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REVIEW 3 major objections 4 minor 38 references

On Tree-Network Distinguishability and Full Identifiability of Phylogenetic Networks

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read For level-1 phylogenetic networks, the leaf-pattern distribution determines the semi-directed topology under JC, K2P, and K3P, and most networks are distinguishable from all trees.

desk verdict Full identifiability for level-1 networks on Θ0 is a genuine step change; the tree-network result is close but Lemma 5.1 needs a real proof before I'd call it a theorem. read the letter →

arxiv 2607.12919 v2 pith:BNVKZD2X submitted 2026-07-14 q-bio.PE

Jari Brits , Niels Holtgrefe , Leo van Iersel , Samuel Martin This is my paper · ORCID
classification q-bio.PE MSC 92D1562R01
keywords phylogeneticnetworksidentifiabilityJukes-CantormodelKimuramodelssite-patterndistributiontrinetinequalitydisplayedtreeslevel-1
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that under the Jukes-Cantor, Kimura 2-parameter, and Kimura 3-parameter substitution models, the semi-directed topology of a level-1 phylogenetic network is fully identifiable from the distribution of leaf patterns, at every parameter point in a biologically reasonable space. It also proves that under JC and K2P, no phylogenetic tree can produce the same leaf-pattern distribution as a network containing a non-trivial blob with at least three incident edges. The only networks that can be mistaken for trees are trees augmented with small 2-blobs. If correct, these results make network inference statistically consistent for level-1 networks and give a formal sense in which reticulate evolution leaves a detectable signature in DNA sequence data.

What carries the argument

The argument uses displayed-tree mixture models and their Fourier parametrization: under group-based models, the network distribution is a convex combination of tree distributions. The load-bearing device is the trinet inequality. For JC, the polynomial Q = q_ACC q_CAC q_CCA − q_AAA q_TCG^2 is zero on every trinet without a non-trivial 3-blob (in particular on a 3-star tree) and strictly positive on every trinet with a non-trivial 3-blob; a K2P analogue uses Q = q_AGG q_GAG q_CCA^2 − q_AAA q_GGA q_TCG^2. Marginalization lemmas transfer these inequalities from 3-leaf subnetworks to the full network, and a separate lemma, stated without proof via a generalization of earlier work, shows that su

What would settle it

Find a trinet with a non-trivial 3-blob and a 3-star tree plus edge parameters in (0,1) for which the JC invariant Q = q_ACC q_CAC q_CCA − q_AAA q_TCG^2 equals zero at a point in the network model; the paper proves this cannot happen. More directly, if any network with a 3-blob and a tree share a point in their leaf-pattern distributions with positive parameters, Corollary 5.8 is false.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.9: two distinct level-1 semi-directed phylogenetic networks, compared modulo the placement of reticulation vertices inside triangles, have disjoint leaf-pattern models on the restricted parameter space for JC, K2P, and K3P. So the network topology can be recovered from exact site-pattern probabilities without generic-position caveats. The second claim, Corollary 5.8, says that under JC and K2P, if one network contains a non-trivial m-blob with m at least 3 and another does not, their models have empty intersection; in particular, no such network is statistically indistinguishable from a tree. The proof works by restriction: explicit polynomial inequalities call

Load-bearing premise

The proof that suppressing a 2-sub-blob leaves the model of the original network contained in the model of the suppressed network is stated without proof, citing a straightforward generalization of a previous section; if that containment fails in some edge case, the lift from trinets to all networks collapses.

