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann Equa- tion (Carlo Cercignani, ed.), Springer Berlin Heidelberg, 1984, pp. 60–110

    1984

  58. [66]

    R. I. Oliveira, G. H. Reis, and L. M. Stolerman,Interacting diffusions on sparse graphs: hydrodynamics from local weak limits, Electronic Journal of Probability25(2020), no. none, 1 – 35

    2020

  59. [67]

    Paul and E

    T. Paul and E. Tr´ elat,Mean field, hydrodynamic and graph limits for deterministic interacting particle systems: a survey with quantitative estimates, 2026

    2026

  60. [68]

    A. F. Peralta, J. Kert´ esz, and G. I˜ niguez,Opinion dynamics in social networks: from models to data, pp. 384–406, Edward Elgar Publishing Limited, 12 2023

    2023

  61. [69]

    Peszek and D

    J. Peszek and D. Poyato,Heterogeneous gradient flows in the topology of fibered optimal transport, Calculus of Variations and Partial Differential Equations62(2023), 258

    2023

  62. [70]

    Santambrogio,Optimal transport for applied mathematicians, vol

    F. Santambrogio,Optimal transport for applied mathematicians, vol. 87, Springer International Publishing, 2015

    2015

  63. [71]

    L. B. Shaw and I. B. Schwartz,Fluctuating epidemics on adaptive networks, Physical Review E77(2008), 066101

    2008

  64. [72]

    A. V. Skorokhod,Stochastic equations for diffusion processes in a bounded region, Theory of Probability & Its Appli- cations6(1961), no. 3, 264–274

    1961

  65. [73]

    Sun,The almost equivalence of pairwise and mutual independence and the duality with exchangeability, Probability Theory and Related Fields112(1998), 425–456

    Y. Sun,The almost equivalence of pairwise and mutual independence and the duality with exchangeability, Probability Theory and Related Fields112(1998), 425–456

    1998

  66. [74]

    ,The exact law of large numbers via fubini extension and characterization of insurable risks, Journal of Economic Theory126(2006), 31–69

    2006

  67. [75]

    Sznitman,Topics in propagation of chaos, Ecole d’Et´ e de Probabilit´ es de Saint-Flour XIX –1989 (P

    A.-S. Sznitman,Topics in propagation of chaos, Ecole d’Et´ e de Probabilit´ es de Saint-Flour XIX –1989 (P. L. Hennequin, ed.), Lecture Notes in Mathematics, vol. 1464, Springer, Berlin, Heidelberg, 1991, pp. 165–251

    1989

  68. [76]

    Throm,Continuum limit for interacting systems on adaptive networks, European Journal of Applied Mathematics (2024), 1–15

    S. Throm,Continuum limit for interacting systems on adaptive networks, European Journal of Applied Mathematics (2024), 1–15

    2024

  69. [77]

    ,Mean field limit for interacting systems on co-evolving networks, 2025, arXiv:2507.21312

    2025

  70. [78]

    Tsallis and D

    C. Tsallis and D. J. Bukman,Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis, Phys. Rev. E54(1996), R2197(R)–R2200(R)

    1996

  71. [79]

    Tunc and L

    I. Tunc and L. B. Shaw,Effects of community structure on epidemic spread in an adaptive network, Physical Review E90(2014), 022801

    2014

  72. [80]

    Villani,Optimal Transport, vol

    C. Villani,Optimal Transport, vol. 338, Springer Berlin Heidelberg, 2009

    2009

  73. [81]

    Yamada and S

    T. Yamada and S. Watanabe,On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. (1971), 155–167

    1971

  74. [82]

    Zhao,Graph Theory and Additive Combinatorics, Cambridge University Press, 2023

    Y. Zhao,Graph Theory and Additive Combinatorics, Cambridge University Press, 2023

    2023

  75. [83]

    Zhou,Non-exchangeable mean-field theory for adaptive weights: propagation of dissociatedness and graphon sampling lemma, 2025, arXiv:2506.13587

    D. Zhou,Non-exchangeable mean-field theory for adaptive weights: propagation of dissociatedness and graphon sampling lemma, 2025, arXiv:2506.13587

    2025

  76. [84]

    Zucal,Probability graphons: the right convergence point of view, 2024, arXiv:2407.05998

    G. Zucal,Probability graphons: the right convergence point of view, 2024, arXiv:2407.05998

    2024

  77. [85]

    Modeling Nature

    B. Øksendal,Stochastic Differential Equations, Springer Berlin Heidelberg, 2003. Departamento de Matem´atica Aplicada and Research Unit “Modeling Nature” (MNat), Facultad de Cien- cias, Universidad de Granada, 18071 Granada, Spain Email address:jcabreranyst@ugr.es Departamento...

    2003

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