{"id":"5a058b07-c465-487c-a7c9-bb048d43ee16","arxiv_id":"2501.09623","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"SIR epidemics on dynamic random graphs converge to epidemics on their time-marked union local limits under a stronger form of dynamic local convergence.","lead":"This paper proves that when a sequence of dynamic random graphs converges to a local limit in a strong 'time-marked union' sense, the spread of an SIR epidemic on the large graphs converges to the epidemic on the limit graph. The result lets epidemic simulations run on simpler limiting trees and identifies which notion of dynamic local convergence is actually needed.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.1 couples SIR with per-edge recovery times, whereas the model assigns one recovery time per vertex; the continuity bridge for Proposition 3.4 is therefore not proven.","rationale":"The reader identified the per-edge versus per-vertex recovery mismatch in Lemma 5.1, and this is indeed the most load-bearing defect in the manuscript. Lemma 5.1 is the only continuity result connecting local time-marked union convergence to convergence of the epidemic local approximation: Proposition 3.4 invokes it directly, and Theorem 1.2 depends on Proposition 3.4. The proof as written couples recovery times edgewise, which is inconsistent with the SIR model's per-vertex recovery times. This is not a stylistic issue: under the edgewise coupling, a vertex can recover before transmitting along one edge while remaining infectious along another, so the coupled process does not have the same law as the model. The proof therefore does not establish continuity of h_{t,r}. The defect is concrete, located in Section 5.2, and serious enough to justify the CONDITIONAL verdict. It is also plausibly fixable: one can couple recovery times vertexwise through the rooted isomorphism and use continuity of D_I to control the δ-perturbed ON/OFF boundary intervals, so I do not see grounds to reject the paper outright. The simulation study provides empirical support but no formal proof, and no code is supplied. Apart from this gap, the overall architecture—first- and second-moment local approximations, limiting approximation, and the transfer argument—is coherent and closely modeled on the static analogue [3]. No other internal inconsistency rose to the same level of concern. Hence the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":44653,"tokens_out":18854,"duration_ms":219019,"concrete_test":"Rewrite Lemma 5.1's proof using the model's per-vertex recovery times. For two rooted marked union graphs within δ in the metric of Definitions 2.8-2.9, couple each vertex v to φ(v) with the same uniform variable and the same recovery time R_v = R_{φ(v)}, couple initial infection statuses symmetrically, and couple each edge's D_I with its image edge. Then check whether the difference in PΛ(t < T^(r)(o)) can be bounded, uniformly over finite paths in B_r, by a sum of terms of the form P(D_I ∈ [x-2δ, x+2δ]), which tends to 0 as δ→0 by continuity of D_I. If such a bound holds, Lemma 5.1 is repairable and the reader's concern is a non-fatal proof defect; if not, the continuity claim behind Theorem 1.2 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central transfer theorem Theorem 1.2 rests on Proposition 3.4, which says that the conditional epidemic expectation EΛ[S_n,r^(ρ)(t) | (G_s^n)] converges to the limiting value s_r(t). The proof of Proposition 3.4 applies local time-marked union convergence to the functional h_{t,r} of Lemma 5.1, so continuity of h_{t,r} on the marked graph space is a load-bearing step. Lemma 5.1's proof couples two epidemics by generating, for every edge e, a pair (D_e^I, D_e^R) and letting infection pass through e exactly when D_e^I ≤ D_e^R and D_e^I falls in an ON interval. This is not the SIR model of Section 1.2 and Section 2.4, where each vertex u receives a single recovery time R_u that governs all incident edges. As written, the coupled process allows a vertex to recover at different times for different edges, which changes infection times and can create discrepancies between the coupled processes that do not exist under the true per-vertex recovery dynamics. Since no argument is given for why the per-edge coupling is equivalent to the per-vertex one, the continuity proof is incomplete. The gap is localized and likely repairable by coupling R_v with R_{φ(v)} through one uniform variable per vertex and then using continuity of D_I plus δ-close marks, but that repair is not present in the manuscript. The manuscript does not flag this missing step; it asserts Lemma 5.1 as proved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a local approximation theory for SIR epidemics on dynamic random graphs. The main theorem (Theorem 1.2) states that if the rooted