Editorial extensions

If this is right

  • Level-1 network reconstruction from sequence data is statistically consistent under JC, K2P, and K3P, up to the semi-directed network and triangle placement.
  • Under JC and K2P, a network with a non-trivial 3-blob can never be inferred as a tree by any method based exactly on leaf-pattern distributions, so reticulation becomes statistically testable.
  • The intersection of two level-1 network models is exactly the union of the models of their maximal shared displayed networks, so residual ambiguity is fully described.
  • The trinet inequality answers a previously open conjecture and extends tree-network distinguishability from low-level networks to arbitrary level under JC and K2P.
  • The combinatorial results transfer to several coalescent-based models, giving identifiability results for classes of galled tree-child networks of arbitrary level.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same reasoning extends to equivariant substitution models, similar full identifiability may hold for models beyond group-based ones, such as the strand-symmetric model; the paper leaves this open.
  • The 2-blob caveat suggests that 2-blobs are statistically invisible under these models—a testable claim: simulate a tree and a 2-blob-augmented tree with the same displayed trees and compare their site-pattern distributions.
  • The K3P gap at arbitrary level looks bridgeable by replacing the single invariant with a simultaneous set of invariants, since the level-1 K3P case already requires four invariants.
  • A practical consequence not discussed in the paper: the trinet inequality could serve directly as a test statistic for detecting reticulation from quartet site patterns.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies full identifiability of phylogenetic network topologies from the leaf-pattern distribution under the JC, K2P, and K3P substitution models. In Section 4 it proves that the semi-directed parameter of a level-1 network is identifiable at every point of the biologically restricted parameter space Θ0, up to the placement of reticulation vertices in triangles; the proof proceeds by establishing trinet/quarnet inequalities and then reducing arbitrary level-1 networks to 3- and 4-leaf restrictions via marginalization. In Section 5 it turns to tree-network distinguishability: for JC and K2P it claims that any trinet with a non-trivial 3-blob is distinguishable from a 3-star tree, and via a suppression lemma for 2-sub-blobs it derives the broader statement (Corollary 5.8) that a network containing a non-trivial m-blob with m≥3 cannot share a leaf-pattern distribution with a network without such a blob, e.g. a tree. The main results are thus: full level-1 identifiability for all three models, and a positive resolution of Conjecture 2.16 of [14] for arbitrary level under JC and K2P, conditional on an unproved structural lemma.

Significance. If the results hold, they constitute a substantial advance over prior generic identifiability results: the level-1 theorem is non-generic and covers K3P as well as JC and K2P, and the tree-network distinguishability result gives a strong, falsifiable signature of reticulation. The paper also explicitly draws consequences for statistical consistency of network inference and for coalescent-based models. The proofs are largely explicit: the invariants are written down as concrete polynomials and the positivity arguments on Θ0 are checkable. However, the Section 5 results currently rest on Lemma 5.1, which is stated without proof and is load-bearing for Corollary 5.2, Lemmas 5.5/5.6, and Corollary 5.8; until that lemma is supplied, the tree-network distinguishability theorem is not fully established.

major comments (3)
  1. [§5.1, Lemma 5.1] Lemma 5.1 is stated without proof and is load-bearing for the main tree-network distinguishability claim. The sentence that it 'follows by a straightforward generalization of the results in Section 5 of [37]' is not an argument. The generalization from 2-blobs to 2-sub-blobs inside a larger k-blob is nontrivial: one must show that marginalizing over all internal states of the sub-blob yields, between the two attachment points, an effective transition matrix that still lies in the JC/K2P/K3P model, including when the sub-blob contains multiple reticulation cycles and when it traps the root. Corollary 5.2, the no-2-sub-blob reductions in Lemmas 5.5 and 5.6, and hence Corollary 5.8 all depend on this. Please provide a full proof or a precise reduction to [37] that covers arbitrary 2-sub-blobs, not just 2-blobs.
  2. [§4.3, Theorem 4.9] The proof of Theorem 4.9 delegates the key combinatorial reduction to [17, Theorem 14(b)] without stating that theorem. This external result is the sole justification for the assertion that any two distinct level-1 networks (modulo triangles and contracting triangles) differ on a 3- or 4-leaf restriction in one of the enumerated ways. The theorem's hypotheses and conclusion should be reproduced or stated precisely, and the exact meaning of 'modulo the placement of the reticulation vertices in any triangles and contracting triangles to single vertices' should be made explicit, so that the reader can verify the theorem applies to every case used in the proof.
  3. [§5.2, Lemma 5.4] The proof of Lemma 5.4 is too compressed for a lemma that carries the induction in Lemmas 5.5 and 5.6. In Case 1, the claim that a non-trivial 2-blob in the basin has at least one attachment point (for otherwise it would be a 2-sub-blob in N) needs a precise boundary-count argument; as written it is not obvious why exactly two vertices of the blob are adjacent to V\W. In Case 2, the notion of a 'lowest' reticulation presupposes a partial order that is not defined for semi-directed networks, and the final step that the blob containing u' or r* is a non-trivial 3-blob requires checking the number of incident edges. Please expand this proof.
minor comments (4)
  1. [§5.1, Lemma 5.1] The lemma says the models are 'on the parameter set Θ0(N)' for both M and M'; presumably M' is on Θ0(N'), the parameter set of the suppressed network. Please clarify.
  2. [§5.2, Lemma 5.5] In the base case k=1, the text says 'we identify edges d and e', and edges e and e''' using the edge labelling from Figure 4, but Figure 4 does not label e' and e''. Please align the notation.
  3. [§2.1, Definition 2.2] The discussion of restriction mentions that 2-blobs can be more complex at higher levels and refers to Section 5; it may be helpful to state explicitly there that 2-sub-blobs will not be suppressed in Definition 2.2.
  4. [Appendix A, Lemma 4.6] Several positivity computations are summarized with expressions such as 'the reader can check'; these are checkable, but a small comment on how the invariant multisets are compared would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the paper's central claims are supported by explicit polynomial invariants and by independent combinatorial lemmas; the unproved 2-sub-blob suppression lemma is a proof gap, not circularity.

