{"id":"1999d1ac-f432-41f3-aeb9-ea02716efaca","arxiv_id":"2502.05156","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For interacting jump processes whose single-particle transitions form an acyclic graph, the hydrodynamic limit of the neighborhood empirical measure equals the solution of a finite ODE system, the Markov local-field forward equations.","lead":"This paper proves a mathematical theorem: for a large class of interacting random processes on tree-like sparse networks, the distribution of a particle's local neighborhood over time is exactly described by a finite system of ordinary differential equations. This gives a tractable alternative to mean-field approximations for models with nonlinear interactions, such as seizure spread in brain networks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The time-marginal 2-MRF proof contains an unjustified conditional-independence implication: after (5.20) the text asserts that conditioning on the post-jump boundary state alone retains independence, which does not follow from the stated pre- and post-jump conditioning; Theorem 4.7 and hence…","rationale":"The reader's weakest_assumption identifies precisely the unjustified implication after (5.20) in the proof of Theorem 4.7, and I agree that this is the most load-bearing concern. The proof of Theorem 2.8 depends on Theorem 4.7 to equate the Markov local-field rates with the rates of the Markovian projection of the local-field equations; if the time-marginal 2-MRF property is not established, the ODE (2.11)-(2.12) is not shown to describe the true space-time marginals of the interacting particle system. The specific logical gap is real: conditional independence of A and B given a pair (C,D) does not imply conditional independence given a function of (C,A,B), and the paper supplies no argument that the particular function f_d2A in (5.20) has the needed structure. A simple probability-space counterexample (A,B independent, D=A XOR B) shows the inference rule is invalid in general. The companion issue in Lemma 5.12, where the stopped state is claimed to encode the full stopped trajectory, reinforces the concern, though the post-(5.20) step is the one explicitly identified by the reader. The paper has other strengths: the ODE well-posedness proof in Section 7 is explicit, the examples are plausible, and the overall framework is promising. But the central theorem is not yet convincingly proven as written. Since the reader already returned CONDITIONAL rather than ACCEPT, my stress-test does not change that verdict; it confirms that the conditional verdict is appropriate pending a fix or a justification of the missing implication.","tokens_in":28748,"tokens_out":11865,"duration_ms":109254,"concrete_test":"Independently re-derive the equality after (5.20) from the explicit transition structure (2.2) and the conditional independence supplied by Lemma 5.12, without invoking the asserted implication. To settle whether the implication holds for the paper's class, compute both sides of E[X_A^-(tau-bar) | X_d2A(tau-bar), X_B^-(tau-bar)] = E[X_A^-(tau-bar) | X_d2A(tau-bar)] for a minimal tree satisfying Assumptions A-D, for example the entrenched majority voter model of Section 3.2.2 on the tree with vertices 0-1-11-111 and 1-12, taking A={12}, B={111}, so that d2A={0,1,11}, using the exact Markov jump rates. If the two conditional expectations differ, the implication after (5.20) is false within the paper's own framework and Theorem 4.7 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 2.8 reduces the hydrodynamic limit to the ODE (2.11)-(2.12) by showing that the Markov local-field rates gamma in (4.7) equal the MLFE rates rho-hat in (4.6) (proof of Theorem 2.8, Section 4.4). That equality is derived from the time-marginal 2-MRF property of Theorem 4.7. In the proof of Theorem 4.7 (Section 5.2), after Lemma 5.12 gives X_A^-(tau-bar) perpendicular X_B^-(tau-bar) given (X_d2A^-(tau-bar), X_d2A(tau-bar)), and after (5.20) expresses X_d2A(tau-bar) as f_d2A(X_d2A^-(tau-bar), X_A^-(tau-bar), X_B^-(tau-bar)), the text says 'This in turn implies' that E[X_A^-(tau-bar) | X_d2A(tau-bar), X_B^-(tau-bar)] = E[X_A^-(tau-bar) | X_d2A(tau-bar)]. This inference is not valid in general: if A,B are independent Bernoulli(1/2), C is identically 0, and D = A XOR B, then A perpendicular B given (C,D) but A and B are dependent given D. No additional property of f_d2A (such as conditional independence of X_d2A^-(tau-bar) and X_d2A(tau-bar) given A,B, or injectivity of the map) is stated or proved. Moreover, Lemma 5.12 itself coarsens the full stopped trajectory to the stopped state at time t, asserting that the two conditioning sigma-algebras coincide; this is also not generally true for a single-jump cadlag path unless the jump time is a function of the pre- and post-jump states. Thus the time-marginal 2-MRF property is not established. Since Theorem 2.8 relies on this property to replace trajectory-conditioned rates by state-conditioned rates, the main ODE description is not rigorously connected to the original dynamics.