{"id":"36a5edbf-efc1-4501-b6d1-de51cff3e6c1","arxiv_id":"2507.02523","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A message-passing source-detection algorithm for hypergraph SI epidemics, augmented by a local neighbor-infection heuristic, outperforms four baseline methods in numerical experiments.","lead":"This paper proposes a new computer method, HDMPN, to find the original infected node in an outbreak on a hypergraph, where interactions can involve groups of people rather than just pairs. The method multiplies a standard likelihood estimate by a local heuristic, and the authors report it beats existing source-detection methods in most simulations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (3) defines ρ_i as the fraction of susceptible neighbors, while the abstract, Sec. 3, and Sec. 5 describe it as the fraction of infectious neighbors; the reported gain is therefore not attributable to the stated mechanism. The paper must specify and evaluate one formula, and the choice matters.","rationale":"I read the paper as an empirical claim: a modified likelihood-maximization method using neighbor infection probability improves source detection on hypergraphs. The key new object is ρ_i, so whatever enters Eq. (4) must be well-defined and must match the stated mechanism. The manuscript contains a direct contradiction: Eq. (3) defines ρ_i through susceptible neighbors, while the abstract, Sec. 3, and Sec. 5 describe it as the fraction of infectious neighbors. This is not a stylistic issue because the two factors rank candidate sources differently, especially for nodes with many infected contacts. The paper's claimed advantage of HDMPN over HDMP is precisely this multiplicative factor, so the empirical result cannot be interpreted until we know which factor was actually used. The reader identified this as the weakest assumption, and I agree. The Eq. (14) index error and missing error bars are real concerns, but they affect presentation and significance testing rather than the identity of the method itself. The proposed two-variant reimplementation directly settles the issue without requiring access to the authors' code, and it would also provide the missing confidence intervals.","tokens_in":16176,"tokens_out":5876,"duration_ms":76066,"concrete_test":"Implement both variants: ρ_i^S = |Γ_S(i)| / (|Γ_I(i)| + |Γ_S(i)|) as printed in Eq. (3), and ρ_i^I = |Γ_I(i)| / (|Γ_I(i)| + |Γ_S(i)|) as described in the abstract and Sec. 5. Run the paper's protocol (R = 1000, λ = 0.5, stopping threshold 10% for N ≤ 1000 and 5% for NDC-classes) on ERH-200, SFH-200, HCL-2.3-200, Algebra, Restaurant-Rev, and NDC-classes. For each variant, report accuracy (Eq. 23), ranking (Eq. 24), and a 95% confidence interval over the R runs, compared with HDMP. If the |Γ_S| variant does not beat HDMP on the majority of these datasets, the printed formula cannot support the central claim. If the |Γ_I| variant reproduces the paper's figures while the |Γ_S| variant does not, then the manuscript's formula is wrong rather than merely misdescribed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that HDMPN outperforms HDMP because the source likelihood is modulated by the fraction of infectious neighbors. But Eq. (3) explicitly defines ρ_i = |Γ_S(i)| / (|Γ_I(i)| + |Γ_S(i)|), i.e., the fraction of susceptible neighbors. The abstract, the Sec. 3 motivation, and Sec. 5 all describe ρ_i as the fraction of infected/infectious neighbors. These are complementary quantities: the infectious fraction is 1 − ρ_i as printed. For a true source in an SI process, by time T the source's neighbors are disproportionately infected, so the printed susceptible fraction is typically small; multiplying the likelihood by it penalizes exactly the nodes that the stated mechanism is supposed to favor. If the experiments used the formula as written, the reported improvement over HDMP is not explained by the paper's mechanism and could be an incidental penalty (e.g., related to degree or local density) that happens to help on the tested hypergraphs. If the experiments used the infectious fraction, then Eq. (3) and the surrounding text do not specify the algorithm that was actually evaluated. No code is released to disambiguate. This is load-bearing because the entire novel component of HDMPN relative to HDMP is the ρ_i factor; an inconsistency in its definition makes the numerical advantage uninterpretable. The index typo in Eq. (14) and the absence of error bars are secondary; they can be repaired without deciding which version of the method produced the figures.