{"id":"3c417006-e9da-4af9-89db-7e5ae999b6af","arxiv_id":"2607.21110","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Non-exchangeable particles with adaptive weights converge to a Vlasov-type equation, with the limit described by vector-valued dynamic extended graphons.","lead":"Large swarms whose members both move and adjust the strength of their connections can be approximated by a continuous equation. This paper proves such a mean-field limit for adaptive, non-symmetric networks, including sparse ones.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's kernel assumptions are too weak: the existence and stability lemmas used in its proof require K1,K2∈L1 and divK1∈L1, which W^{1,∞} does not imply.","rationale":"The reader identified the kernel integrability gap as the weakest assumption, and the manuscript text confirms it. The proof of the main theorem invokes Proposition 2 to produce the limit f and Lemma 11 to compare f_N with f. Proposition 2 explicitly requires K_i∈L1 and divK1∈L1; Lemma 11 requires K1∈L∞∩W^{1,1}, divK1∈L∞, and K2∈L1. The estimates (36)–(40) that make these lemmas work are all L1-kernel estimates via Young's inequality. Assumption (3) only gives W^{1,∞} kernels, which do not decay and need not have finite L1 norm. The paper contains no truncation or approximation argument that would allow non-L1 kernels to be handled. Thus the central existence-uniqueness-stability chain is broken for the stated assumptions. This is a precise, load-bearing concern. It is also repairable: adding the missing L1 hypotheses (or a controlled truncation argument) would restore the proof, which is why the proper verdict remains CONDITIONAL rather than REJECT. No ad hominem or theatrical judgment is needed; the issue is purely a hypothesis/lemma mismatch. The reader's weakest_assumption is the same, so agreement is full and the verdict need not change.","tokens_in":46089,"tokens_out":5188,"duration_ms":51957,"concrete_test":"Construct admissible data under (3) with K1(x)=e1 (nonzero constant vector) and K2(x)=1. For these kernels, recompute the estimates in Proposition 2 and Lemma 11. In particular, (36) gives ∥V_i[f]∥ ≤ ∥K_i∥_{L1}∥f∥, (40) gives ∥divV1[f]∥ ≤ ∥divK1∥_{L1}(·), and the contraction constants in (44)–(46) contain ∥K1∥_{L1}, ∥divK1∥_{L1}, and ∥K2∥_{L1}; all are infinite for these kernels. Then check whether any alternative part of the paper supplies finite bounds or an explicit truncation argument for such non-L1 kernels. If not, the proof of Theorem 1 fails for this admissible example, confirming the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem is stated under assumption (3), which only requires K1∈W^{1,∞}(R^d;R^d), K2∈W^{1,∞}(R^d;R). However, the proof of Theorem 1 relies directly on Proposition 2 and Lemma 11, whose hypotheses are strictly stronger. In §3.2 the weak solution definition starts 'Let K1,K2∈L1(R^d)', and Proposition 2 assumes K_i∈L1(R^d) and divK1∈L1(R^d). Lemma 10 assumes K1∈W^{1,1}(R^d), divK1∈L∞(R^d), K2∈L1(R^d); Lemma 11 assumes K1∈L∞∩W^{1,1}, divK1∈L∞, K2∈L1. The estimates (36)–(40) that underpin both the solvability and the stability arguments all scale with ∥K_i∥_{L1} or ∥divK1∥_{L1}. For a generic W^{1,∞} kernel these norms are infinite — e.g. any nonzero constant kernel is in W^{1,∞} but not in L1. No truncation, periodization, or approximation argument is provided to bridge this gap. Consequently, the existence of the limit object f and the stability estimate comparing f_N with f are not established for the admissible kernel class in Theorem 1. This is an internal hypothesis mismatch, not a disagreement with an external consensus: the theorem as stated is unproven, though it is plausibly repairable by strengthening (3) to include L1-integrability conditions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces vector-valued dynamic extended graphons to derive a mean-field limit for a non-exchangeable, non-conservative particle system with adaptive weights, equation (1). The main result, Theorem 1, asserts that under assumptions (2)–(8) there exists a weak solution f in L∞((0,t*); W((W^{1,1}∩W^{1,∞})(R^d))) to the limiting equation (9), and that the weighted empirical measure converges in expectation, in the flat metric, to the double graphon integral of f, up to a subsequence. The proof strategy combines a propagation-of-independence result for an auxiliary particle system, a compactness theory for vector-valued extended graphons based on tree-indexed observables, and stability estimates for the limiting PDE.","tokens_in":46497,"tokens_out":5308,"duration_ms":55770,"significance":"If established, the result would be a valuable contribution: it gives a mean-field limit for adaptive-weight systems under fairly mild conditions on the initial connection matrix, covering sparse graphs, and it substantially extends the graphon framework of [14] to vector-valued measures and