{"id":"9345a808-262f-441c-9d5c-4ec3b95a46fa","arxiv_id":"2608.06175","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Spreading on temporal networks is shown to be exactly reachability on event graphs for specially defined SIS variants, yielding closed-form epidemic thresholds.","lead":"This paper maps certain epidemic processes on time-varying networks to path-following in static event graphs, and derives formulas for when outbreaks can grow. It offers a way to analyze spreading on temporal networks with graph algorithms instead of many simulations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The analytical R0 threshold rests on a locally tree-like event-graph assumption that Appendix C shows fails for undirected networks; the abstract's unrestricted claim is therefore unsupported.","rationale":"The reader's weakest-assumption analysis identifies exactly the load-bearing condition: the closed-form R0 and threshold equations require the event graph to be locally tree-like and branches to be independent. I agree with that assessment. The paper's own Appendix C is the strongest evidence: a concrete, internally produced demonstration that the analytical threshold underestimates the observed onset in undirected networks. This directly undermines the abstract's blanket claim of thresholds for arbitrary degree and inter-event time distributions. The concern does not, however, invalidate the entire framework. The exact lrSIS-to-reachability mapping in Appendix A is a proof and remains correct; the numerical/component-analysis approach retains its computational value; and the directed configuration-model results show nontrivial agreement with simulations. Therefore the appropriate response is a conditional acceptance that narrows the scope to directed random networks (or fixes the theory for undirected ABAB motifs), tightens the abstract, and releases code for reproducibility. I do not see a reason to move the reader's verdict to rejection, because the core mapping is sound and the flaw is in the genericity of the analytic threshold, not in the basic construction.","tokens_in":10359,"tokens_out":18726,"duration_ms":236010,"concrete_test":"Reproduce the Appendix C experiment on undirected k-regular temporal networks with Poisson link activations, but use the exact lrSIS reachability mapping rather than frSIS. Increase system size from N = 10^4 to N = 10^5 and vary k and the dimensionless activity lambda*dt. Measure the epidemic onset from quasi-stationary density and compare it with the prediction of the undirected R0 formula (the Poisson analogue of Eq. C3). If the observed onset remains significantly above the predicted threshold beyond finite-size error, then the tree-like assumption is falsified for undirected networks and the analytic threshold cannot be claimed as general. As a secondary check, add a fraction of reciprocal directed edges to the directed configuration model; if the threshold mismatch grows with that fraction, the directed-network claim also needs qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central analytic result is Eqs. (2)-(5): R0 = F_R(dt) * R_stat^0 and the resulting threshold. The derivation in Appendix B is a branching-process calculation that explicitly assumes the event graph is locally tree-like, and it models each static edge as contributing at most one independent branch (H(y) = (1-p) + p y). This independence assumption is load-bearing: without it, the generating function does not give the epidemic threshold. Appendix C is a direct counterexample to the assumption being generic. For undirected k-regular temporal networks, Eq. (C3), derived from the same tree-like logic, underestimates the observed frSIS onset, and the paper attributes this to ABAB motifs that make branches converge. Since undirected contact networks are a standard case, and since the abstract advertises thresholds for temporal networks with arbitrary degree and inter-event time distributions without a directedness caveat, the central claim as stated is not established. The concern is not merely empirical: the event graph is not tree-like whenever a static node has multiple in-events and a shared out-event, a local configuration that is not suppressed in the directed configuration model by network sparsity. Thus Eq. (2) rests on an unverified asymptotic limit, and the paper's own Appendix C shows that the limit is not universally valid.