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes HDMPN, a dynamic-message-passing algorithm for source detection in SI epidemics on hypergraphs. The method multiplies the likelihood Prob(V_I|H,i) by a heuristic neighbor-infection probability rho_i, derives message-passing recursions for the hypergraph SI model, and compares against HDMP, HCC, HBC, and HMCSM on synthetic (ERH, SFH, HCL) and empirical hypergraphs using accuracy and ranking metrics.","tokens_in":16488,"tokens_out":10639,"duration_ms":116235,"significance":"The problem is timely and relevant: source detection in higher-order interaction structures is underexplored, and an efficient DMP-based method would be a useful tool. The paper contributes a DMP formulation for hypergraph SI dynamics, a detailed numerical protocol, and a comparison on multiple synthetic and empirical hypergraphs. If the rho_i factor is correctly specified and the numerical advantage is statistically reliable, the method offers a practical improvement for source detection in hypergraphs.","major_comments":[{"comment":"Equation (3) defines rho_i as |Gamma_S(i)|/(|Gamma_I(i)|+|Gamma_S(i)|), the fraction of susceptible neighbors, while the abstract, the motivation in Section 3, and the concluding Section 5 all describe rho_i as the fraction of infectious neighbors. Because rho_i is the only novel mechanism distinguishing HDMPN from HDMP, this contradiction is load-bearing: if the printed formula was used, the factor penalizes nodes with many infected neighbors, and the reported improvement cannot be attributed to the stated mechanism; if the infectious fraction was used, the paper does not specify the actual algorithm evaluated. The authors must correct the definition or the text, report which variant was run, and re-evaluate the comparisons.","section":"Section 3, Eq. (3)"},{"comment":"The claim that 'HDMPN consistently outperforms all baseline methods on all hypergraphs and for all N values' is not supported by the displayed evidence: no error bars or confidence intervals are provided, and the HDMPN-HDMP differences appear small; additionally, the same section reports that on the NDC-classes hypergraph HBC is slightly superior to HDMPN, contradicting the 'all hypergraphs' statement. The authors should report standard errors or significance tests (e.g., binomial confidence intervals for accuracy), and qualify the claim to match the NDC-classes exception.","section":"Section 4.4, Figs. 2-4"},{"comment":"Equation (14) states Prob(q_k(t)=S|D_i) = P_i^S(0) prod_{v_j in partial_i\\k} theta_{j->i}(t); by Lemma 1 (Eq. (7)) the correct expression should be P_k^S(0) prod_{v_j in partial_k\\i} theta_{j->k}(t). The printed version swaps the roles of i and k; as written, the recursion cannot be reproduced. Please correct the equation and check the subsequent steps for the same index swap.","section":"Equation (14)"}],"minor_comments":[{"comment":"Tuning alpha for HCC/HBC on the same test data (alpha chosen to maximize accuracy for each dataset) should be explicitly labeled as an in-sample optimization; this is favorable to the paper's claims but should be stated for methodological clarity.","section":"Section 4.4"},{"comment":"The caption entry '(l) HCL, (N, Theta) = (40, 2.0)' appears to be a typo for (400, 2.0); please check all N values in the captions.","section":"Fig. 4 caption"},{"comment":"The HCL power-law exponent is denoted by both Theta (in figure captions) and gamma (in the text); please use a single notation throughout.","section":"Figures 3 and 4 / text"},{"comment":"The abstract says 'in most cases' while the introduction says 'consistently outperforms'; align the strength of the claims with the actual results, including the NDC-classes exception.","section":"Abstract and Introduction"},{"comment":"No code or data repository is linked; since the numerical protocol is detailed, providing code would strengthen reproducibility.","section":"General"},{"comment":"Equation (8) writes Prob(sum ... | D_j) without specifying that the quantity is the probability that the sum equals zero; please clarify this notation.","section":"Eq. (8)"},{"comment":"The phrase 'higher-order neighbor infection probability' in Section 1 differs from 'neighbor infection probability' used in Section 3; unify the terminology.","section":"Section 1 and Section 3"}],"recommendation":"major_revision","confidential_remarks":"The central issue is a definitional contradiction in the main algorithmic contribution, which is fixable but requires re-running the experiments with a clearly specified rho_i. The paper cites relevant literature, including works by the same group, appropriately. If the corrected experiments continue to show a consistent advantage, the paper would be a useful contribution to the field of source detection on higher-order networks."