non-conservative dynamics. The introduction of tree-indexed observables in the vector-valued setting and the propagation-of-independence argument are genuine technical contributions, and the paper is careful to formulate explicit convergence rates in intermediate steps. However, the central theorem is not proven for the kernel class stated in assumption (3), because the existence and stability results on which the proof relies require stronger L1-type integrability of the kernels. The argument is plausibly repairable by strengthening (3), but as it stands the main claim is not established.","major_comments":[{"comment":"Theorem 1 is stated under assumption (3), which only requires K1 ∈ W^{1,∞}(R^d;R^d) and K2 ∈ W^{1,∞}(R^d;R). However, the proof relies on Proposition 2 and Lemma 11, whose hypotheses are strictly stronger. Proposition 2 assumes Ki ∈ L1(R^d) and divK1 ∈ L1(R^d); Lemma 11 assumes K1 ∈ L∞∩W^{1,1}, divK1 ∈ L∞, and K2 ∈ L1. The key estimates (36)–(40) are all expressed in terms of ∥Ki∥_{L1} and ∥divK1∥_{L1}. W^{1,∞} does not imply L1 on R^d — for example, a nonzero constant kernel is in W^{1,∞} but not in L1. No truncation, periodization, or approximation argument is supplied to bridge this gap. Consequently, existence of the limit f and the stability comparison between f_N and f are not established for the admissible kernel class. This is an internal hypothesis mismatch, not a disagreement with an external consensus.","section":"Theorem 1 and §3.2–3.4"},{"comment":"The main stability estimate, Lemma 11, depends essentially on Lemma 9, which is imported verbatim from the companion preprint [13, Lemma 6] with only the comment that the proof is unaffected by the changed hierarchy indexing. Since [13] is a companion, not-yet-published preprint by the same research group, this creates a serious self-containedness gap: the reader cannot verify the central stability input. The same remark applies to Lemma 12, taken from [13, Lemma 3]. The paper should either provide full proofs, state these as assumptions with an accessible reference, or justify why the companion result can be used in this context.","section":"§3.4, Lemma 9 and Lemma 11"},{"comment":"In the final step of Theorem 1, the author asserts that f_N belongs to L∞((0,t*); W(H^1(R^d))) by Lemma 10, and then applies Lemma 11 to f_N and f. But Lemma 10 is proved under hypotheses K1∈W^{1,1}, divK1∈L∞, K2∈L1 (see the statement of Lemma 10 and the estimates in its proof). These hypotheses are not implied by assumption (3). Thus the regularity needed to apply the stability estimate is unavailable for the stated kernel class. This is not a minor technicality: the L2-stability comparison in Lemma 11 uses the H1 estimate of Lemma 10 in an essential way, e.g. through estimate (68).","section":"Proof of Theorem 1, application of Lemma 10 and Lemma 11"},{"comment":"The definition of weak solution at the beginning of §3.2 starts with 'Let K1,K2 ∈ L1(R^d), ν≥0', while the main theorem assumes only W^{1,∞}. This internal inconsistency should be resolved in the main statement. If the intended theorem is for integrable kernels, assumption (3) should be strengthened accordingly; if W^{1,∞} is the intended class, a genuinely new argument avoiding the L1-norm estimates (36)–(40) is required.","section":"§3.2, weak solution definition"}],"minor_comments":[{"comment":"Typo: 'mean-filed limit' should be 'mean-field limit'.","section":"Abstract"},{"comment":"The introduction states that the paper establishes 'well-posedness' of the limiting problem (9), but Proposition 2 only proves existence (via fixed point), not uniqueness. Lemma 11 gives uniqueness only for the integrated observable ∫∫ f, not for the full graphon f. The wording should be adjusted.","section":"§1, paragraph after Theorem 1"},{"comment":"The display for g_ij under independence is notationally garbled; the density f^{X_i^0} appears without its argument and the expression '(7) reduces to sup ... < ∞' omits the factor E w_ji^0, which is present in the preceding formula. This should be cleaned up.","section":"Remark 1"},{"comment":"In the estimate of E|Σ_i G_ik|^2, the transition from the conditional independence of off-diagonal terms to the final bound is compressed. The reader must reconstruct that the diagonal term is handled separately and that terms with k=l are included in the sum of squares. A short clarification would improve readability.","section":"§2, Lemma 2 proof"},{"comment":"In the proof of Theorem 1, after (71) the phrase 'there exists some λ>0 small enough' should specify the dependence of λ on the initial data norms, because Lemma 11 uses λ in the logarithmic term. This is a minor clarity issue.","section":"§3.5"}],"recommendation":"major_revision","confidential_remarks":"The paper's main idea is promising and the technical apparatus is substantial, but the mismatch between the theorem's kernel assumptions and the assumptions used in the existence and stability lemmas is load-bearing. I would recommend major revision rather than rejection because the gap appears fixable by adding L1-integrability hypotheses to (3) or by developing new estimates for non-integrable kernels. The paper also leans heavily on the companion preprint [13] for two central lemmas; the editors may wish to consider whether such reliance is acceptable and whether [13] should be published or included in the paper. I did not find evidence of fabricated entities or circular reasoning: the limit equation is not assumed and there are no fitted parameters."