{"id":"b8c88c73-97f3-4129-acdb-66ba2cc300bf","arxiv_id":"2507.23056","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The displayed tree phylogenetic network model is a submodel of DAG graphical models, and this perspective yields new nonidentifiability results for stacked reticulations and 2-blobs.","lead":"This math paper shows that a standard evolutionary network model, the displayed tree model, is a special case of a broader class of statistical models called DAG graphical models. Using this connection, it proves that some network features, like stacked hybridization events or 2-blobs, cannot be determined from sequence data under this model.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.5's proof omits the reverse containment, which is load-bearing for Theorem 7.8 and the 2-blob collapse; the gap is repairable with splittability but must be stated.","rationale":"The reader's conditional verdict is appropriate. The strongest claim, Theorem 5.1, is proven with both directions and only requires multiplicatively closed, convex, and splittable models. For general equivariant models, splittability is proven via the identity matrix, so it is not a hidden weakness for the paper's stated scope. The genuine issue is the incomplete proof of Proposition 4.5, which is later invoked in Proposition 5.4 and Theorem 7.8. This is a concrete, textual gap rather than a failure of the central construction. A one-paragraph repair using splittability closes the hole, so the conditional verdict stands unchanged. The paper should add the reverse containment and clarify that Proposition 4.5 depends essentially on splittability.","tokens_in":16799,"tokens_out":23636,"duration_ms":282291,"concrete_test":"Repair the proof of Proposition 4.5 by explicitly writing the reverse containment: given any M_ac in T M, invoke Definition 3.6 with k=1 and M_1=M_ac to obtain N in T M with M_ac N^-1 in T M; set M_ab=M_ac N^-1, M_bc=N, and check that this produces a valid parameter assignment in the subdivided graph yielding the same conditional distribution pG1,c|A. If the requirement that M_ab and M_bc be valid needs any hypothesis beyond splittability (e.g., convexity), then Theorem 7.8's use of Proposition 4.5 is invalid and the rank bound needs a different argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The displayed tree model perspective and Theorem 5.1 appear sound under the stated hypotheses. The manuscript's real soft spot is the proof of Proposition 4.5 (edge subdivision). It shows only that the subdivided-edge distribution pG2,c|A is a special case of pG1,c|A by taking M_ac = M_ab M_bc; it never shows the reverse containment. Splittability is exactly the tool needed to reverse this (for any M_ac choose N with M_ac N^-1 in T M, set M_bc=N and M_ab=M_ac N^-1), but the proof stops after the forward direction. Proposition 5.4 uses Proposition 4.5 to contract degree-2 vertices inside a 2-blob, and Theorem 7.8 uses it to justify subdividing cut-edges before applying d-separation. As written, those results are not fully supported. The gap does not affect the core stacked-reticulation contraction (Theorem 5.1), whose reverse direction is proven in detail, so the central nonidentifiability claim survives.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a DAG-graphical-model perspective on the displayed tree phylogenetic network model. It introduces local modifications of DAGs and a splittability condition on sets of transition matrices, then proves that, under multiplicative closure, convex closure, and splittability, contracting certain hidden edges—including stacked reticulation edges—does not change the family of distributions. This leads to nonidentifiability results for stacked reticulations and 2-blobs. The paper also derives dimension formulas for reticulation conditional distributions under the general Markov model and proves rank bounds on flattenings that generalize classical tree-model results.","tokens_in":16903,"tokens_out":13979,"duration_ms":162905,"significance":"If the results are fully established, the paper offers a valuable unifying framework: displayed tree models are constrained DAG graphical models, and local modification arguments yield model-uniform statements instead of case-by-case algebraic computations. The splittability property is cleanly defined and proven for general equivariant and open equivariant models, and the central stacked-reticulation contraction (Theorem 5.1) is proved in detail. The rank conditions in Section 7 are a genuine generalization of tree flattening results. The main weakness is an incomplete proof in Proposition 4.5, which is local and repairable but affects later results that rely on the equality of the two distribution families.","major_comments":[{"comment":"The proof of Proposition 4.5 only establishes one containment: it shows that p_{G',c|A}(x_c|x_A) from the subdivided-edge graph is a special case of p_{G,c|A}(x_c|x_A) by taking M_ac = M_ab M_bc. It never proves the reverse containment, although the proposition states that the two DAGs produce the same family of distributions. This is not merely a presentational shortcut: Proposition 5.4 and Theorem 7.8 both rely on the equality of the two families. The missing direction is straightforward using splittability: for any M_ac in the model, choose N in the transition-matrix set with M_ac N^{-1} in the model, set M_ab = M_ac N^{-1} and M_bc = N, and leave the remaining parent terms and reticulation weights unchanged. Please add this argument, or alternatively restate the proposition as a one-sided containment and adjust the later results