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a useful incremental DMP adaptation for hypergraph source detection, but the key heuristic (rho_i) has a definition/text mismatch that makes the reported improvement uninterpretable until the authors pick one formula and verify the experiments.\n\nWhat's genuinely new: the DMP equations are carefully derived for a hypergraph SI model, with the eta_ki term capturing shared hyperedge proportions, and the authors provide a coherent likelihood formulation. The numerical study covers three synthetic models and three empirical hypergraphs, with reasonable baselines (HCC, HBC, HMCSM). The complexity analysis is clear. That's real work, and the code, if released, would be a useful resource.\n\nSoft spots, in order of severity.\n\nFirst, Eq. (3) defines rho_i as the fraction of susceptible neighbors, but the abstract, Section 3, and conclusion all describe it as the fraction of infectious neighbors. These are complementary: for a true source at time T, neighbors are disproportionately infected, so the printed formula penalizes exactly the nodes the text says it favors. If the experiments used the printed formula, the improvement over HDMP cannot be attributed to the stated mechanism; if they used the infectious fraction, the paper doesn't specify the algorithm actually evaluated. No code is released to disambiguate. This is load-bearing since rho_i is the only new ingredient.\n\nSecond, Eq. (14) has an index error (should be P_k^S(0) and partial_k\\i, not P_i^S(0) and partial_i\\k). Clearly a typo, but it needs fixing.\n\nThird, there are no error bars or significance tests on the accuracy differences, and the claim of consistent outperformance on all hypergraphs is contradicted by the paper's own statement that HBC is slightly superior on NDC-classes. That's an overreach.\n\nFourth, no code or data are provided beyond 'available on reasonable request.'\n\nThe locally tree-like assumption is standard for DMP and the paper doesn't hide it; I wouldn't over-weight that.\n\nOverall: the paper deserves a serious referee—the topic is timely, the DMP derivation is sound, and the central flaw is fixable. But I would not cite or rely on the performance claim until the authors clarify the rho_i definition and re-run (or re-verify) the experiments. I'd consider it for a reading group precisely because the discrepancy is instructive.\n\nRecommendation: send to peer review with major revision expected; the authors should be asked to specify one formula, correct Eq. (14), add error bars, and temper the conclusion.","headline":"Useful incremental DMP for hypergraph source detection, but a definition/text mismatch in the key heuristic makes the headline gain uninterpretable until fixed.","tokens_in":17042,"tokens_out":3912,"would_cite":false,"duration_ms":39621,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that adding a neighbor-infection heuristic to dynamic message passing consistently improves source detection for SI outbreaks on hypergraphs over plain likelihood maximization and three baselines.","keywords":["Hypergraph","Source detection","Epidemic processes","Dynamic message passing","Higher-order interactions","Maximum likelihood estimation","Susceptible-infectious model"],"falsifier":"Run the identical $R = 10^3$ comparisons with $\\rho_i$ computed two ways—susceptible-neighbour fraction as in Eq. (3) and infectious-neighbour fraction as described in the text—and also on a hypergraph with a planted short loop through the true source; if the two definitions give materially different accuracy and the infectious-fraction version does not beat HDMP, the paper's stated mechanism is wrong, and if HDMPN's edge disappears on the looped hypergraph, the locally tree-like approximation is the load-bearing part.","tokens_in":15948,"feed_emoji":"🦠","tokens_out":8093,"duration_ms":87899,"temperature":0.7,"pith_summary":"Source detection—finding which node started an outbreak from a snapshot of who is infected—has mostly been solved for ordinary pairwise networks, but real contact often happens in groups captured by hypergraphs. This paper claims that for a stochastic susceptible-infectious process on a hypergraph, a message-passing estimate of each node's source likelihood can be improved by multiplying it by a local heuristic factor $\\rho_i$, defined in Eq. (3) as