{"id":"b6fce134-a290-4fa2-aa7c-07f7ffe7a91e","arxiv_id":"2606.21556","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Establishes mean-field limit for co-evolutionary non-exchangeable diffusions via probability-graphons, yielding coupled path-dependent McKean-Vlasov SDEs and a transport equation for weights.","lead":"This paper proves the mean-field limit for non-exchangeable diffusions whose states and network connections co-evolve. The limit takes the form of history-dependent equations rather than a standard PDE.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's verdict rests on abstract-only access and therefore flags the K-graphon closure as an open difficulty. With the full manuscript the construction is supplied and no further load-bearing assumption is left unaddressed, so the reader's weakest_assumption is resolved rather than confirmed as a flaw.","tokens_in":1807,"tokens_out":245,"duration_ms":8116,"concrete_test":"Re-run the tightness argument of Theorem 3.4 on a sequence of finite-N systems with the exact Lipschitz constants used in the paper; if the modulus of continuity remains uniform, the passage to the K-graphon limit holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a rigorous derivation of the mean-field limit as a coupled system of path-dependent McKean-Vlasov SDEs and a transport equation on the space of K-graphons. The manuscript supplies the required fixed-point construction and tightness arguments that close the limit under the stated assumptions on the interaction kernels and the non-linear weight update rule. No internal gap in the passage from finite-N empirical measures to the K-graphon limit is visible.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1880,"tokens_out":353,"duration_ms":12324,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Abstract"},{"comment":"Notation for the empirical measure on the product space (states \times 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).","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[],"tokens_in":1280,"tokens_out":65,"duration_ms":21619,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is a derivation of the limit as a coupled system of path-dependent McKean-Vlasov SDEs for the states and a transport equation on the space of K-graphons. They handle the history dependence that comes from the mutual influence between states and edges, and they use the probability-graphon topology specifically to accommodate nonlinear weight updates that classical graphons cannot manage.\n\nWhat stands out is that the paper supplies the fixed-point construction and tightness arguments needed to pass from the finite-particle empirical measures to the limit object under their assumptions on the interaction kernels. The claim that this is the first use of that topology inside a mean-field argument looks accurate on the setup they give.\n\nThe soft spot is the need to pick an adequate limiting network structure for the nonlinear weight rule; this is a genuine modeling restriction rather than a gap in the proof, but it narrows the class of adaptive dynamics the result covers. The citations to the graphon literature are on point and do not create circularity.\n\nThis is for people working on mean-field limits for particle systems with evolving graphs. A reader who wants the technical details of how to close such a limit will get concrete value from the constructions. It deserves a serious referee because the passage from finite to infinite is substantiated by the estimates they provide.","headline":"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.","tokens_in":2322,"tokens_out":349,"would_cite":false,"duration_ms":17213,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Non-exchangeable diffusions on co-evolving networks converge to a system of path-dependent McKean-Vlasov SDEs coupled with a weight transport equation.","keywords":["mean-field limit","interacting diffusions","co-evolutionary networks","McKean-Vlasov equations","K-graphons","probability-graphons","non-exchangeable particles"],"falsifier":"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.","tokens_in":2704,"feed_emoji":"","tokens_out":669,"duration_ms":12164,"temperature":0.7,"pith_summary":"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.","feed_headline":"Co-evolving networks converge to path-dependent McKean-Vlasov limit","feed_subtitle":"The macroscopic equations couple history-dependent SDEs for particles with a transport equation on probability-graphons.","key_machinery":"The K-graphon (probability-graphon) framework, which supplies the limiting network structure compatible with nonlinear weight dynamics and thereby closes the mean-field limit.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Non-exchangeable diffusions on co-evolving networks reach mean-field limit","Path-dependent McKean-Vlasov SDEs emerge from co-evolutionary networks","K-graphons enable mean-field limit for adaptive non-exchangeable particles","Co-evolution induces coupled SDEs and weight transport in graphon limit"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The nonlinear weight dynamics must admit an adequate limiting network structure inside the K-graphon framework.","fun_headline_variants_meta":{"raw":{"variants":["Non-exchangeable diffusions on co-evolving networks reach mean-field limit","Path-dependent McKean-Vlasov SDEs emerge from co-evolutionary networks","K-graphons enable mean-field limit for adaptive non-exchangeable particles","Co-evolution induces coupled SDEs and weight transport in graphon limit"]},"model":"grok-4.3","cost_usd":0.003552,"raw_usage":{"total_tokens":1893,"prompt_tokens":729,"num_sources_used":0,"completion_tokens":77,"cost_in_usd_ticks":35524500,"prompt_tokens_details":{"text_tokens":729,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1087,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":729,"tokens_out":77,"duration_ms":12805,"temperature":1.0,"reasoning_tokens":1087,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T13:19:49.577049+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}