full rationale

I walked the derivation chain for the two main results. Theorem 4.9 (full identifiability of level-1 networks) is proved by reducing to 3- and 4-leaf cases via Theorem 3.6, using explicit invariants such as Q_{12|34}=q_AAAA q_TTTT - q_AATT q_TTAA and the K3P invariants Q_1,...,Q_4. These are not defined in terms of the models' disjointness; they are explicit polynomials, and the positivity/vanishing statements are verified by direct substitution into the parameterization maps. The appeals to [17, Theorem 14(b)], [25, Lemma 1 and Theorem 1], and [26, Lemma 5.4] are parameter-free combinatorial facts about networks, not restatements of the model-identifiability claim, so they do not make the argument circular even though some coauthors overlap. Corollary 5.8 (tree-network distinguishability) rests on the trinet inequalities (Lemmas 5.5 and 5.6), whose inductive proofs use Lemma 2.6 (proved in the paper) and the combinatorial Lemma 5.4 (also proved). Lemma 5.1, suppressing 2-sub-blobs, is stated without proof and is load-bearing, but it is asserted to follow from the external [37, Section 5], not from the paper's own conclusions; the paper even flags that equality in part (i) is open. A missing proof is a correctness risk, not a circular reduction. I find no step in which a prediction or identifiability result is equivalent to its input by construction, and no fitted parameter is renamed as a prediction. Score 0 reflects that the derivation is self-contained at the level of the model invariants, with the only caveat being an unproved external lemma.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper's central results are derived from invariant polynomials and combinatorial reductions; it introduces no new free parameters or entities. It does rely on a chain of external structural lemmas from the same research community, one of which (Lemma 5.1) is asserted without proof.