time-marked union graphs associated with a dynamic graph sequence converge locally in probability, then the empirical SIR proportions (S_n, I_n, R_n) converge in probability to the corresponding root probabilities in the limiting rooted time-marked union graph under the same SIR dynamics. The proof strategy follows Alimohammadi et al. [3]: restrict the epidemic to r-neighborhoods, control first and second moments (Propositions 3.1 and 3.2), approximate the limit by finite radii (Proposition 3.3), and transfer via a continuous bounded functional h_{t,r} of the marked union graph (Proposition 3.4, Lemma 5.1). The paper also introduces the time-marked union convergence framework, illustrates it on dynamic Erdős-Rényi, random intersection, and configuration models, and includes simulation comparisons.","tokens_in":44960,"tokens_out":6735,"duration_ms":71409,"significance":"If the results are correct, Theorem 1.2 provides a useful and essentially parameter-free transfer principle: dynamic epidemics on large graphs can be read off from their local limits. The paper's main conceptual contribution—identifying the insufficiency of snapshot-based dynamic local convergence and introducing local time-marked union convergence, with a counterexample—is valuable. The detailed treatment of the metric space and the backward process is a strength. The central derivation, however, currently rests on a continuity proof that couples the wrong recovery mechanism; until this is repaired, the main theorem is not established. The simulations are suggestive but do not substitute for the missing proof.","major_comments":[{"comment":"The continuity proof of h_{t,r} couples the two SIR processes by generating, for every edge e, a pair (D_e^I, D_e^R) and letting infection pass through e exactly when D_e^I ≤ D_e^R and D_e^I falls in an ON interval of e. This is not the SIR model of Section 1.2, where each vertex u has a single recovery time R_u governing all incident edges, and it is also not the backward process of Section 2.4 and Algorithm 1, which draws one R_u per vertex. Under the proposed coupling a vertex can recover at different times for different edges, which changes infection times and can create differences between the two coupled processes that the true per-vertex dynamics would not produce. Since no argument shows that the per-edge coupling is equivalent to per-vertex recovery, the claimed continuity of h_{t,r} is not proved. This is load-bearing: Proposition 3.4 applies local time-marked union convergence to this functional, and Theorem 1.2 relies on Proposition 3.4. A repair is plausible—couple R_v and R_{φ(v)} through one uniform variable per vertex and then use the same δ-close ON/OFF mark argument—but that repair is absent from the manuscript.","section":"§5.2, Lemma 5.1"},{"comment":"The bound on Var_n(X'_K), and hence Proposition 3.2, rests on the assertion that marked local convergence in probability implies E_n[(P^{(G_n)}_{r,\\bar s_K}(\\tilde H_{[K]}) - p^{(n)}_r(\\tilde H_{[K]}))^2] is small for all finite sequences of neighbourhood types. The manuscript states this as an 'analogous consequence' of [3, Appendix C.3] without proof. In the dynamic setting this is not immediate: the marks are continuous random objects, the empirical frequencies are random variables indexed by n, and one needs convergence in L^2 rather than just in probability. Since this inequality is used to control the variance of EΛ[S_{n,r}^{(ρ)}(t)|(G_s^n)] in (4.44)-(4.47), it should be proved in the present framework or replaced by a precise reference that covers the marked, converging-in-probability case.","section":"§4.2, Step 2e"}],"minor_comments":[{"comment":"The lemma statement says h_{t,r} is 'bounded and continuous in t', but the proof establishes continuity of the functional on the space of rooted time-marked union graphs under the metric of Definitions 2.8-2.9; the statement should be reworded accordingly, and the domain should be the marked graph space rather than G⋆.","section":"§5.2, Lemma 5.1"},{"comment":"There are several typographical issues: 'dominates convergence theorem' should be 'dominated convergence theorem' (Section 5.2), 'Cachy-Schwarz' should be 'Cauchy-Schwarz' (Section 4.2, Step 2e), and 'c` adl` ag' should be typeset correctly.","section":"Throughout"},{"comment":"The counterexample in Remark 2.14 is described only heuristically; since it motivates the paper's main conceptual choice of the stronger convergence notion, it would help to specify the graph dynamics and the limit object in a few more sentences.","section":"§2.3, Remark 2.14"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is a clean conditional transfer statement, and the gap in Lemma 5.1 appears localized and repairable. The paper would be suitable for this journal once the coupling