{"id":"c401d98d-1a8a-49d7-9c5a-ecc78e02f1b6","arxiv_id":"2507.21312","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Interacting particle systems on co-evolving networks, with memory in the interaction, converge to a deterministic mean-field limit described by a characteristic equation on path space.","lead":"This paper proves a mean-field limit for large particle systems whose interaction network evolves in time along with the particle states, including interactions that depend on the entire past history of the particles. The result covers adaptive network models where edge weights change nonlinearly, and the continuum limit is a measure on path space rather than on state space.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The bridge from the co-evolving system (1) to the abstract nonlocal system (6) is unverified: for natural globally Lipschitz F, the derived K fails the stated boundedness assumption (11), so the advertised application to nonlinear weight dynamics is not established.","rationale":"Proposition 2.4's Dobrushin estimate and Theorem 2.6 are internally consistent for K satisfying the stated hypotheses; Lemma 2.2's Gronwall estimate is correct, and path-space convergence can be obtained from the flow estimates, so I do not see a fatal flaw in the conditional theorem. The load-bearing gap is the bridge from the co-evolving model (1) to the abstract nonlocal system (6). Lemma 1.1 only assumes F globally Lipschitz; it does not imply the derived K satisfies (11)-(13). The counterexample F=w (which satisfies Lemma 1.1) gives K_t(a,f,g)=a e^t C(f(t),g(t)), so (11) fails. For any F with linear growth in the state arguments, the flow can grow with the sup-norms of the trajectories, again violating (11). The paper neither restricts (11) to the bounded range of W nor verifies that the induced K meets the hypotheses. Since Theorem 2.6 and the advertised applicability to nonlinear co-evolving networks rely on this reduction, the central claim is only conditional. I agree with the reader's conditional verdict; the concern is addressable by adding a reduction lemma with explicit hypotheses on F and C, and by modifying (11) or restricting a and trajectory sup-norms. The non-Lipschitz section (Theorem 3.5) is delegated to [24, Thm 3.11] and would also need a full proof, but that is secondary to the reduction gap.","tokens_in":16392,"tokens_out":16438,"duration_ms":202800,"concrete_test":"Set C≡1 and F_t(w,a,b)=w, which satisfies Lemma 1.1's global Lipschitz hypothesis. Then the induced interaction is K_t(a,r_t f,r_t g)=a e^t. Substituting into assumption (11) gives sup_{t∈[0,T],a∈R}|a|e^t=∞, so the reduction from (1) to (6) does not preserve the standing assumptions. This settles that Lemma 1.1's hypotheses alone do not imply (11), and identifies the missing hypothesis: restrict a to the bounded range of W (and, for general F, impose uniform boundedness of the flow on bounded trajectory sets) and re-run the proof of Proposition 2.1 under this weakened assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result Theorem 2.6 is conditional on K satisfying (11)-(13), but the paper's advertised class of co-evolving networks enters only through Lemma 1.1, which replaces weights by the flow map Φ. No statement proves that the induced K_t(a,r_t f,r_t g)=Φ_t[a,f,g]C(f(t),g(t)) satisfies (11)-(13) for any F beyond the linear-in-w example (7). This is not merely a missing verification. If F_t(w,a,b)=w (globally Lipschitz, so Lemma 1.1 applies), then Φ_t[a,f,g]=a e^t, and |K_t|=|a|e^t|C(f(t),g(t))|; the supremum over a∈R in (11) is infinite, so (11) fails as stated. Even if (11) is restricted to the bounded range of the Lipschitz graphon W (as it must be), the paper does not state or prove the needed modification. For F with linear growth in a,b, e.g. F=w+a, the flow contains ∫e^{t-s}|f(s)|ds and is unbounded over all f,g∈C([0,T]), again contradicting (11). Since (11) is used in Proposition 2.1 and in the Dobrushin estimates (Propositions 2.3-2.4) via the weighted fixed-point norm, the conditional theorem may hold, but its applicability to the original model (1) is unproved. The abstract's claim that the result applies to 'a large class of systems on co-evolving networks including non-linear weight dynamics' therefore goes beyond what is demonstrated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies mean-field limits for interacting particle systems on co-evolving networks. The adaptive weights are eliminated by solving the weight ODE through a flow map, yielding the nonlocal-in-time