assumptions (7)
  • domain assumption Fourier parameterization of group-based substitution models: q-coordinates are monomials for trees and mixtures over displayed trees for networks (Eq. 3-4).
    Foundation of algebraic phylogenetics; cited to [22,35,36]; not proved in paper.
  • domain assumption The restricted parameter space Θ₀: transition matrices are positive definite with eigenvalues in (0,1), and mixing parameters δ_i in (0,1).
    Biologically reasonable; excludes trivial mixing and zero branch lengths.
  • standard math Marginalization maps restricted networks: for multiplicatively closed models, marginalizing leaves gives a point in the model of the restricted tree/network (Prop 3.3, Cor 3.4, Theorem 3.6).
    Proved in the paper, but depends on multiplicative closure and on the displayed-tree restriction equality [18, Prop A.1].
  • domain assumption [17, Theorem 14(b)]: distinct level-1 networks can be separated by a 4-leaf restriction in one of three ways.
    External structural result invoked in Theorem 4.9; not proved in this paper.
  • domain assumption [25, Lemma 1 & Theorem 1]: a semi-directed network without cherries has a leaf adjacent to a reticulation.
    Used in Lemma 5.3; external structural result.
  • ad hoc to paper [37, Section 5] and Lemma 5.1: suppressing 2-sub-blobs preserves model containment/equality.
    Lemma 5.1 is stated without proof, asserted to follow by straightforward generalization of [37]. This is load-bearing for Cor 5.2 and Lemmas 5.5/5.6.
  • domain assumption [7, Lemma 17] and [26, Lemma 5.4]: every edge in an m-blob lies on an up-down path and every non-cut edge in a simple network is in an excellent cycle.
    Used in Lemma 5.7 to establish the existence of a 3-blob in some trinet; external results.

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Pith. "Pith review of On Tree-Network Distinguishability and Full Identifiability of Phylogenetic Networks." pith.science (2026). https://pith.science/paper/BNVKZD2X

@misc{pith2026260712919,
  author       = {Pith},
  title        = {Pith review of: On Tree-Network Distinguishability and Full Identifiability of Phylogenetic Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BNVKZD2X}},
  note         = {Machine review of arXiv:2607.12919}
}
abstract

Phylogenetic networks generalize phylogenetic trees to evolutionary histories that include reticulate events such as recombination, horizontal gene transfer, and hybridization. Under a Markov model of nucleotide substitution, a phylogenetic network determines a distribution of leaf-patterns. Here, we study the identifiability of the network topology from this distribution under the Jukes-Cantor (JC), Kimura 2-parameter (K2P), and Kimura 3-parameter (K3P) models. Our first result is that the semi-directed network parameter of a level-1 phylogenetic network (modulo redirecting triangles) is fully identifiable under all three models, on a biologically reasonable parameter space in which substitution rates are probabilistic and mixing parameters are non-trivial (i.e., not 0 or 1). In contrast to the generic identifiability established in prior work, this holds at every point of the parameter space, not merely off of a measure-zero subset. Our second result distinguishes phylogenetic networks from phylogenetic trees, on the same parameter space, under JC and K2P. We prove that no phylogenetic network and phylogenetic tree can induce the same leaf-pattern distribution unless the network is a tree, possibly augmented with certain substructures called $2$-blobs. This means the presence of reticulate evolution creates, in most cases, a detectable signature in the leaf-pattern distribution. More broadly, these results have consequences for identifiability beyond the models and network classes studied here, including for several coalescent-based models.

Figures

Figures reproduced from arXiv: 2607.12919 by the authors.