is corrected and the L^2 consequence in Proposition 3.2 is justified. The examples are largely delegated to [40] and [52], but this does not affect the central claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. First, the main result is genuinely new: it transfers SIR epidemic convergence to dynamic random graphs using a new notion they call local time-marked union convergence, and it shows that the more static snapshot version of dynamic local convergence is not enough. The counterexample in Remark 2.14 makes that point cleanly. Second, the paper has a real proof gap in Lemma 5.1, the continuity of the conditional infection-time functional h_{t,r}. The coupling used to prove continuity draws a recovery time per edge, whereas the SIR model in Section 1.2 assigns one recovery time per vertex. So the coupled process is not the model. That makes Proposition 3.4, and hence Theorem 1.2, conditional as written.\n\nThe paper is otherwise well-organized and honest. The technical framework for the time-marked union graph, the metric on marks, and the backward epidemic process are laid out carefully. The examples — dynamic ER, dynamic random intersection graphs, and the rewiring configuration model — add value, though the last is only a heuristic convergence claim, which the authors acknowledge. The simulation study supports the approximation claim and gives a nice illustration of the (1-ρ)^r error bound. The moment bounds and local approximation arguments in Sections 3 and 4 look sound to me; I don't see hidden circularity.\n\nThe Lemma 5.1 issue is the main problem. It is localized and likely repairable: couple recovery times per vertex rather than per edge, exploit the isomorphism between the two finite balls, and use continuity of D_I plus the fact that marks are δ-close. But that repair isn't in the manuscript, and the authors don't flag the mismatch. A referee should ask for it before accepting.\n\nMinor quibbles: no simulation code is provided, and the claim that the marked graph space is Polish is asserted without proof. Neither is a big deal.\n\nOverall, this is a substantial extension of the static local-epidemic theory, and the new convergence notion is likely to be useful beyond this paper. With the lemma fixed, I'd accept it. I'd send it to a serious referee with a request to verify the repaired proof. I would bring it to a reading group only after the fix, since the current gap could derail a session.","headline":"A genuine extension of the static local-epidemic transfer to dynamic graphs, with a real but repairable gap in Lemma 5.1's proof.","tokens_in":45452,"tokens_out":4011,"would_cite":true,"duration_ms":40945,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C80","60F05","92D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"SIR epidemics on dynamic random graphs converge to the epidemic on their local time-marked union limit, so the course of an outbreak on a huge evolving network can be studied through a limiting marked graph or tree.","keywords":["dynamic random graphs","local convergence","time-marked union graph","SIR epidemic","backward process","dynamic local convergence","random graph limits"],"falsifier":"Run the SIR model with per-vertex recovery times on two sequences of dynamic graphs whose rooted time-marked union balls are within metric distance $\\delta$ but whose ON/OFF schedules differ near a recovery deadline, and check whether the root infection probability at time $t$ differs by an amount that does not vanish as $\\delta \\to 0$; a non-vanishing difference would falsify the continuity step. More directly, simulate the epidemic on a dynamic Erdős–Rényi graph and on its limiting time-marked union tree, but in the finite graph give each vertex one common recovery time while in the coupled construction assigning independent recovery times per edge; if the infection proportions differ systematically as $n$ grows, the coupling in Lemma 5.1 is invalid and the theorem's proof has a gap.","tokens_in":44461,"feed_emoji":"🦠","tokens_out":5450,"duration_ms":55849,"temperature":0.7,"pith_summary":"This paper proves that the course of an SIR epidemic on a large dynamic random graph is determined by the local, time-marked structure around a typical vertex. The main theorem states: if the dynamic graphs converge in probability in the local time-marked union sense to a limiting marked union graph with law $\\mu$, then for every time $t \\in [0,T]$ the epidemic proportion vector converges in probability to the root-status probabilities $s(t), i(t), r(t)$ of the same SIR dynamics on the limit. The result extends the known static-graph local limit theorem to networks whose edges switch on and off over time. A constructive point of the paper is that ordinary dynamic local convergence is not enough for epidemics