system (6) driven by a functional K. Under the boundedness and Lipschitz assumptions (11)-(13), the author constructs characteristic equations (21), proves several Dobrushin-type estimates, and states convergence of the empirical measures on path space (Theorem 2.6). Section 3 extends the setting to non-Lipschitz graphons satisfying (26) and states a second convergence theorem (Theorem 3.5). The paper also relates the mean-field limit to a continuum equation in the spirit of [40].","tokens_in":16749,"tokens_out":10770,"duration_ms":124538,"significance":"If completed, the result would be a useful extension of graphon mean-field theory to adaptive and co-evolving networks with memory. The characteristic-flow approach and the use of Wasserstein distances are standard and appropriate, and the strategy is clearly laid out; Example 1.3 is instructive. The paper also makes the right structural choice in isolating the regularity of K as the key hypothesis. However, the advertised scope currently exceeds the proved statements: the bridge from the co-evolving system (1) to the abstract system (6) is not verified against assumptions (11)-(13), and Theorem 3.5 is delegated to [24] rather than proved. Both issues are, in principle, fixable within the manuscript's scope, but they are load-bearing for the central claims.","major_comments":[{"comment":"The abstract claims that the result applies to 'a large class of systems on co-evolving networks including non-linear weight dynamics', but no statement verifies that the flow-map-induced functional K_t(a,r_t f,r_t g)=Φ_t[a,f,g]C(f(t),g(t)) from (1)/(5) satisfies (11)-(13). This is not a purely technical omission: for globally Lipschitz F_t(w,a,b)=w and bounded non-zero C, one has Φ_t[a,f,g]=a e^t, so sup_{a∈R}|K_t(a,r_t f,r_t g)|=∞ and (11) fails. Even for Example 1.3, K_0(a,f,g)=a C(f_0,g_0), which violates (11) unless the first argument is restricted to a bounded interval. Since (11) is used in Proposition 2.1 and in the Dobrushin estimates, Theorem 2.6 as stated does not cover the original co-evolving network system (1). The author should either verify (11)-(13) under explicit hypotheses on F and C, or state and prove a version of the theorem in which the first argument ranges over the bounded set of initial weights (which is natural because all evaluations use a=W(x,y)).","section":"Section 1.1, assumptions (11)-(13), Lemma 1.1"},{"comment":"Theorem 3.5 is advertised as a second main result, yet its proof is entirely delegated to [24, Theorem 3.11]. The same applies to Propositions 3.1 and 3.3, which are stated without proofs, and the weak equation (33) is asserted without derivation. A journal proof of a main theorem cannot consist of a reference to a different paper's argument, especially since the present setting has a nonlocal-in-time functional K and a measure-valued path-space formulation that are not identical to the static Kuramoto setting in [24]. At minimum, a full proof or a rigorous, self-contained proof sketch of the key estimates and of the convergence step is needed.","section":"Theorem 3.5 and Section 3.3"},{"comment":"The proof of Theorem 2.6 says that the convergence dist(µ^N,µ)→0 'is an immediate consequence of Proposition 2.4'. Proposition 2.4, however, is a statement about time-t marginals µ_t=(et,Id)#µ and does not by itself give convergence in P(C([0,T])×I). The missing step is to use the characteristic representation ϕ^{N,k}=Z^{x_{N,k}}_·(ϕ^{N,k,in},µ^N_0), Proposition 2.3 at t=T, and Lemma 2.2 to estimate the C([0,T])-distance between the two push-forwards; this argument should be written out. The one-sentence derivation of the weak equation (22) should also be expanded.","section":"Theorem 2.6, proof, and Proposition 2.4"}],"minor_comments":[{"comment":"In the chain of inequalities, several displayed lines omit absolute values: for instance the line after '≤ ζ_in − ζ~_in + ...' should estimate |Z^x_t(ζ_in)−Z^{˜x}_t(ζ~_in)|, not the signed difference.","section":"Lemma 2.2, proof"},{"comment":"In the proof of Proposition 3.4, the first displayed estimate after the definition of dist_I uses e^{2L_K T} in the first term; the surrounding derivation and Proposition 3.3 suggest the intended factor is e^{L_K t}, and the final estimate should be derived consistently.","section":"Proposition 3.4, proof"},{"comment":"The mode of convergence in Theorem 3.5 is not specified: the statement 'µ^{n,m,·} converges to µ^· as n,m→∞' should say whether this is convergence in probability, in L^1, or almost surely with respect to the