Figure 1
Figure 1. Three rooted phylogenetic networks T , N1, and N2 on the set of taxa X = {1, 2, 3, 4, 5}. T is also a rooted phylogenetic tree, whereas N1 and N2 are not. N1 can be obtained from T by augmenting it with a 2-blob (in bold). N2 cannot be obtained by augmenting a rooted phylogenetic tree with 2-blobs (since it contains a 3-blob). Hence, by Corollary 5.8 we can distinguish T and N1 from N2, but not T from N1. In Section… view at source ↗
Figure 2
Figure 2. (a) A 4-leaf, level-2 binary rooted phylogenetic network on the set of taxa X = {1, 2, 3, 4}. Reticulation edges are drawn as dashed lines. (b) The corresponding semi￾directed phylogenetic network. In this case, this consists of a single 4-blob (containing all internal vertices, highlighted in bold) and leaves adjacent to the blob. that form part of the cycle. See N2 of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Three semi-directed phylogenetic networks on the set of taxa X = {1, 2, 3, 4}. Reticulation edges are drawn with dashed lines. The network N1 is a level-2 network, N2 is level-1, and N3 is level-0 (i.e., a tree). Both N2 and N3 are displayed networks of N1, whilst N3 is also a displayed network of N2. In terms of the partial order, we have N1 > N2 > N3. Since it is minimal, the network N3 is a displayed tree of N1 a… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: A leaf and edge-labelled 3-sunlet network. Example 2.4. Consider the semi-directed phylogenetic network N in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: The four level-1 quarnet topologies. From left to right: a quartet tree, a single triangle, a double triangle, a 4-sunlet. Note that reticulation edges are not identifiable in triangles. In the 4-sunlet, reticulation edges are dashed. Only the 4-sunlet does not have a …
Figure 6
Figure 6. Figure 6: A leaf and edge-labelled directed single-triangle quarnet. Then for a point p ∈ M2 we have Q12|34(p) = aTbTcTdT(δfT + (1 − δ)kThT) − cTdTeThT × aTbTeT(δfThT + (1 − δ)kT) = aTbTcTdT(δfT + (1 − δ)kThT) − aTbTcTdTe 2 ThT(δfThT + (1 − δ)kT) = aTbTcTdT(δfT + (1 − δ)kThT − e…
Figure 7
Figure 7. Figure 7: Two 4-sunlets with circular orders (a) (1,2,3,4) and (b) (1,3,2,4). Note that N2 is directed and edge-labelled to determine the parameterization. Now, let N1 be the 4-sunlet with circular order (1, 2, 3, 4) and let N2 be the 4-sunlet with circular order (1, 3, 2, 4), a…
Figure 8
Figure 8. Figure 8: Three trinets with leaf set {x, y, z}. N1 and N2 both have a 2-sub-blob (in bold). The 2-sub-blob in N2 traps the root and the one in N1 does not. The trinet N3 is the one obtained from N1 or N2 by suppressing their respective 2-sub-blobs. The following result follows …
Figure 9
Figure 9. Figure 9: Left: A level-5 trinet N with the vertices W in the basin for x and the vertices U in the funnel for x. The three thick red vertices are the attachment points of the basin. Right: The two displayed networks N ′′ = N − e ′ and N ′ = N − e ′′ . Lemma 5.4. Let N be a trin…
Figure 10
Figure 10. Figure 10: (A) A phylogenetic tree T , focussed on the leaf n, its parent vertex p(n), and the incident edges leading to the subtrees S1 and S2. (B) The corresponding restricted tree T ′ = T |[n−1]. Subtrees S1 and S2 remain unchanged. is zero on M1 and strictly positive on M2. …
Figure 11
Figure 11. Figure 11: All single-triangle quarnets without the split 12|34. First we show that Q12|34 is strictly positive on the quarnets in [PITH_FULL_IMAGE:figures/full_fig_p030_11.png]
Figure 12
Figure 12. Figure 12: All double-triangle quarnets without the split 12|34. As in the single-triangle case, we have that (a) can be obtained from (b) with the leaf-permutation (34), so it is sufficient to prove that Q12|34 is strictly positive on the quarnet in (a). The parameterization is…

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Works this paper leans on

38 extracted references · 2 linked inside Pith

  1. [14]

    Identifiability of phylogenetic level-2 networks under the Jukes-Cantor model.bioRxiv 2025.04.18.649493, 2025

    Aviva K Englander, Martin Frohn, Elizabeth Gross, Niels Holtgrefe, Leo van Iersel, Mark Jones, and Seth Sullivant. Identifiability of phylogenetic level-2 networks under the Jukes-Cantor model.bioRxiv 2025.04.18.649493, 2025

    2025

  2. [37]

    Phylogenetic network models as graphical models.arXiv preprint arXiv:2507.23056, 2025

    Seth Sullivant. Phylogenetic network models as graphical models.arXiv preprint arXiv:2507.23056, 2025. ON FULL IDENTIFIABILITY OF PHYLOGENETIC NETWORKS 27

    arXiv 2025

  3. [1]

    Beyond level-1: identifiability of a class of galled tree-child networks.Bulletin of Mathematical Biology, 87(11):1–26, 2025

    Elizabeth S Allman, C´ ecile An´ e, Hector Ba˜ nos, and John A Rhodes. Beyond level-1: identifiability of a class of galled tree-child networks.Bulletin of Mathematical Biology, 87(11):1–26, 2025