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Markov interacting pure jump processes on unimodular Galton-Watson trees, assuming a finite state space and an acyclic directed transition graph for single-particle transitions (Assumption D). The main result, Theorem 2.8, claims that the time-t law of the root together with its neighbors is exactly the solution of a finite coupled ODE system, equations (2.11)-(2.12). The proof combines a local-field equation characterization from a companion paper, a new time-marginal second-order Markov random field property, a Markovian projection theorem for pure jump processes, and a Lipschitz well-posedness analysis for the ODE. The paper also derives a hydrodynamic-limit corollary on configuration-model graphs and illustrates the approximation with simulations for a seizure-spread model, an entrenched majority voter model, and a thresholded Hawkes model.","tokens_in":1768,"tokens_out":3639,"duration_ms":231023,"significance":"If the main theorem is correct, this is a significant advance: it gives a finite-dimensional autonomous ODE description of the hydrodynamic limit for a class of interacting jump processes with nonlinear, non-pairwise-linear rates, going beyond the SIR/SIER results that rely on pairwise linear structure. The ODE is derived from the given rates without fitted parameters. The claimed time-marginal 2-MRF property and the general Markovian projection result are potentially of independent interest. The paper is well organized and the simulation studies are a useful sanity check. However, the proof of the time-marginal 2-MRF property has a load-bearing gap, and the main theorem depends on results from an unpublished companion paper, so the central claim is not fully established in the present manuscript.","major_comments":[{"comment":"The assertion 'This in turn implies' is not justified. Lemma 5.12 gives conditional independence of X_A^-(tau-bar) and X_B^-(tau-bar) given both X_{d2A}^-(tau-bar) and X_{d2A}(tau-bar), but equation (5.20) only defines X_{d2A}(tau-bar) as a function of X_{d2A}^-(tau-bar), X_A^-(tau-bar), and X_B^-(tau-bar). Conditional independence given a pair does not imply conditional independence given one component when that component is a function of the other variables. For example, with independent Bernoulli variables C, A, B and D = (A XOR C, B XOR C), A is independent of B given (C,D) but not given D alone. This gap affects the equality of the rates gamma and rho-hat in the proof of Theorem 2.8, and hence Theorem 4.7 is not established by the given argument.","section":"Section 5.2, after Eq. (5.20)"},{"comment":"The proof states that conditioning on the stopped augmented process at time t is the same as conditioning on the full stopped trajectory because there are no jumps on the double boundary before tau-bar. For t < tau-bar this is correct, but at t = tau-bar the stopped trajectory includes the time of the first jump, which is not a measurable function of the pre- and post-jump states alone. Thus the sigma-algebra generated by Y_{d2A}^{tau-bar}(tau-bar) is generally smaller than that generated by Y_{d2A}^{tau-bar}[tau-bar], and conditional independence does not automatically transfer from the larger to the smaller sigma-algebra. An additional argument is required at the jump time.","section":"Section 5.2, Lemma 5.12"},{"comment":"In the proof of Proposition 2.6, the denominator D in the definition of Psi-bar has indicators {b_empty = a1, b1 = a0}, while delta(q) immediately below is defined using {c_empty = a0, c1 = a1}. Unless a0 = a1, the inequalities delta(q) <= D <= d_max delta(q) do not follow. The Lipschitz bound therefore is not derived as written. Please correct the definition of delta(q) or the indicators in (7.1) and redo the argument.","section":"Section 7, Eq. (7.1)"},{"comment":"The proof of Theorem 2.8 relies on Theorem 4.3 and Proposition 4.10, which are deferred to the companion paper [22], listed as 'in preparation.' Theorem 4.8 also invokes well-posedness 'from considerations equivalent to Proposition 4.10.' Hence the main theorem is not self-contained and cannot be fully verified from the present manuscript. The authors should provide the missing proofs or clarify exactly which results are assumed and make the companion paper available.","section":"Sections 4.1.1 and 