random initial data.","section":"Theorem 3.5, Section 3.3"},{"comment":"There are notation inconsistencies between the subscripts (k−1)m+ℓ used for the particle states and the initial data ϕ^{N,kℓ,in}; these indices should be unified to avoid ambiguity.","section":"Equations (30)-(31)"},{"comment":"Theorem 1.5, which provides existence and uniqueness for the discrete system (6), is stated without proof ('we omit the details'); since the fixed-point argument is short and this statement underpins the later characteristic representation, a brief proof should be included.","section":"Theorem 1.5"}],"recommendation":"major_revision","confidential_remarks":"I see no circularity or problematic citation pattern: the only self-citation, [40], appears in the secondary continuum-limit discussion, not in the main derivation. My recommendation is based on the gap between the advertised scope and the statements actually proved: the verification of (11)-(13) for the co-evolving network model and the proof of Theorem 3.5 are both missing, but neither seems fatal if the author supplies the necessary arguments in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe core of this paper—a mean-field limit for the abstract non-local system (6) on path space—is new, and it is likely correct. The advertised application to co-evolving networks, however, is not established. The bridge from the weight dynamics (1) to the abstract system is never checked against the assumptions, and for natural choices of F it fails them.\n\nWhat the paper does well: it places the empirical measure on C([0,T]) × I, defines characteristics through a measure-dependent functional, and adapts the standard Dobrushin machinery to that setting. For Lipschitz graphons, Theorem 2.6 is plausible and the intermediate estimates (Props 2.1–2.4) are coherent, modulo minor typos. The abstract framework is a real contribution to mean-field limits with memory.\n\nNow the soft spots, in proportion.\n\nFirst and most important, the reduction from (1) to (6). Lemma 1.1 provides a flow map, but the induced K_t(a, r_t f, r_t g) = Φ_t[a,f,g] C(f(t),g(t)) is never shown to satisfy assumptions (11)–(13). It does not, as stated. If F(w)=w, then Φ = a e^t and K is unbounded in a over R, so (11) fails. If F(w,a,b)=w+a, the flow contains an integral of f, and K is unbounded in f over C([0,T]). The only example, (7), is linear in the weight. So the abstract's claim of \"non-linear weight dynamics\" goes beyond what is proved. This is fixable—restrict initial weights to the bounded graphon range and re-state (11) on that range—but it must be done.\n\nSecond, Theorem 3.5, advertised as a main result, is not proved; it is delegated to [24]. That is a summary of a proof, not a proof. It might be acceptable for a secondary extension, but it is a headline result here.\n\nThird, the proof of Theorem 2.6 is two sentences. Convergence in path-space distance requires Proposition 2.3, not just the time-marginal estimate in Proposition 2.4. It works, but the inference is not immediate as written.\n\nWho should read this? Researchers in mean-field limits, graphons, and adaptive networks. The abstract system is worth studying, and the path-space formulation is a genuine new tool. But the paper overstates its applicability. I would send it to peer review, with a request to either prove the reduction under a bounded-range condition or soften the claims.\n\nBest,","headline":"A genuinely new mean-field limit for a non-local path-space system, but the advertised co-evolving network application is not actually verified.","tokens_in":17237,"tokens_out":6863,"would_cite":true,"duration_ms":73602,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C22","35Q83"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that particle systems on co-evolving networks have a well-defined mean-field limit: the empirical trajectory measures converge to a deterministic path-space measure determined by a characteristic flow.","keywords":["mean-field limit","co-evolving networks","adaptive networks","graphons","non-local-in-time dynamics","Dobrushin estimate","Wasserstein distance","interacting particle systems"],"falsifier":"Take a bounded, Lipschitz $K_t$ and the graphon $W(x,y)=xy$, solve the characteristic equation (21) numerically, and simulate the discrete system (6) for increasing $N$ with the same initial empirical data; if the Wasserstein distance between the empirical path measure and the characteristic-flow measure does not tend to zero on $[0,T]$, the central claim of Theorem 2.6 