    2025

  4. [2]

    Identifiability of level-1 species networks from gene tree quartets.Bulletin of Mathematical Biology, 86(9):110, 2024

    Elizabeth S Allman, Hector Ba˜ nos, Marina Garrote-Lopez, and John A Rhodes. Identifiability of level-1 species networks from gene tree quartets.Bulletin of Mathematical Biology, 86(9):110, 2024

    2024

  5. [3]

    The tree of blobs of a species network: identifiability under the coalescent.Journal of Mathematical Biology, 86, 2023

    Elizabeth S Allman, Hector Ba˜ nos, Jonathan D Mitchell, and John A Rhodes. The tree of blobs of a species network: identifiability under the coalescent.Journal of Mathematical Biology, 86, 2023

    2023

  6. [4]

    Phylogenetic invariants

    Elizabeth S Allman and John A Rhodes. Phylogenetic invariants. In Olivier Gascuel and Mike Steel, editors,Reconstructing Evolution. Oxford University Press, 2007. 26 ON FULL IDENTIFIABILITY OF PHYLOGENETIC NETWORKS

    2007

  7. [5]

    Phylogenetic ideals and varieties for the general Markov model.Advances in Applied Mathematics, 40:127–148, 2008

    Elizabeth S Allman and John A Rhodes. Phylogenetic ideals and varieties for the general Markov model.Advances in Applied Mathematics, 40:127–148, 2008

    2008

  8. [6]

    Anomalous networks under the multispecies coalescent: theory and prevalence.Journal of Mathematical Biology, 88(3), 2024

    C´ ecile An´ e, John Fogg, Elizabeth S Allman, Hector Ba˜ nos, and John A Rhodes. Anomalous networks under the multispecies coalescent: theory and prevalence.Journal of Mathematical Biology, 88(3), 2024

    2024

Show all 38 references
  1. [7]

    Identifying species network features from gene tree quartets under the coalescent model.Bulletin of Math- ematical Biology, 81(2):494–534, 2019

    Hector Ba˜ nos. Identifying species network features from gene tree quartets under the coalescent model.Bulletin of Math- ematical Biology, 81(2):494–534, 2019

    2019

  2. [8]

    Statistical learning with phylogenetic network invariants

    Travis Barton, Elizabeth Gross, Colby Long, and Joseph Rusinko. Statistical learning with phylogenetic network invariants. Bulletin of the Society of Systematic Biologists, 4(1), 2026

    2026

  3. [9]

    Springer-Verlag, 1998

    Jacek Bochnak, Michel Coste, and Marie-Fran¸ coise Roy.Real Algebraic Geometry, volume 36 ofA Series of Modern Surveys in Mathematics. Springer-Verlag, 1998

    1998

  4. [10]

    Group-based phylogenetic models on 3-sunlet networks.Bulletin of Mathematical Biology, 87, 2025

    Shelby Cox, Elizabeth Gross, and Samuel Martin. Group-based phylogenetic models on 3-sunlet networks.Bulletin of Mathematical Biology, 87, 2025

    2025

  5. [11]

    The Pfaffian structure of CFN phylogenetic networks.Journal of Mathematical Biology, 92(1), 2026

    Joseph Cummings, Elizabeth Gross, Benjamin Hollering, Samuel Martin, and Ikenna Nometa. The Pfaffian structure of CFN phylogenetic networks.Journal of Mathematical Biology, 92(1), 2026

    2026

  6. [12]

    Semialgebraic conditions for identifying triangles in phylogenetic net- works.arXiv preprint arXiv:2606.26673, 2026

    Bryan Currie, Aviva K Englander, Jose A Esparza-Lozano, Elizabeth Gross, Max Hill, Colby Long, Devon Olds, Kawika O’Connor, Udani Ranasinghe, and Christin Sum. Semialgebraic conditions for identifying triangles in phylogenetic net- works.arXiv preprint arXiv:2606.26673, 2026