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things. First, this paper does something genuinely new: it treats mean-field limits for non-exchangeable particle systems with adaptive weights under sparse-graph assumptions, which was open. The vector-valued dynamic extended graphon construction is a real extension of [14], and the weighted empirical measure is the right object to capture network activity. Second, the main theorem as stated is not proven. The proof leans on Proposition 2 and Lemma 11, which require K1, K2 ∈ L1 and div K1 ∈ L1, while assumption (3) only gives K1, K2 ∈ W^{1,∞}. That is a load-bearing mismatch. The estimates in (36)–(40) scale with L1 norms, and a nonzero constant kernel is in W^{1,∞} but not L1. No truncation or approximation argument is supplied. So existence of the limiting PDE and the stability bound are not established for the stated kernel class.\n\nWhat is good: the overall architecture is coherent, the tree observables and Vlasov hierarchy are adapted carefully, and the local limit theorem (Theorem 3) is solid. The reliance on the companion preprint [13] for several lemmas is mildly uncomfortable — some results are imported without showing the promised adaptation — but that is secondary to the kernel gap.\n\nThe gap looks repairable: strengthen (3) to include L1 integrability of K1, K2 and div K1, or add a truncation argument. I would bet the proofs go through with only cosmetic changes. As written, my advice to an editor would be to invite a revision, not desk-reject. This deserves a serious referee: the problem is important, the framework is new, and the flaw is identifiable and fixable. I would not cite the theorem as stated yet; I would cite the graphon machinery with caution.","headline":"Genuinely new extension of the Jabin–Poyato–Soler program to adaptive weights, but the main theorem's kernel assumptions don't match the proof; the gap looks repairable.","tokens_in":46943,"tokens_out":1882,"would_cite":false,"duration_ms":20944,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q83","46G10","35Q70","35R02","35Q49","35R06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the weighted empirical measure of a non-exchangeable, adaptively weighted particle system converges to a limit density f that solves a single nonlocal, non-conservative transport equation, and that limits of sparse gr","keywords":["mean-field limit","adaptive weights","non-exchangeable particle systems","extended graphons","vector-valued measures","non-conservative dynamics","sparse graphs","Vlasov hierarchy"],"falsifier":"Take d=1, K1≡1, K2≡0, A=0. Then K1 belongs to W^{1,∞}(R) but not to L1(R), so assumption (3) holds while the proof's L1 hypotheses fail. The limiting PDE becomes ∂t f + M(t,η)∂x f = 0 with M(t,η)=∫∫ f(t,y,η,dζ)dy conserved in time, giving an explicit solution. Check whether the weighted empirical measure converges to that explicit solution: if convergence fails for this kernel, the theorem as stated is false; if it holds, the gap is in the proof rather than the claim.","tokens_in":45960,"feed_emoji":"🕸️","tokens_out":6016,"duration_ms":67105,"temperature":0.7,"pith_summary":"This paper tries to establish a mean-field limit for a large system of non-exchangeable agents whose interaction weights evolve together with their states and whose total influence is not conserved. It introduces a four-variable object f(t,x,ξ,η), a vector-valued dynamic extended graphon, that encodes how much agent ξ's state distribution is weighted toward agent η at time t. The main theorem asserts that the double weighted empirical measure converges, in expectation and flat metric, to the twice-integrated graphon, where f solves a single nonlocal transport equation with a linear growth term. If correct, this means the limiting description tracks network-weighted activity rather than raw population counts, so small, hyper-connected subgroups can dominate the collective behavior. This extends the sparse-graph mean-field regime from conservative to non-conservative adaptive dynamics.","feed_headline":"Adaptive-weight particle swarms converge to a single limit equation","feed_subtitle":"A network-weighted density, not just population counts, maps the many-agent system to one nonlocal PDE.","key_machinery":"The