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a structural framework for spreading on temporal networks: it maps a class of SIS-type processes to reachability in temporal event graphs (EGs). Two 'globally consistent' processes are introduced, lrSIS and frSIS, whose recovery clocks reset upon infection, and Appendix A proves exact isomorphisms between these processes and EG reachability together with an upper bound relating lrSIS to standard fixed-recovery-time SIS. Using branching-process generating functions under a locally tree-like EG assumption, the paper derives the basic reproduction number R0 = F_R(dt) R_stat^0 (Eq. 2) and threshold conditions (Eqs. 3-5) for directed configuration-model temporal networks with independent renewal link dynamics, and validates them on Poisson and Lomax inter-event-time models and on three empirical temporal networks, claiming significant computational advantages from single-sweep EG component analysis. Appendix C reports that the undirected generalization fails: ABAB motifs make EG branches converge, so the analytical threshold underestimates the observed onset.","tokens_in":10619,"tokens_out":18393,"duration_ms":181694,"significance":"The framework is a genuine contribution if the directed-network results are confirmed. The equivalence and upper-bound proofs in Appendix A are clean and correct, and the upper-bound theorem gives the constructed lrSIS/frSIS processes a substantive connection to the standard SIS model rather than leaving them as purely definitional objects. The R0 factorization is elegant and falsifiable: it reduces to the known static threshold as dt grows, and Eqs. (4)-(5) give explicit, testable predictions for Poisson and bursty dynamics that the simulations in Fig. 3 reportedly match. The computational reduction, building the EG once and obtaining outcomes for all initial conditions from a single component sweep, is real and is benchmarked on empirical data. The paper is also honest: it reports the Helsinki overestimate and, in Appendix C, its own failure case. The principal limitations are the scope of the analytical claim (directed networks only; the abstract overstates this), the unverified tree-likeness assumption, and the definitional nature of lrSIS/frSIS as processes constructed for EG compatibility; these are fixable with qualification and additional verification.","major_comments":[{"comment":"The abstract states that epidemic thresholds are derived 'for temporal networks with arbitrary degree and inter-event time distributions' without any directedness caveat. The actual derivation (Eqs. (1)-(5) and Appendix B) is restricted to directed configuration-model static networks with independent link event processes, and Appendix C shows that the natural undirected analogue, Eq. (C3), 'significantly underestimates the onset of the frSIS process' on k-regular undirected networks because ABAB motifs make branches converge (Fig. 5). Undirected contact networks are a standard case, so the headline claim in the abstract, the introduction, and the discussion is not established; the threshold result should be stated as valid for directed temporal networks, with the undirected breakdown reported in the main text.","section":"Abstract; Appendix C"},{"comment":"The branching-process derivation of Eq. (2) assumes the EG is locally tree-like and branches are independent, but the paper provides no argument that this holds for the directed configuration model, and Appendix C identifies a concrete violation mechanism: a static node with multiple in-events and a shared out-event creates convergent EG branches, a local configuration not suppressed by network sparsity. The close agreement claimed in Fig. 3 is numerical evidence for particular parameter settings, not verification of an asymptotic limit. The authors should add a convergence argument or an explicit structural check (e.g., merge statistics or a comparison of Eq. (2) against a non-tree computation) for the directed case, and phrase the 'striking' agreement (Fig. 3 caption) as a validated approximation with stated scope.","section":"Section 'If the EG is locally tree-like...'; Appendix B"},{"comment":"The main text invites the reader to 'see Appendix C' for generalization to undirected events, but Appendix C shows that only the process definition and the local branching factor generalize: the analytical threshold 'significantly underestimates the onset' (Fig. 5b). This internal mismatch between the main-text claim and the appendix result should be resolved by rewording the main text to state that the process mapping and numerical component analysis generalize, while the closed-form threshold does not.","section":"Section 'A temporal event graph...'"},{"comment":"The generating function H(y) = 1 - F_R(dt) + F_R(dt)y in Eq. (1)/(B1) models each static link as contributing at most one EG branch. This corresponds to the frSIS constrained event graph D_C (at most one admissible event per link), not to the unconstrained EG to which lrSIS maps exactly, where multiple dt-adjacent events on one static link create multiple EG edges; at the parameters of Fig. 2 the expected count lambda*dt reaches order 1. Since the same dashed line from Eq. (4) is overlaid on both the lrSIS and frSIS phase diagrams (Fig. 2b,c), the text should state explicitly which event-graph variant and which process the analytic R0 describes, and justify the at-most-one-branch reduction for lrSIS.","section":"Eq. (1); Appendix B; Fig. 2"}],"minor_comments":[{"comment":"Simulation results are reported without error bars or confidence intervals; in Fig. 4, where outcomes are averaged over randomly sampled observation windows, the spread across windows should be reported to support the 'remarkable' agreement claim.","section":"Figs. 3 and 