is false.","tokens_in":16197,"feed_emoji":"🕸️","tokens_out":10260,"duration_ms":115162,"temperature":0.7,"pith_summary":"The paper proves that a large family of interacting particle systems whose interaction network co-evolves with the particles has a well-defined infinite-size limit. The key move is to integrate out the edge-weight dynamics: the coupling strength at each link becomes a functional of the whole past trajectories of the two particles, turning the model into a non-local-in-time system. Under a global Lipschitz condition on that functional, the empirical measures of the $N$-particle system converge in Wasserstein distance to a deterministic path-space measure built from a characteristic flow. The same conclusion holds for graph sequences whose limit is a non-Lipschitz graphon, provided the weights are defined by block averages. This matters because adaptive networks appear in neuroscience, opinion dynamics, and epidemics, and here their continuum description is shown to be memory-retaining rather than state-only.","feed_headline":"Co-evolving network weights still allow a mean-field limit","feed_subtitle":"A flow-map argument turns link dynamics into memory, giving a path-space continuum limit for particle systems.","key_machinery":"The carrying mechanism is the characteristic flow $Z^x_t(\\zeta^{\\mathrm{in}}, \\mu_0)$, defined as the unique solution of the non-local equation (21); it transports an initial particle state $\\zeta^{\\mathrm{in}}$ at continuum location $x$ through the mean-field interaction with the whole initial measure $\\mu_0$. Stability of this flow is quantified by the Dobrushin estimate of Proposition 2.4, which bounds the Wasserstein distance between the pushed-forward measures at time $t$ by a constant times the initial distance, with the constant growing like $e^{2 L_K t}$. This estimate, together with the flow-map reduction of Lemma 1.1 that replaces co-evolving weights by the trajectory functional $K_t$, is what converts the finite-particle system into a contraction argument on path space. A graphon $W\\colon I\\times I\\to\\mathbb{R}$ serves as the continuum model of the network: it is a symmetric function whose values give the coupling weights between continuum locations.","core_discovery":"On the paper's own terms, the discovery is that co-evolution of the network is not an obstruction to a mean-field limit: it merely turns the model into one with memory. More precisely, Theorem 2.6 states that if the interaction functional $K_t$ satisfies (11)-(13) and the initial graph is sampled from a symmetric Lipschitz graphon $W$, then $\\mu^N = \\frac1N \\sum_{k=1}^N \\delta_{(\\phi^{N,k}_\\cdot, x_{N,k})}$ satisfies $\\mathrm{dist}(\\mu^N, \\mu)\\to 0$, where $\\mu = (Z^\\cdot_\\cdot(\\cdot,\\mu_0), \\mathrm{Id})_\\#\\mu_0$ and $Z$ solves the characteristic equation (21). Theorem 3.5 extends this to non-Lipschitz graphons satisfying (26), using block-averaged weights. The limiting measure satisfies the weak equations (22) and (33), so the continuum limit is a measure on path space $C([0,T], \\mathbb{R}^d)\\times I$, and it is the unique fixed point of the characteristic flow.","pith_inferences":["An implicit consequence of the flow-map reduction is that the framework should cover nonlinear edge-weight rules, not just the linear example (7), whenever the solution map of the weight equation is Lipschitz in the endpoint histories.","The path-space form of the limit suggests that correlation functions of the adaptive couplings, which depend on two or more times, can be extracted directly from $\\mu$; state-space mean-field limits would lose this information.","Because Proposition 2.9 identifies the mean-field limit with a delta measure along the continuum solution for deterministic initial data, the two approximation routes, mean field and continuum limit, can be cross-checked numerically, with the Dobrushin estimate providing a quantitative error bound.","A natural testable extension is to let the sampled graphons converge only in $L^1$, as in the non-Lipschitz part, and to check whether the convergence rate is governed by the term $\\|W^N-W\\|_{L^1(I^2)}$ appearing in Proposition 3.3."],"forward_implications":["Every admissible co-evolving system has a unique finite-$N$ solution, and the empirical measure converges to the characteristic-flow limit with error controlled by the initial sampling error and the factor $e^{2L_K T}$.","The infinite-particle limit is a measure on continuous paths, so the memory built into the adaptive couplings survives the