the fraction of the candidate node's neighbours that are still susceptible at the observation time. The resulting algorithm, HDMPN, is tested against the unmodified likelihood method (HDMP), two hypergraph centrality scores, and a Monte Carlo baseline; the paper's central numerical claim is that HDMPN consistently identifies the true source more often and ranks it higher on synthetic and empirical hypergraphs. A sympathetic reader would care because group-based transmission changes epidemic dynamics, and a cheap multiplier that sharpens likelihood-based source inference is a directly usable priority-ranking tool for outbreak containment.","feed_headline":"Neighbor-weighted likelihood beats baselines finding patient zero","feed_subtitle":"On synthetic and real hypergraphs, the extra local factor consistently raises source-ranking accuracy over plain dynamic message passing.","key_machinery":"The carrying object is the 'neighbor infection probability' $\\rho_i$, defined in Eq. (3) as $|\\Gamma_S(i)| / (|\\Gamma_I(i)| + |\\Gamma_S(i)|)$, the fraction of a node's neighbours that are still susceptible at time $T$, multiplied onto the dynamic-message-passing likelihood in Eq. (4) (though the text calls $\\rho_i$ the fraction of infectious neighbours). The likelihood itself is computed through a recursion over $\\theta_{k \\to i}(t)$ and $\\varphi_{k \\to i}(t)$, the probabilities that an infection has not passed from $k$ to $i$ by time $t$, optionally with $k$ infectious, conditioned on $i$ staying susceptible; Lemma 1 and Eq. (11) close the recursion by writing $\\mathrm{Prob}(q_i(t) = S \\mid D_j)$ as the product of $\\theta$ messages over neighbours, which is what makes the computation tractable but also assumes the hypergraph is locally tree-like. The heuristic does the work: it reweights candidates by their local snapshot context before the likelihood is compared.","core_discovery":"The paper's central claim is that the source of a hypergraph SI epidemic can be found more reliably by augmenting maximum-likelihood source inference with a per-node structural heuristic. Conditioned on a candidate source $v_i$, HDMP computes $\\mathrm{Prob}(V_I \\mid H, i)$ by recursive message-passing variables $\\theta_{k \\to i}(t)$ and $\\varphi_{k \\to i}(t)$, under a locally tree-like independence approximation; HDMPN then scores each infectious candidate by $\\rho_i$ times that likelihood and selects $\\arg\\max_{v_i \\in V_I}$, hoping the factor $\\rho_i$ captures whether the candidate's local neighbourhood looks like a plausible infection frontier. In simulations on three synthetic hypergraph models (ERH, SFH, HCL) and three empirical hypergraphs (Algebra, Restaurant-Rev, NDC-classes), the paper reports that HDMPN consistently beats HDMP, HCC, HBC, and HMCSM in accuracy and ranking, with gains especially visible at smaller sizes, and it notes one exception: HBC ranks the true source slightly better on NDC-classes. An unresolved textual inconsistency is that Eq. (3) defines $\\rho_i$ as the fraction of susceptible neighbours, whereas the abstract and conclusion describe the modulating factor as the fraction of infectious neighbours.","pith_inferences":["The text's description of $\\rho_i$ as the fraction of infectious neighbours conflicts with Eq. (3)'s susceptible-neighbour fraction; a direct next experiment is to implement both definitions and compare accuracy, which would either fix the mechanism or show which local signal the improvement actually comes from.","I would expect the multiplier's value to depend on the infection probability $\\lambda$ and the stopping threshold: near the epidemic threshold, susceptible-neighbour counts may be the most informative, whereas in a large outbreak the likelihood alone may dominate, an implication the paper does not test explicitly.","The same modulation idea could be carried to SIR/SEIR dynamics or simplicial contagion models by replacing $\\rho_i$ with an analogue computed from recovered or exposed states; the paper mentions these models as future work but does not claim the transfer.","If the locally tree-like approximation fails in clustered empirical hypergraphs yet HDMPN still wins, the heuristic may be acting as a robustness correction rather than as part of an exact likelihood, which would be a different and interesting explanation."],"forward_implications":["If the numerical claim holds, HDMPN gives a drop-in replacement for HDMP: the same $O(N^2 T)$ runtime plus an $O(N)$ precomputation, so the accuracy gain costs almost nothing in scaling.","The consistent advantage over