accordingly.","section":"Section 4, Proposition 4.5"},{"comment":"Both Proposition 5.4 and Theorem 7.8 depend on the equality stated in Proposition 4.5, not merely on the containment proven there. In Proposition 5.4, the step 'We can assume that all degree 2 vertices within the 2-blob have been contracted by using Proposition 4.5' inherits the gap described above. In Theorem 7.8, if only the containment from the subdivided graph G' to the original graph G were available, then a distribution arising from G need not be representable in G', and the d-separation/rank argument would not apply to all distributions in the model. Once the reverse containment of Proposition 4.5 is supplied, both arguments are sound, but as written they are not fully supported.","section":"Section 5, Proposition 5.4 and Section 7, Theorem 7.8"}],"minor_comments":[{"comment":"In the first display of the proof, the second sum is missing the factor pi^c_a: the term should be pi^c_a M_ac(x_c|y_a), not M_ac(x_c|y_a).","section":"Section 6, proof of Proposition 6.1"},{"comment":"The proof states that matrix rank is upper-semicontinuous and uses this to conclude that generic parameter values have rank at least the rank at a particular point. The correct fact is that rank is lower-semicontinuous; the wording should be corrected.","section":"Section 7, proof of Theorem 7.11"},{"comment":"The theorem assumes only that b has outdegree 1, but the reverse construction divides by delta^c_b, which can be zero if A1 union A3 is empty. Since the intended application is to stacked reticulations, where b has indegree greater than one and hence has parents, the theorem statement should either include that assumption explicitly or discuss this edge case.","section":"Section 5, Theorem 5.1"},{"comment":"The dimension computation for the Jukes-Cantor model is asserted without details. Please include the calculation or a reference, since the contrast between dimensions 8 and 9 is used to argue that two networks do not yield the same family.","section":"Example 4.7"},{"comment":"The text cites [6] as the source of generalizations of Theorem 6.2 and for level-1 identifiability results, but [6] is listed as 'In preparation'. If these citations are not load-bearing, consider marking them as pointers; if the results are needed, state the relevant statements in the paper.","section":"References [6,7]"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a solid contribution, and the central stacked-reticulation contraction is proved carefully. The Proposition 4.5 gap is real but strictly local and easily repaired using the splittability assumption already stated in the proposition. I recommend major revision rather than rejection; after the missing reverse containment is added, the remaining issues appear minor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine contribution, not a repackaging. Sullivant shows the displayed tree model is a DAG graphical model submodel, proves a local modification theorem, and uses it to show stacked reticulations and 2-blobs are not identifiable under the displayed tree model for equivariant Markov models. Theorem 5.1 is the centerpiece and it is proven in detail; the reverse direction uses splittability, which is a real assumption but is verified for general equivariant and open equivariant models. The 2-blob collapse (Prop 5.4) follows by repeated application. The dimension loss at reticulation nodes (Thm 6.2) is also new and clean. The flattening rank bounds (Thms 7.8 and 7.11) extend known tree results to networks under the displayed tree model.\n\nWhat is genuinely new: the contraction theorem for stacked reticulations, the 2-blob unidentifiability, and the rank bounds. The DAG reformulation is useful conceptually and the paper is clearly written.\n\nSoft spots: the proof of Proposition 4.5 (edge subdivision) only shows one containment. It shows distributions from the subdivided graph are special cases of the original by multiplying transition matrices, but never proves that every distribution in the original model arises from a subdivision. Splittability is exactly the missing ingredient; the stress-test note is right that the reverse containment is straightforward but omitted. As written, this affects Theorem 7.8 and Proposition 5.4, which rely on Proposition 4.5. The gap is repairable and does not threaten the main stacked-reticulation result, whose proof is self-contained.\n\nMinor: the heavy use of splittability is a non-obvious closure condition. It holds for the standard equivariant models, but for arbitrary rate-matrix models it may fail; the paper is clear about this, so it is a stated assumption rather than a hidden flaw. The self-citations [6,7] are only pointers, not load-bearing.\n\nCitation pattern looks fine: standard references, no suspicious self-citation inflation.