4"},{"comment":"The affiliation contains a typo: 'Aalto Univerisity' should read 'Aalto University'.","section":"Author affiliations"},{"comment":"The Lomax model requires alpha > 2 for a finite mean (t0 = mu(alpha-2) > 0); the parameter range should be stated explicitly at Eq. (5).","section":"Eq. (5)"},{"comment":"The parenthetical remark after Eq. (3) that the threshold takes a different functional form if p depends on kin or kout is correct but unexplained; a one-line example would help the reader gauge the scope of the result.","section":"Eq. (3)"},{"comment":"The statement that the insets compare SIS and frSIS densities is informative, but the caption could also state the number of independent realizations and network sizes used in the numerical R0 measurements.","section":"Fig. 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The manuscript builds naturally on the authors' prior event-graph program (refs. [25,26,28]), and the incremental contribution, the process-level equivalence in Appendix A and the R0 factorization, is substantive and fits physics.soc-ph. Provided the abstract is qualified to directed networks and the tree-likeness assumption is either verified or explicitly bounded, I see no novelty or scope problem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a genuine step forward for temporal network epidemiology. The new pieces are the global consistency notion, the lrSIS and frSIS processes, and the exact mapping of those processes to reachability on event graphs. The R0 formula R0 = F_R(δt) R_stat^0 is clean, and for Poisson link dynamics it reduces to the known static SIS threshold, which is a nice sanity check. The derivation in Appendix B is careful under its stated assumptions, and the simulation agreement for directed configuration-model networks is convincing. The empirical tests on Twitter, flights, and Helsinki transport add useful evidence, and the runtime comparison makes the practical advantage concrete.\n\nThe soft spots are real but mostly fixable. The abstract promises thresholds for 'arbitrary degree and inter-event time distributions' with no directedness caveat, and that is not what the paper establishes. Appendix C shows that for undirected networks the tree-like generating-function logic breaks down because ABAB motifs make branches converge, shifting the predicted threshold below the observed onset. The stress test is right: Eq. (2) rests on a locally tree-like event-graph assumption that is not generic. The paper does flag the limitation honestly in Appendix C, but the abstract and main-text phrasing need to match the actual domain. A second issue is mild definitional circularity: lrSIS is defined so the event-graph mapping is exact, and the claim that this tells us about standard SIS rests on the upper-bound proof and simulations, not on the mapping itself. That is acceptable if stated clearly, and it mostly is. Missing error bars on simulation symbols and no released code are minor but should be addressed.\n\nNet assessment: the directed-network framework is solid and useful; the undirected extension is a known-broken approximation that the authors themselves diagnose. If the paper is scoped honestly, it deserves publication. I would send it to a serious referee; the core idea is sound and the analytic result is a real contribution, but the claims need tightening before acceptance.","headline":"A useful mapping of temporal SIS variants to event-graph reachability, with a clean R0 for directed random networks, but the abstract overclaims generality that Appendix C itself undermines.","tokens_in":11153,"tokens_out":1619,"would_cite":true,"duration_ms":16155,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For a class of SIS processes, temporal spreading is exactly reachability in one event graph.","keywords":["temporal networks","event graphs","spreading processes","SIS model","epidemic threshold","percolation","global consistency","basic reproduction number"],"falsifier":"Run the lrSIS process on a directed configuration-model temporal network with a specified joint in/out degree distribution and a known inter-event time distribution, and compare the measured onset to $δt_c$ satisfying $F_R(δt_c) = 1/R_{stat}^0$; a significant mismatch would falsify the branching-process derivation. Conversely, on an undirected $k$-regular network with Lomax inter-event times, the measured onset should sit above the analytical $R_0 = 1$ line because ABAB motifs make branches converge, as seen in the paper's Fig. 5.","tokens_in":10143,"feed_emoji":"🦠","tokens_out":8040,"duration_ms":73219,"temperature":0.7,"pith_summary":"The paper aims to put spreading on temporal networks on the same footing that percolation gave to static networks: instead of simulating an epidemic many times, one builds a temporal event graph whose nodes are contact events and whose directed links connect events that could pass an infection, then reads outbreak size from reachable components. The central claim is that a class of SIS-type processes, defined as globally consistent, are exactly reachability on a single event graph