because infections travel along indirect temporal connections; the stronger time-marked union convergence is introduced to capture those. If the theorem is right, simulating an epidemic on a huge evolving network can be replaced by simulating the limiting marked tree or graph.","feed_headline":"Epidemics on evolving graphs reduce to their local limits","feed_subtitle":"Local time-marked union convergence pins down the full SIR epidemic curve on dynamic random graphs.","key_machinery":"The carrying object is the time-marked union graph: the graph containing every edge that was ever active during $[0,T]$, with each edge marked by the full sequence of its ON and OFF intervals. Convergence of these marked rooted graphs in probability, called local time-marked union convergence, is the hypothesis of the main theorem. On top of this object, the paper runs a backward epidemic process that assigns recovery times to vertices, transmission times to edges, and initial infection marks to vertices, then computes the infection time of the root as the minimum over all temporally feasible paths from initially infected vertices. The bridge between finite graphs and the limit is the functional $h_{t,r}$ giving the probability that the root remains uninfected by time $t$ when the epidemic is confined to the $r$-neighbourhood; the argument needs this functional to be continuous and bounded on the space of rooted time-marked union graphs.","core_discovery":"The central claim, Theorem 1.2, is that local time-marked union convergence transfers epidemic dynamics from finite dynamic random graphs to their infinite local limits. Concretely, for a sequence of dynamic graphs $(G_n^s)_{s\\in[0,T]}$ whose rooted time-marked union graphs converge in probability to a limiting time-marked union graph with law $\\mu$, and for an SIR epidemic with arbitrary continuous infection and recovery time distributions started from independently infected vertices with probability $\\rho>0$, the proportion vector $(S_n^{(\\rho)}(t), I_n^{(\\rho)}(t), R_n^{(\\rho)}(t))$ converges in probability to $(s(t), i(t), r(t))$, where $s(t), i(t), r(t)$ are the probabilities that the root of the limiting graph is susceptible, infected, or recovered at time $t$ under the same SIR dynamics. The proof proceeds by comparing the true epidemic with an epidemic restricted to $r$-neighbourhoods of vertices, bounding the first and second moments of the approximation error, and then passing to the limit through a continuous functional of the time-marked union graph. The paper also shows that this stronger form of convergence is genuinely needed, since edges that switch off can still transmit infection through indirect temporal paths that ordinary snapshot local convergence misses.","pith_inferences":["A testable extension is to check whether the continuity step survives when recovery times are truly per-vertex rather than per-edge; if the gap in the coupling cannot be closed, the theorem may still hold under an extra condition such as short ON periods or rare switching.","The counterexample structure behind Remark 2.14 suggests that time-marked union convergence is likely necessary for any process whose evolution depends on paths that are never simultaneously present in the snapshot graph, not just SIR epidemics.","For edge dynamics in which each edge activates only once, such as the dynamic Erdős–Rényi model in the paper, time-marked union convergence may reduce to a simpler dynamic local convergence criterion, which would be a cheaper hypothesis to verify in applications.","The same proof architecture might transfer to SEIR or other monotone compartmental models, provided the per-vertex recovery-time coupling issue is resolved and the relevant status functional is continuous on the same marked space."],"forward_implications":["For any dynamic random graph model that converges in the local time-marked union sense, the full SIR epidemic curve on the finite graph is approximated by the corresponding root-status probabilities on the limiting marked graph or tree.","The dynamic Erdős–Rényi graph converges locally in the time-marked union sense to a Poisson branching tree with mean offspring $\\gamma(1+T)$ and edge marks given by an explicit joint distribution, so simulations of the epidemic can be run on that tree instead of on the $n$-vertex graph.","Dynamic random intersection graphs and the rewiring configuration model also admit time-marked union tree limits, making epidemic simulation on clustered dynamic networks computationally cheaper.","The first-moment error bound $(1-\\rho)^r$ shows that larger initial infection proportions and deeper local neighbourhoods improve the quality of the local approximation, with accuracy decaying exponentially in the neighbourhood radius.","Simulations reported in the paper