\n\nWho this is for: mathematical phylogeneticists and anyone using displayed tree models for inference. The warning that stacked reticulations and 2-blobs won't be recoverable matters empirically. The paper deserves a serious referee; after fixing the Proposition 4.5 gap (or replacing reliance on it with an explicit splittability argument), I'd take it as a solid publication.","headline":"Solid theory paper that recasts displayed-tree network models as DAG submodels and proves real nonidentifiability results; one repairable gap in Proposition 4.5 should be fixed before publication.","tokens_in":17520,"tokens_out":1955,"would_cite":true,"duration_ms":20508,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["92D15","62R01"],"pacs":[],"model":"deepseek-v4-flash","headline":"Stacked reticulations are statistically invisible in displayed tree network models.","keywords":["displayed tree model","phylogenetic networks","graphical models","local modifications","splittability","stacked reticulations","flattening ranks","equivariant models"],"falsifier":"For a concrete four-state model satisfying the three conditions, such as the general Markov model with strictly positive transition matrices, compute the dimension of the displayed-tree distribution family for the left and middle networks in Figure 1.1; the paper predicts the two dimensions are exactly equal, so a pair with different dimensions under such a model would contradict Theorem 5.1.","tokens_in":16506,"feed_emoji":"🧬","tokens_out":5492,"duration_ms":64418,"temperature":0.7,"pith_summary":"This paper shows that the displayed tree model of evolution on a phylogenetic network is a special case of a directed graphical model, and it uses that connection to prove when two different networks produce the same probability distributions on observed leaves. The central result is that an edge from a reticulation vertex with no other outgoing edges can be contracted without changing the model's distribution family, provided the substitution matrices are multiplicatively closed, closed under convex combinations, and splittable. Since stacked reticulations are exactly such edges, networks that differ only by stacked reticulations cannot be distinguished from sequence data under these models. The same machinery shows that 2-blobs are invisible and yields rank bounds on flattenings of the probability tensor that generalize classic results for phylogenetic trees. A sympathetic reader cares because knowing which network features are identifiable is a precondition for inferring reticulate evolution from data.","feed_headline":"Stacked reticulations are invisible in displayed tree network models","feed_subtitle":"Recasting network models as DAGs shows stacked reticulations and 2-blobs leave identical leaf distributions.","key_machinery":"The central object is a local modification of a DAG: a triple (A, B, C) of vertex sets in which every edge into B comes from A or B, every edge out of B goes to B or C, and every edge into C comes from A, B, or C. Theorem 4.4 says that two DAGs that are local modifications with the same conditional family p_{C|A} give the same joint family once B is hidden. Splittability, defined in Definition 3.6, is the closure property on transition matrices that lets the proof reverse an edge contraction: for any finite list of matrices M_i in the model, there must exist a single matrix N in the model such that every M_i $N^{{-1}}$ is also in the model. Splittability holds for general equivariant and open equivariant phylogenetic models, which is why the contraction results apply uniformly across model types rather than only to group-based models or the general Markov model.","core_discovery":"The displayed tree phylogenetic network model sits as a natural submodel of the graphical model associated to a directed acyclic graph, and this representation carries the paper's main results. Theorem 5.1 establishes that if the transition model is multiplicatively closed, closed under convex combinations, and splittable, then contracting an edge b -> c where b has outdegree 1 does not change the family of leaf distributions when b is hidden. Because a stacked reticulation is precisely such an edge from one reticulation to another, networks that differ only by stacked reticulations are distributionally equivalent. The same argument, applied repeatedly, implies that a 2-blob can be replaced by a single edge without changing the model family. The paper also derives linear relations among conditional distributions at a reticulation node that cause a dimension loss of (m-1)k parameters, and it uses d-separation to bound flattening ranks of the leaf probability tensor by k^(#E) for any edge cutset E.","pith_inferences":["A practical consequence the paper does not spell out is that any inference pipeline treating stacked reticulations as distinct evolutionary hypotheses is fitting statistically equivalent models, so likelihood-based support for one network over the other cannot come from the displayed-tree model alone.","The flattening rank bounds suggest an implementable model check: compute the flattening of the observed site-pattern tensor for a candidate network and compare its rank to k^(#E); a violation would indicate the network or the substitution model is wrong.","The local-modification technique may transfer to other hidden-variable graphical models beyond phylogenetics, wherever a conditional distribution factors through an intermediate variable that can be contracted under a suitable closure condition.","The dimension loss at reticulation nodes suggests that even when a reticulation's position in the graph is identifiable, the mixing proportions and branch lengths entering that node may not be separately estimable from displayed-tree data."],"forward_implications":["Two binary phylogenetic networks that differ only by a stacked reticulation yield exactly the same family of leaf distributions under multiplicatively closed, convex, splittable models, so those networks cannot be distinguished by displayed-tree data.","Any 2-blob in a network can be replaced by a single edge without changing the displayed-tree distribution family, under the same conditions, meaning blob structure is not identifiable from the displayed-tree model alone.","For the general Markov model on k states with a reticulation node having m parents, the space of conditional