for fixed recovery time, so their prevalence and epidemic threshold follow from component analysis rather than simulation. For random temporal networks the basic reproduction number factorises as $R_0 = F_R(δt) R_{stat}^0$, where $F_R$ is the residual inter-event time CDF and $R_{stat}^0$ is the static network reproduction number; this gives explicit threshold conditions for Poisson and bursty link dynamics. A last-contact reinforcing SIS process (lrSIS) realises this mapping exactly and is an upper bound for standard fixed-recovery-time SIS, with matching epidemic onsets in simulations and on three empirical temporal networks. The analytical threshold assumes independent, locally tree-like branches; the paper shows this assumption fails for undirected networks when alternating motifs make branches converge.","feed_headline":"Epidemic size and threshold fall out of one event graph","feed_subtitle":"A new mapping makes temporal contact patterns a static graph whose components give outbreak size and onset without simulation.","key_machinery":"The load-bearing object is the temporal event graph (EG): a static directed acyclic graph whose nodes are the events of the temporal network and whose directed edges connect temporally adjacent events, weighted by waiting time. The recovery time $δt$ thresholds these edges (bond percolation) and the infection probability $β$ removes events (site percolation), so a spreading process with waiting-time limit $δt$ traverses exactly the paths kept in the pruned EG. For random temporal networks the argument runs through the excess out-degree generating function $G_1(y)$, whose derivative at 1 gives $R_0 = F_R(δt) \\langle k_{in} k_{out}\\rangle / \\langle k_{in}\\rangle$, separating temporal statistics (the residual CDF $F_R$) from static structure. The global consistency condition specifies which processes can be coded by a single EG; lrSIS and frSIS satisfy it, standard fixed-recovery SIS does not, and lrSIS nevertheless upper-bounds standard SIS.","core_discovery":"The paper's discovery is that spreading on temporal networks is isomorphic to reachability in a temporal event graph whenever the process is globally consistent: all possible realisations from any initial condition live in a single static directed acyclic graph. The lrSIS process, in which an infected node's fixed recovery clock resets after every incoming infectious event, satisfies this condition, and a proof in Appendix A shows that the infected events in lrSIS are exactly the nodes in the event-graph out-component of the seed. As a direct corollary, the basic reproduction number for random temporal networks is $R_0 = F_R(δt) R_{stat}^0$, so the epidemic threshold is determined by the residual waiting-time distribution and the static degree structure alone; for Poisson link activation this reproduces the known fixed-recovery-time SIS threshold. Because event-graph out-components are ordered in time, they carry infection times and transmission routes, not only final outbreak size, making the correspondence stronger than the static percolation mapping. On empirical Twitter, flight, and public-transport networks, the numerical $R_0 = 1$ condition from the event graph predicts the observed epidemic onset well, with a noted overestimate on the public transport data.","pith_inferences":["The global-consistency criterion is likely the right organising principle for a general theory: any dynamics (random walks, complex contagions, opinion or behaviour spread) that admits a globally consistent variant automatically becomes a structural reachability problem on the same kind of event graph.","The ABAB-motif failure suggests a testable correction for undirected networks: if one prunes alternating two-link returns in the event graph, the analytical threshold may be recovered, but the graph would then depend on process history, so the exactness is lost; an effective branching-factor correction might retain simplicity.","The factorisation $R_0 = F_R(δt) R_{stat}^0$ implies a design principle: temporal heterogeneity and static topology act multiplicatively, so a network designer could compensate for burstiness (small $α$) by increasing static connectivity, and the threshold line from the paper's Eq. (5) gives the trade-off.","In empirical networks with locally dense regions, a globally subcritical network can still have supercritical subpopulations, so onset may occur before $R_0 = 1$; a local out-component spectrum of the event graph could quantify this."],"forward_implications":["On random temporal networks, the epidemic threshold can be calculated in closed form from the inter-event time distribution and static degree distribution, with no epidemic simulation.","For Poisson link activation the threshold reduces to $δt_c = -(1/λ)\\log(1 - 1/R_{stat}^0)$, the same condition as fixed-recovery-time SIS on a static network, so temporal and static analyses coincide in this limit.","Because the event graph encodes the whole time-respecting path structure, its out-components give not only final outbreak size but also infection times and transmission