    2026 arXiv

  7. [13]

    On the ideals of equivariant tree models.Mathematische Annalen, 344:619–644, 2009

    Jan Draisma and Jochen Kuttler. On the ideals of equivariant tree models.Mathematische Annalen, 344:619–644, 2009

    2009

  8. [15]

    Phylogenetic algebraic geometry

    Nicholas Eriksson, Kristian Ranestad, Bernd Sturmfels, and Seth Sullivant. Phylogenetic algebraic geometry. In Ciro Cilliberto, Anthony V. Geramita, Brian Harbourne, Rosa Maria Mir´ o-Roig, and Kristian Ranestad, editors,Projective varieties with unexpected properties. de Gruy...

    2005

  9. [16]

    Invariants of some probability models used in phylogenetic inference.The Annals of Statistics, 21(1):355–377, 1993

    Steven N Evans and T P Speed. Invariants of some probability models used in phylogenetic inference.The Annals of Statistics, 21(1):355–377, 1993

    1993

  10. [17]

    Reconstructing semi-directed level-1 networks using few quarnets.Journal of Computer and System Sciences, 152, 2025

    Martin Frohn, Niels Holtgrefe, Leo van Iersel, Mark Jones, and Steven Kelk. Reconstructing semi-directed level-1 networks using few quarnets.Journal of Computer and System Sciences, 152, 2025

    2025

  11. [18]

    Bounds on the sequence length sufficient to reconstruct binary level-1 phylogenetic networks under the CFN model.Annals of Combinatorics, 2026

    Martin Frohn, Niels Holtgrefe, Leo van Iersel, Mark Jones, and Steven Kelk. Bounds on the sequence length sufficient to reconstruct binary level-1 phylogenetic networks under the CFN model.Annals of Combinatorics, 2026

    2026

  12. [19]

    Dimensions of level-1 group-based phylogenetic networks.Bulletin of Mathematical Biology, 86, 2024

    Elizabeth Gross, Robert Krone, and Samuel Martin. Dimensions of level-1 group-based phylogenetic networks.Bulletin of Mathematical Biology, 86, 2024

    2024

  13. [20]

    Distinguishing phylogenetic networks.SIAM Journal on Applied Algebra and Geometry, 2(1):72–93, 2018

    Elizabeth Gross and Colby Long. Distinguishing phylogenetic networks.SIAM Journal on Applied Algebra and Geometry, 2(1):72–93, 2018

    2018

  14. [21]

    Distinguishing level-1 phylogenetic networks on the basis of data generated by Markov processes.Journal of Mathematical Biology, 83, 2021

    Elizabeth Gross, Leo van Iersel, Remie Janssen, Mark Jones, Colby Long, and Yukihiro Murakami. Distinguishing level-1 phylogenetic networks on the basis of data generated by Markov processes.Journal of Mathematical Biology, 83, 2021

    2021

  15. [22]

    Michael D Hendy and David Penny. Complete families of linear invariants for some stochastic models of sequence evolution, with and without the molecular clock assumption.Journal of Computational Biology, 3(1):19–31, 1996

    1996

  16. [23]

    Identifiability in phylogenetics using algebraic matroids.Journal of Symbolic Computation, 104:142–158, 2021

    Benjamin Hollering and Seth Sullivant. Identifiability in phylogenetics using algebraic matroids.Journal of Symbolic Computation, 104:142–158, 2021

    2021

  17. [24]

    Distinguishing phylogenetic level-2 networks with quartets and inter-taxon quartet distances.Bulletin of Mathematical Biology, 87(12):1–35, 2025

    Niels Holtgrefe, Elizabeth S Allman, Hector Ba˜ nos, Leo van Iersel, Vincent Moulton, John A Rhodes, and Kristina Wicke. Distinguishing phylogenetic level-2 networks with quartets and inter-taxon quartet distances.Bulletin of Mathematical Biology, 87(12):1–35, 2025

    2025

  18. [25]

    Characterizing semi-directed phylogenetic networks and their multi-rootable variants.Theory in Biosciences, 145(1), 2026