load-bearing object is the vector-valued extended graphon space W(X*) = L∞ξ(Mη(X*)) ∩ L∞η(Mξ(X*)), here with X*=(L1∩L∞)(R^d); it lets f be a measure in the label variables while remaining a function in the state variable x. The proof machinery consists of (i) an auxiliary independent system that propagates independence, (ii) piecewise-constant graphon approximations fN built from the auxiliary measures, (iii) observables τ(T,f) indexed by rooted directed trees, which satisfy a non-exchangeable Vlasov hierarchy, and (iv) a stability estimate comparing two solutions through a logarithmic bound on the tree observables. The tree observables reduce the mean-field convergence to a compactness-","core_discovery":"The central claim is that the entire adaptive-weight empirical measure has a deterministic continuum limit: there is a nonnegative f in L∞((0,t*); W((W^{1,1}∩W^{1,∞})(R^d))) that weakly solves ∂t f + div(f V1[f]) = f(A − V2[f]), with Vk[f](t,x,η)=∫ Kk(x−y)∫0^1 f(t,y,η,dζ)dy, and the random double sum (1/N)Σi,j wji(t,x)δ_Xi(t,x) converges in expectation with respect to the flat metric to ∫∫ f(t,·,ξ,dη)dξ, up to a subsequence. The paper's interpretation is that f is a network-weighted joint density: integrating out the label variables gives the global state profile weighted by interaction strengths, not the ordinary population density.","pith_inferences":["The most consequential open repair is the L1 gap: if the kernels are truncated to compact support and the convergence constants are shown independent of the truncation, the theorem would hold under its stated W^{1,∞} hypotheses.","The same graphon formalism should yield quantitative propagation of chaos for higher-order correlations, since the tree observables solve a closed hierarchy.","For opinion dynamics, the limit suggests a testable prediction: a tiny but densely connected minority shifts the aggregate outcome in proportion to weighted activity, so interventions should target connectivity-weighted influence, not headcount.","A numerical experiment with a constant kernel K1, which satisfies W^{1,∞} but not L1, could separate failure of the theorem from failure of the proof route."],"forward_implications":["The weighted empirical measure, not the plain one, is the right macroscopic variable: convergence identifies network-activity clusters as the drivers of the limiting behavior.","The limit equation is non-conservative: the term f(A − V2[f]) lets total weighted influence grow with time, so the continuum description is not a standard conservative Vlasov equation.","Sparse initial graphs are covered: only row and column sums of the initial weights and the conditional densities gij are controlled, not the total number of connections.","The tree observables give access to correlation margins of the limit beyond the first marginal, through a closed hierarchy.","The initial interaction matrix is encoded in f through the initial data, so different network topologies can produce genuinely different macroscopic limits."],"fun_headline_variants":["Adaptive-weight swarms converge to one nonlocal PDE","Non-exchangeable particles: mean-field limit via graphons","Sparse graphs still allow a deterministic swarm limit","Graphons yield a unified PDE for adaptive swarms","Mean-field limit for adaptive weights on sparse graphs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The theorem assumes only bounded Lipschitz kernels K1 and K2, but the existence and stability proofs require K1, div K1, and K2 to be integrable on all of R^d; no truncation or approximation step is supplied to bridge that gap.","fun_headline_variants_meta":{"raw":{"variants":["Adaptive-weight swarms converge to one nonlocal PDE","Non-exchangeable particles: mean-field limit via graphons","Sparse graphs still allow a deterministic swarm limit","Graphons yield a unified PDE for adaptive swarms","Mean-field limit for adaptive weights on sparse graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000425,"raw_usage":{"total_tokens":1946,"prompt_tokens":603,"completion_tokens":1343,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":347,"completion_tokens_details":{"reasoning_tokens":1267}},"tokens_in":347,"tokens_out":1343,"duration_ms":12760,"temperature":1.0,"reasoning_tokens":1267,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T08:27:01.538105+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take d=1, K1≡1, K2≡0, A=0. Then K1 belongs to W^{1,∞}(R) but not to L1(R), so assumption (3) holds while the proof's L1 hypotheses fail. The limiting PDE becomes ∂t f + M(t,η)∂x f = 0 with M(t,η)=∫∫ f(t,y,η,dζ)dy conserved in time, giving an explicit solution. Check whether the weighted empirical measure converges to that explicit solution: if convergence fails for this kernel, the theorem as stated is false; if it holds, the gap is in the proof rather than the claim.","supporting_citations":[],"review_version":1}