routes for any seed.","The lrSIS process is an upper bound for standard fixed-recovery SIS and shows the same epidemic onset, so event-graph analysis locates conventional SIS thresholds even where the processes are not exactly isomorphic.","Each globally consistent process can be computed for all initial conditions from one event-graph component sweep, giving a large computational saving as the number of realisations grows."],"supporting_citations":[{"why":"Supplies the static percolation mapping and the generating-function method for the reproduction number that the temporal result extends.","marker":"[3]"},{"why":"Shows that fixed infectious period makes SIR exactly isomorphic to percolation, the static analogue of the lrSIS exactness.","marker":"[4]"},{"why":"Defines temporal adjacency and the event graph as the representation of all time-respecting paths.","marker":"[23]"},{"why":"Introduces weighted temporal event graphs, the basis for thresholding links by waiting time $δt$.","marker":"[24]"},{"why":"Establishes that the connectivity transition of event graphs belongs to the directed percolation universality class.","marker":"[25]"},{"why":"Provides the efficient computation of temporal connected components that underlies the computational advantage claimed for event graphs.","marker":"[28]"},{"why":"Gives the residual waiting-time CDF $F_R(δt)$ for independent link activation, which enters the $R_0$ factorisation.","marker":"[33]"},{"why":"Derives the threshold condition for fixed recovery time SIS on static networks, recovered as the Poisson limit of the temporal result.","marker":"[34]"}],"fun_headline_variants":["One event graph component sizes up outbreaks and onset","No simulations: event-graph components reveal epidemic size","Temporal network spread mapped to static graph reachability","Epidemic threshold from a single event graph component"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analytical $R_0$ and threshold equations assume that the event graph is locally tree-like so that spreading branches from an event are independent; the paper's own Appendix C shows that in undirected networks alternating paths between two links make branches converge, shifting the measured onset away from the predicted threshold.","fun_headline_variants_meta":{"raw":{"variants":["One event graph component sizes up outbreaks and onset","No simulations: event-graph components reveal epidemic size","Temporal network spread mapped to static graph reachability","Epidemic threshold from a single event graph component"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000487,"raw_usage":{"total_tokens":2358,"prompt_tokens":864,"completion_tokens":1494,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":480,"completion_tokens_details":{"reasoning_tokens":1432}},"tokens_in":480,"tokens_out":1494,"duration_ms":13301,"temperature":1.0,"reasoning_tokens":1432,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:16:50.853815+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the lrSIS process on a directed configuration-model temporal network with a specified joint in/out degree distribution and a known inter-event time distribution, and compare the measured onset to $δt_c$ satisfying $F_R(δt_c) = 1/R_{stat}^0$; a significant mismatch would falsify the branching-process derivation. Conversely, on an undirected $k$-regular network with Lomax inter-event times, the measured onset should sit above the analytical $R_0 = 1$ line because ABAB motifs make branches converge, as seen in the paper's Fig. 5.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the static percolation mapping and the generating-function method for the reproduction number that the temporal result extends."},{"cited_title":"Kenah and J","cited_arxiv_id":null,"evidence_quote":"Shows that fixed infectious period makes SIR exactly isomorphic to percolation, the static analogue of the lrSIS exactness."},{"cited_title":"Mellor, Journal of complex networks6, 639 (2018)","cited_arxiv_id":null,"evidence_quote":"Defines temporal adjacency and the event graph as the representation of all time-respecting paths."},{"cited_title":"Weighted temporal event graphs,","cited_arxiv_id":null,"evidence_quote":"Introduces weighted temporal event graphs, the basis for thresholding links by waiting time $δt$."},{"cited_title":"Badie-Modiri, A","cited_arxiv_id":null,"evidence_quote":"Establishes that the connectivity transition of event graphs belongs to the directed percolation universality class."},{"cited_title":"Badie-Modiri, M","cited_arxiv_id":null,"evidence_quote":"Provides the efficient computation of temporal connected components that underlies the computational advantage claimed for event graphs."},{"cited_title":"Kivel¨ a and M","cited_arxiv_id":null,"evidence_quote":"Gives the residual waiting-time CDF $F_R(δt)$ for independent link activation, which enters the $R_0$ factorisation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the threshold condition for fixed recovery time SIS on static networks, recovered as the Poisson limit of the temporal result."}],"review_version":1}