    Niels Holtgrefe, Katharina T Huber, Leo van Iersel, Mark Jones, and Vincent Moulton. Characterizing semi-directed phylogenetic networks and their multi-rootable variants.Theory in Biosciences, 145(1), 2026

    2026

  19. [26]

    When are quarnets sufficient to reconstruct semi-directed phylogenetic networks?Bulletin of Mathematical Biology, 87(10), 2025

    Katharina T Huber, Leo van Iersel, Mark Jones, Vincent Moulton, and Leonie Veenema-Nipius. When are quarnets sufficient to reconstruct semi-directed phylogenetic networks?Bulletin of Mathematical Biology, 87(10), 2025

    2025

  20. [27]

    Cambridge University Press, 2010

    Daniel H Huson, Regula Rupp, and Celine Scornavacca.Phylogenetic Networks. Cambridge University Press, 2010

    2010

  21. [28]

    Evolution of protein molecules

    Thomas H Jukes and Charles R Cantor. Evolution of protein molecules. In H N Munro, editor,Mammalian Protein Metabolism, chapter 24. Academic Press, 1969

    1969

  22. [29]

    A simple method of estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences.Journal of Molecular Evolution, 16:111–120, 1980

    Motoo Kimura. A simple method of estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences.Journal of Molecular Evolution, 16:111–120, 1980

    1980

  23. [30]

    Estimation of evolutionary distances between homologous nucleotide sequences.Proceedings of the National Academy of Sciences of the United States of America, 78:454–458, 1981

    Motoo Kimura. Estimation of evolutionary distances between homologous nucleotide sequences.Proceedings of the National Academy of Sciences of the United States of America, 78:454–458, 1981

    1981

  24. [31]

    Inference of phylogenetic networks from sequence data using composite likelihood.Systematic Biology, 74(1):53–69, 2025

    Sungsik Kong, David L Swofford, and Laura S Kubatko. Inference of phylogenetic networks from sequence data using composite likelihood.Systematic Biology, 74(1):53–69, 2025

    2025

  25. [32]

    Algebraic invariants for inferring 4-leaf semi- directed phylogenetic networks.Systematic Biology, 75(4):657–672, 2026

    Samuel Martin, Niels Holtgrefe, Vincent Moulton, and Richard M Leggett. Algebraic invariants for inferring 4-leaf semi- directed phylogenetic networks.Systematic Biology, 75(4):657–672, 2026

    2026

  26. [33]

    Secant varieties of Segre-Veronese varieties.Algebra and Number Theory, 6(8):1817–1868, 2012

    Claudiu Raicu. Secant varieties of Segre-Veronese varieties.Algebra and Number Theory, 6(8):1817–1868, 2012

    2012

  27. [34]

    Identifying circular orders for blobs in phylogenetic networks

    John A Rhodes, Hector Banos, Jingcheng Xu, and C´ ecile An´ e. Identifying circular orders for blobs in phylogenetic networks. Advances in Applied Mathematics, 163 Part A, 2025

    2025

  28. [35]

    Toric ideals of phylogenetic invariants.Journal of Computational Biology, 12(4):457– 481, 2005

    Bernd Sturmfels and Seth Sullivant. Toric ideals of phylogenetic invariants.Journal of Computational Biology, 12(4):457– 481, 2005

    2005

  29. [36]

    American Mathematical Society, 2018

    Seth Sullivant.Algebraic statistics, volume 194 ofGraduate Studies in Mathematics. American Mathematical Society, 2018

    2018

  30. [38]

    Lie Markov models.Journal of Theoretical Biology, 298:16–31, 2012

    Jeremy G Sumner, Jes´ us Fern´ andez-S´ anchez, and Peter D Jarvis. Lie Markov models.Journal of Theoretical Biology, 298:16–31, 2012. 28 ON FULL IDENTIFIABILITY OF PHYLOGENETIC NETWORKS AppendixA.Further Proofs In this appendix we give proofs left out of the main document, an...

    2012

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Reviewed August 2, 2026 · model on record in the stance chip above.