4.4"}],"minor_comments":[{"comment":"In the definition of rho-hat for v not equal to the root, the conditioning is written as Xhat_empty(t-) = y_v and Xhat_1(t-) = y_empty, which appears swapped relative to the conditional expectation in (4.9), where the conditioning is on X_empty[t] = y_empty[t] and X_v[t] = y_v[t]. Please clarify the intended convention or correct the typo.","section":"Section 4.4, Eq. (4.6)"},{"comment":"In the seizure-propagation rate (3.3), the rate for state y(s-) = 1 contains a division by d - (1 + sum alpha^-_w 1{x_w=0}); please clarify that the denominator is positive under the intended graph and parameter assumptions, or state how the zero case is handled.","section":"Section 3.2.1"},{"comment":"The exposition of the Lipschitz proof would benefit from explicitly defining delta(q) with the same ordering of the root and neighbor states used in the denominator D, and from stating upfront that the permutation symmetry in Remark 2.9 is not being used to swap the root and a neighbor.","section":"Section 7"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the unproven conditional-independence implication in Section 5.2; this is a proof gap rather than a novelty or scope issue. The authors should be encouraged to either complete the argument or state the additional structural condition under which independence after Eq. (5.20) is retained. The reliance on the unpublished companion paper [22] is also a serious verification issue. The paper is otherwise well written and the proposed ODE description is potentially valuable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plain English: for interacting jump processes on Galton-Watson trees with acyclic transition graphs, the paper claims that the time-marginal law of a root neighborhood solves a finite ODE system. That is a genuine generalization of prior work, which only covered pairwise-linear SIR/SEIR. The ODE is explicit, and the well-posedness proof in Section 7 is self-contained and fine. The Markovian projection section is also a solid contribution, providing a general mimicking result for pure jump processes.\n\nThe main soft spot is the proof of Theorem 4.7, the time-marginal 2-MRF property. The step after (5.20) asserts that conditional independence of X_A^- and X_B^- given (X_∂2A^-, X_∂2A) implies independence given just X_∂2A. That is not true in general, and the stress-test counterexample with XOR is exactly right. The text does not provide any additional structure (like injectivity or a Markov property of the boundary) to justify it. This is load-bearing: Theorem 2.8 depends on replacing trajectory-conditioned rates by state-conditioned rates. Also, Lemma 5.12 claims that conditioning on the stopped trajectory of the double boundary is the same as conditioning on its time-t value because the boundary has no jumps before τ; that is only true if the jump time is a deterministic function of the pre- and post-jump states, which is not argued. So the time-marginal 2-MRF is not established as written.\n\nAnother issue is the heavy reliance on the companion paper [22], still listed as in preparation, for both the local-field equation (Theorem 4.3) and well-posedness of the Markov local-field equation (Proposition 4.10). That makes it hard for a reader to verify the foundations. The simulations are illustrative only, with no error bars or code, but that is minor and clearly labeled.\n\nIf the gap in Section 5.2 is closed, the paper would be a strong contribution. As it stands, the main theorem is not rigorously connected to the original dynamics. Still, the idea and the structure are worth refereeing; a good referee could help fix the proof. I would send it to a serious journal with the expectation of major revision.","headline":"The paper genuinely extends the Markov local-field ODE reduction to acyclic transition graphs, but the proof of the key time-marginal 2-MRF property has a real gap that needs fixing.","tokens_in":29718,"tokens_out":2067,"would_cite":false,"duration_ms":19554,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60J74","60J27"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that for interacting jump processes on sparse graphs with acyclic single-particle transitions, the law of a typical vertex neighborhood at every time is exactly the solution of a finite coupled ODE system.","keywords":["interacting particle systems","hydrodynamic limits","local-field equations","Markovian projection","Markov random fields","sparse random graphs","Galton-Watson trees","jump processes"],"falsifier":"Construct a concrete finite example satisfying Assumptions A through D in which $X_A(\\bar\\tau)$ and $X_B(\\bar\\tau)$ are conditionally independent given both $(X_{\\partial^2 A}(\\bar\\tau-), X_{\\partial^2 A}(\\bar\\tau))$ but not given $X_{\\partial^2 A}(\\bar\\tau)$ alone; such a counterexample would invalidate the key implication around equation (5.20) and with it the claimed time-marginal 2-MRF property.","tokens_in":28479,"feed_emoji":"🧠","tokens_out":6136,"duration_ms":56930,"temperature":0.7,"pith_summary":"This paper aims to make the hydrodynamic limit of interacting jump processes on sparse random graphs tractable. It claims that when the single-particle transition graph is acyclic, the time-marginal law of the root neighborhood on a unimodular Galton-Watson tree—the local limit of configuration-model networks—is governed exactly by a finite coupled system of ordinary differential equations, the Kolmogorov forward equations of a Markov local-field process. The payoff is that one can compute the macroscopic empirical distribution of vertex states without simulating the full non-Markovian dynamics, even when interactions are nonlinear and not pairwise additive. The proof combines a local-field equation characterization, a Markovian projection that matches time marginals, and a newly established time-marginal second-order Markov random field property. If the theorem is right, a class of previously intractable spreading processes reduces to an ODE initial-value problem.","feed_headline":"A finite ODE predicts spreading dynamics on sparse networks","feed_subtitle":"Under an acyclicity condition, root-neighborhood marginals obey a finite system of ODEs.","key_machinery":"The central object is the Markov local-field equation (MLFE), an SDE whose jump rates are conditional expectations of the local-field rates given only the present root-neighborhood state rather than the full trajectory. Its forward Kolmogorov equations become the ODE system (2.11)–(2.12). The passage from the non-Markovian local-field equation to the MLFE rests on two ingredients: a Markovian projection theorem for pure jump processes, which constructs a Markov process with the same time marginals, and a time-marginal second-order Markov random field property, which says that conditional on the present state of the double boundary of a set, the inside and outside of that set are independent. The acyclic transition graph assumption is what allows the authors to derive this marginal 2-MRF property from a trajectorial version up to stopping times.","core_discovery":"Under Assumptions A, B, C and D, the time marginals of the root neighborhood are characterized by the Markov local-field equation, whose jump rates depend only on the current configuration and its one-time law. The paper proves that the corresponding forward Kolmogorov ODE system (2.11)–(2.12) has a unique solution, and that this solution equals $\\mathrm{L}(X_{V_1^\\theta}(t))$ for every $t \\ge 0$. Consequently, for configuration-model graphs converging to a unimodular Galton-Watson tree, the empirical distribution of vertex states and root-neighborhood configurations converges in probability to this ODE solution. The key structural assumption is that the directed transition graph of a single particle is acyclic, which forces the double boundary of any set to jump only finitely many times; this is what lets the authors pass from a trajectorial 2-MRF property to a time-marginal 2-MRF property, the ingredient that makes the Markovian projection autonomous.","pith_inferences":["Beyond the paper: the time-marginal 2-MRF mechanism, if made fully rigorous, would likely extend to any finite-state process in which the double boundary of every set makes only finitely many jumps, not necessarily under a globally acyclic transition graph.","Beyond the paper: the Markovian projection theorem stated for general bounded trajectory-dependent jump rates is a standalone tool that could be applied to other history-dependent intensity models outside the local-field setting.","Beyond the paper: the ODE approximation algorithm of Section 3.1 invites a quantitative finite-size error analysis, since the simulations in the paper suggest accuracy at modest graph sizes but provide no rigorous convergence-rate guarantee."],"forward_implications":["For configuration-model graphs with a fixed finite-support degree sequence, the empirical distribution of vertex states converges in probability to the ODE solution, so the hydrodynamic limit is explicitly computable.","The result applies to nonlinear, non-pairwise-linear interactions: the seizure-propagation model, the entrenched majority voter model, and thresholded multivariate Hawkes processes all satisfy the acyclicity assumption and are covered.","The trajectory laws of the local-field and Markov local-field processes can differ; only