distributions has dimension (k-1)(m(k-1)+1), losing (m-1)k dimensions relative to the number of parameters used to describe it.","If removing a set of edges E separates the leaves into groups A and B, then every distribution in the model satisfies rank Flat(A,B)(P) <= k^(#E).","If one displayed tree T has parsimony score l_T(A|B), then a generic distribution in the model has rank Flat(A,B)(P) at least min(k^(#A), k^(#B), k^(l_T(A|B)))."],"supporting_citations":[{"why":"Supplies the factorization, d-separation, and conditional independence facts for DAG graphical models on which the paper's representation and Section 7 arguments rest.","marker":"[12]"},{"why":"Provides the background and definitions for equivariant phylogenetic models, including the closure properties under multiplication and convex combinations that the contraction theorems assume.","marker":"[5]"},{"why":"Cited as the source of generalizations of Theorem 6.2 to general equivariant models, including the reparametrization of a two-parent reticulation conditional distribution.","marker":"[6]"},{"why":"Supports the interpretation of the displayed tree model as a limiting family of the network multispecies coalescent, motivating why identifiability in this model matters.","marker":"[13]"},{"why":"Provides the classic flattening rank results for phylogenetic tree models that Theorem 7.8 generalizes to the displayed tree network setting.","marker":"[3]"},{"why":"Used in the proof of Theorem 7.11 for the lower bound on flattening rank via parsimony score on a tree.","marker":"[4]"},{"why":"Also used in Theorem 7.11 to supply the parsimony-based lower bound for the rank of a tree flattening.","marker":"[14]"},{"why":"Prior work on distinguishing phylogenetic networks under the displayed tree model, providing context for the nonidentifiability results in this paper.","marker":"[9]"},{"why":"Prior level-1 network identifiability results for Markov processes, which the present paper's uniform treatment generalizes and contrasts with.","marker":"[10]"}],"fun_headline_variants":["Stacked reticulations and 2-blobs are invisible in tree models","Displayed tree models cannot see stacked reticulations","DAG recasting hides stacked reticulations in networks","Tree network models blind to reticulation stacks","Graphical model view exposes stacked reticulation ambiguity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The set of allowed substitution matrices must be splittable: for any finite list of matrices M_i in the model, some single matrix N in the model must exist such that every M_i $N^{{-1}}$ is also in the model, and this is what lets the proof undo a contracted edge.","fun_headline_variants_meta":{"raw":{"variants":["Stacked reticulations and 2-blobs are invisible in tree models","Displayed tree models cannot see stacked reticulations","DAG recasting hides stacked reticulations in networks","Tree network models blind to reticulation stacks","Graphical model view exposes stacked reticulation ambiguity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000186,"raw_usage":{"total_tokens":1270,"prompt_tokens":834,"completion_tokens":436,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":360}},"tokens_in":450,"tokens_out":436,"duration_ms":5664,"temperature":1.0,"reasoning_tokens":360,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:06:01.207482+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete four-state model satisfying the three conditions, such as the general Markov model with strictly positive transition matrices, compute the dimension of the displayed-tree distribution family for the left and middle networks in Figure 1.1; the paper predicts the two dimensions are exactly equal, so a pair with different dimensions under such a model would contradict Theorem 5.1.","supporting_citations":[{"cited_title":"Lauritzen,Graphical models, Oxford Statistical Science Series, vol","cited_arxiv_id":null,"evidence_quote":"Supplies the factorization, d-separation, and conditional independence facts for DAG graphical models on which the paper's representation and Section 7 arguments rest."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cited as the source of generalizations of Theorem 6.2 to general equivariant models, including the reparametrization of a two-parent reticulation conditional distribution."},{"cited_title":"Rhodes, Hector Baños, Jingcheng Xu, and Cécile Ané,Identifying circular orders for blobs in phylo- genetic networks, Adv","cited_arxiv_id":null,"evidence_quote":"Supports the interpretation of the displayed tree model as a limiting family of the network multispecies coalescent, motivating why identifiability in this model matters."},{"cited_title":"Allman and John A","cited_arxiv_id":null,"evidence_quote":"Provides the classic flattening rank results for phylogenetic tree models that Theorem 7.8 generalizes to the displayed tree network setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Used in the proof of Theorem 7.11 for the lower bound on flattening rank via parsimony score on a tree."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Also used in Theorem 7.11 to supply the parsimony-based lower bound for the rank of a tree flattening."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior work on distinguishing phylogenetic networks under the displayed tree model, providing context for the nonidentifiability results in this paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior level-1 network identifiability results for Markov processes, which the present paper's uniform treatment generalizes and contrasts with."}],"review_version":1}