indicate that on graphs with low clustering, edge dynamics accelerate the epidemic curve compared with a static graph of the same stationary degree distribution."],"supporting_citations":[{"why":"Supplies the static-graph theorem and the first- and second-moment proof template that this paper adapts to the dynamic setting.","marker":"[3]"},{"why":"Introduces the dynamic local convergence framework and proves time-marked union convergence for the dynamic random intersection graph, which the paper extends and uses for examples.","marker":"[40]"},{"why":"Presents an alternative notion of dynamic local convergence that the paper distinguishes from its stronger time-marked union convergence.","marker":"[29]"},{"why":"Provides the Polish-space background for rooted graphs and the stable neighbourhood structure consequences used in the second-moment bound.","marker":"[37]"},{"why":"One of the original sources of local convergence of graphs, the concept this paper generalizes to dynamic graphs.","marker":"[12]"},{"why":"The other foundational reference for local weak convergence, supplying the objective-method perspective underlying local limits.","marker":"[2]"},{"why":"Provides the weak-convergence and tightness criteria for càdlàg processes used in the formulation of dynamic local weak convergence.","marker":"[14]"}],"fun_headline_variants":["Epidemics on evolving graphs reduce to local limits","Dynamic graph SIR follows local limit dynamics","Time-marked convergence predicts SIR on moving nets","Local limits steer outbreaks on time-varying graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result rests on the assumption that tiny differences in edge activity schedules cause only tiny differences in infection probabilities; the paper's proof of this couples two epidemics by giving each edge its own recovery time, whereas the actual SIR model gives each person one common recovery time shared by all their edges, so the coupled process is not the model and the continuity proof as written is incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Epidemics on evolving graphs reduce to local limits","Dynamic graph SIR follows local limit dynamics","Time-marked convergence predicts SIR on moving nets","Local limits steer outbreaks on time-varying graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000163,"raw_usage":{"total_tokens":1246,"prompt_tokens":954,"completion_tokens":292,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":233}},"tokens_in":570,"tokens_out":292,"duration_ms":3889,"temperature":1.0,"reasoning_tokens":233,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:50:43.390926+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the SIR model with per-vertex recovery times on two sequences of dynamic graphs whose rooted time-marked union balls are within metric distance $\\delta$ but whose ON/OFF schedules differ near a recovery deadline, and check whether the root infection probability at time $t$ differs by an amount that does not vanish as $\\delta \\to 0$; a non-vanishing difference would falsify the continuity step. More directly, simulate the epidemic on a dynamic Erdős–Rényi graph and on its limiting time-marked union tree, but in the finite graph give each vertex one common recovery time while in the coupled construction assigning independent recovery times per edge; if the infection proportions differ systematically as $n$ grows, the coupling in Lemma 5.1 is invalid and the theorem's proof has a gap.","supporting_citations":[{"cited_title":"Alimohammadi, C","cited_arxiv_id":null,"evidence_quote":"Supplies the static-graph theorem and the first- and second-moment proof template that this paper adapts to the dynamic setting."},{"cited_title":"van der Hofstad, M","cited_arxiv_id":null,"evidence_quote":"Introduces the dynamic local convergence framework and proves time-marked union convergence for the dynamic random intersection graph, which the paper extends and uses for examples."},{"cited_title":"Dort and E","cited_arxiv_id":null,"evidence_quote":"Presents an alternative notion of dynamic local convergence that the paper distinguishes from its stronger time-marked union convergence."},{"cited_title":"van der Hofstad","cited_arxiv_id":null,"evidence_quote":"Provides the Polish-space background for rooted graphs and the stable neighbourhood structure consequences used in the second-moment bound."},{"cited_title":"Aldous and J","cited_arxiv_id":null,"evidence_quote":"The other foundational reference for local weak convergence, supplying the objective-method perspective underlying local limits."},{"cited_title":"Billingsley","cited_arxiv_id":null,"evidence_quote":"Provides the weak-convergence and tightness criteria for càdlàg processes used in the formulation of dynamic local weak convergence."}],"review_version":1}