limit.","The limiting weak equation (22) provides a continuum evolution law for co-evolving networks, enabling study of stationary states and long-time behaviour without simulating all $N$ particles.","For non-Lipschitz network limits, the block-averaged weight construction still yields convergence of local empirical measures, extending the theory beyond smooth graphons.","Stability under perturbations holds: nearby initial empirical measures produce nearby mean-field limits on any finite time horizon, at an explicitly quantified rate."],"supporting_citations":[{"why":"It supplies the graphon-based mean-field analysis for the Kuramoto model on graphs that Section 2 follows for Lipschitz networks.","marker":"[9]"},{"why":"It is the classical origin of the Dobrushin estimate that Proposition 2.4 adapts to the non-local setting.","marker":"[14]"},{"why":"It provides the characteristic-flow method and fixed-point arguments used to prove existence and Lipschitz regularity of the limit.","marker":"[18]"},{"why":"It is the source of the non-Lipschitz graphon approximation and the proof scheme behind Theorem 3.5.","marker":"[24]"},{"why":"It gives the earlier continuum limit for adaptive networks whose relationship to the mean-field limit is discussed in Section 2.3.","marker":"[40]"},{"why":"It supplies the Wasserstein-1 distance and the Kantorovich-Rubinstein duality used in the stability estimates.","marker":"[41]"},{"why":"It is the prior mean-field and graph-limit treatment of time-varying weights that motivates the model class in (1).","marker":"[1]"}],"fun_headline_variants":["Co-evolving networks still yield a mean-field limit","Mean-field limit persists for co-evolving network weights","Path-space mean-field limit via graphon memory","Co-evolution becomes memory: mean-field limit holds","Adaptive networks: mean-field limit via memory"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the co-evolving interaction can be encoded by a bounded, globally Lipschitz functional $K_t$ after the edge-weight dynamics is integrated out; if a concrete weight rule produces a flow map that violates the Lipschitz bound (13), the convergence proof and both theorems fall through.","fun_headline_variants_meta":{"raw":{"variants":["Co-evolving networks still yield a mean-field limit","Mean-field limit persists for co-evolving network weights","Path-space mean-field limit via graphon memory","Co-evolution becomes memory: mean-field limit holds","Adaptive networks: mean-field limit via memory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1401,"prompt_tokens":893,"completion_tokens":508,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":450}},"tokens_in":509,"tokens_out":508,"duration_ms":5552,"temperature":1.0,"reasoning_tokens":450,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:55:19.749096+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a bounded, Lipschitz $K_t$ and the graphon $W(x,y)=xy$, solve the characteristic equation (21) numerically, and simulate the discrete system (6) for increasing $N$ with the same initial empirical data; if the Wasserstein distance between the empirical path measure and the characteristic-flow measure does not tend to zero on $[0,T]$, the central claim of Theorem 2.6 is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the classical origin of the Dobrushin estimate that Proposition 2.4 adapts to the non-local setting."},{"cited_title":"On the dynamics of large particle systems in the mean field limit","cited_arxiv_id":null,"evidence_quote":"It provides the characteristic-flow method and fixed-point arguments used to prove existence and Lipschitz regularity of the limit."},{"cited_title":"Medvedev","cited_arxiv_id":null,"evidence_quote":"It is the source of the non-Lipschitz graphon approximation and the proof scheme behind Theorem 3.5."},{"cited_title":"Continuum limit for interacting systems on adaptive networks","cited_arxiv_id":null,"evidence_quote":"It gives the earlier continuum limit for adaptive networks whose relationship to the mean-field limit is discussed in Section 2.3."},{"cited_title":"Springer-Verlag, Berlin,","cited_arxiv_id":null,"evidence_quote":"It supplies the Wasserstein-1 distance and the Kantorovich-Rubinstein duality used in the stability estimates."},{"cited_title":"Mean-field and graph limits for collective dynamics models with time-varying weights","cited_arxiv_id":null,"evidence_quote":"It is the prior mean-field and graph-limit treatment of time-varying weights that motivates the model class in (1)."}],"review_version":1}