three baselines means likelihood-based inference, not centrality, is the right backbone for hypergraph source detection in the tested regime, and a local multiplier can be a useful correction to it.","The improvement is largest on smaller hypergraphs (accuracy above 0.2 at $N=100$ and above 0.1 at $N=400$), so the method is best suited to early-warning or moderate-size contact systems rather than very large populations.","Because the algorithm assumes one source, complete final states, and a single homogeneous $\\lambda$, a correct claim leaves multi-source and partial-observation variants open rather than solved."],"supporting_citations":[{"why":"It introduces the factor-graph DMP framework for hypergraph source inference that HDMP extends to node-to-node infection.","marker":"Yu et al. 2024"},{"why":"It defines the hypergraph SI model whose final-state snapshots HDMPN is tested on.","marker":"Suo et al. 2018"},{"why":"It provides the soft-boundary Monte Carlo source estimator that the HMCSM baseline extends to hypergraphs.","marker":"Antulov-Fantulin et al. 2015"},{"why":"It supplies the Algebra and Restaurant-Rev empirical hypergraphs used in the numerical comparisons.","marker":"Amburg et al. 2020"},{"why":"It supplies the NDC-classes empirical hypergraph on which the ranking exception is observed.","marker":"Benson et al. 2018"},{"why":"It supplies the higher-order contagion motivation and the claim that group interactions change spreading dynamics.","marker":"Battiston et al. 2020"},{"why":"It defines s-distance and s-walks used by the HCC and HBC centrality baselines.","marker":"Aksoy et al. 2020"},{"why":"It provides the Erdős–Rényi hypergraph generation model used for synthetic tests.","marker":"Surana et al. 2022"}],"fun_headline_variants":["Group-aware source detection beats standard epidemic baselines","Hypergraph message passing sharpens patient zero search","Neighbor-weighted likelihood improves epidemic source ranking","Higher-order interactions yield better patient zero detection","New algorithm uses group structure to trace epidemic origins"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole method leans on a heuristic score $\\rho_i$ that must carry real information about who the source is, and on the approximation that the hypergraph has few short loops so that infection routes around a node can be treated independently; Eq. (3) defines $\\rho_i$ as the fraction of susceptible neighbours while the text describes it as the fraction of infectious neighbours, so the mechanism is not yet pinned down.","fun_headline_variants_meta":{"raw":{"variants":["Group-aware source detection beats standard epidemic baselines","Hypergraph message passing sharpens patient zero search","Neighbor-weighted likelihood improves epidemic source ranking","Higher-order interactions yield better patient zero detection","New algorithm uses group structure to trace epidemic origins"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000316,"raw_usage":{"total_tokens":1798,"prompt_tokens":962,"completion_tokens":836,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":767}},"tokens_in":578,"tokens_out":836,"duration_ms":10070,"temperature":1.0,"reasoning_tokens":767,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:27:50.201330+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the identical $R = 10^3$ comparisons with $\\rho_i$ computed two ways—susceptible-neighbour fraction as in Eq. (3) and infectious-neighbour fraction as described in the text—and also on a hypergraph with a planted short loop through the true source; if the two definitions give materially different accuracy and the infectious-fraction version does not beat HDMP, the paper's stated mechanism is wrong, and if HDMPN's edge disappears on the looped hypergraph, the locally tree-like approximation is the load-bearing part.","supporting_citations":[{"cited_title":"Physica A 495, 475–487 (2018) https://doi.org/10.1016/j.physa.2017.12.108","cited_arxiv_id":null,"evidence_quote":"It defines the hypergraph SI model whose final-state snapshots HDMPN is tested on."},{"cited_title":"Physical Review Letters 114(24), 248701 (2015) https://doi.org/10.1103/PhysRevLett.114.248701","cited_arxiv_id":null,"evidence_quote":"It provides the soft-boundary Monte Carlo source estimator that the HMCSM baseline extends to hypergraphs."},{"cited_title":"EPJ Data Science 9(1), 16 (2020) https://doi.org/10.1140/epjds/s13688-020-00231-0","cited_arxiv_id":null,"evidence_quote":"It defines s-distance and s-walks used by the HCC and HBC centrality baselines."}],"review_version":1}