the time marginals are asserted to coincide.","On a $\\kappa$-regular tree, the ODE reduces from $m^{\\kappa+1}$ to $m\\binom{\\kappa+m-1}{m-1}$ equations, giving a dimension reduction.","Finite graphs can be approximated by solving the ODE with the empirical degree distribution substituted for $\\theta$, as in Section 3.1."],"supporting_citations":[{"why":"Supplies well-posedness of the interacting jump SDE and the hydrodynamic-limit convergence of empirical measures to the root law on the limit tree.","marker":"[21]"},{"why":"Derives the local-field equation that autonomously characterizes the law of the root neighborhood as the solution of a trajectory-dependent jump SDE.","marker":"[22]"},{"why":"Establishes the trajectorial second-order Markov random field property that underlies the local-field derivation and serves as the starting point for the time-marginal version.","marker":"[20]"},{"why":"Introduces Markov local-field equations for jump processes on regular trees, the special case that the present paper extends to general unimodular Galton-Watson trees.","marker":"[23]"},{"why":"Provides the regular-tree special case, including the derivation of the forward Kolmogorov ODE system and the hydrodynamic-limit corollary used here.","marker":"[19]"},{"why":"Gives the martingale-problem existence result for Markovian projection of jump semimartingales, which the paper adapts to obtain the mimicking process.","marker":"[32]"}],"fun_headline_variants":["Finite ODEs predict spreading on sparse random graphs","Acyclic jumps yield finite ODEs for network spreading","Hydrodynamic limits on sparse graphs reduce to finite ODEs","Sparse network dynamics tamed by a finite ODE system"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the claim that once one knows the boundary's state right after a jump, the inside and outside of any separated region are independent—the paper asserts this follows from a two-sided conditioning but supplies no proof, and if it fails the main theorem does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Finite ODEs predict spreading on sparse random graphs","Acyclic jumps yield finite ODEs for network spreading","Hydrodynamic limits on sparse graphs reduce to finite ODEs","Sparse network dynamics tamed by a finite ODE system"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000245,"raw_usage":{"total_tokens":1508,"prompt_tokens":890,"completion_tokens":618,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":549}},"tokens_in":506,"tokens_out":618,"duration_ms":5838,"temperature":1.0,"reasoning_tokens":549,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:03:12.472559+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a concrete finite example satisfying Assumptions A through D in which $X_A(\\bar\\tau)$ and $X_B(\\bar\\tau)$ are conditionally independent given both $(X_{\\partial^2 A}(\\bar\\tau-), X_{\\partial^2 A}(\\bar\\tau))$ but not given $X_{\\partial^2 A}(\\bar\\tau)$ alone; such a counterexample would invalidate the key implication around equation (5.20) and with it the claimed time-marginal 2-MRF property.","supporting_citations":[{"cited_title":"185, 1 –63","cited_arxiv_id":null,"evidence_quote":"Supplies well-posedness of the interacting jump SDE and the hydrodynamic-limit convergence of empirical measures to the root law on the limit tree."},{"cited_title":"in preparation","cited_arxiv_id":null,"evidence_quote":"Derives the local-field equation that autonomously characterizes the law of the root neighborhood as the solution of a trajectory-dependent jump SDE."},{"cited_title":"Interacting Jump Processes Preserve Semi-Global Markov Random Fields on Path Space","cited_arxiv_id":"2210.09253","evidence_quote":"Establishes the trajectorial second-order Markov random field property that underlies the local-field derivation and serves as the starting point for the time-marginal version."},{"cited_title":"preprint","cited_arxiv_id":null,"evidence_quote":"Introduces Markov local-field equations for jump processes on regular trees, the special case that the present paper extends to general unimodular Galton-Watson trees."},{"cited_title":"Ganguly, Non-Markovian interacting particle systems on large sparse graphs: Hydrodynamic limits and marginal characterizations, Ph.D","cited_arxiv_id":null,"evidence_quote":"Provides the regular-tree special case, including the derivation of the forward Kolmogorov ODE system and the hydrodynamic-limit corollary used here."},{"cited_title":"Larsson and S","cited_arxiv_id":null,"evidence_quote":"Gives the martingale-problem existence result for Markovian projection of jump semimartingales, which the paper adapts to obtain the mimicking process."}],"review_version":1}