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arxiv: 2606.26673 · v1 · pith:QIUBYJXDnew · submitted 2026-06-25 · 🧬 q-bio.PE

Semialgebraic Conditions for Identifying Triangles in Phylogenetic Networks

Bryan Currie , Aviva K. Englander , Jose A. Esparza-Lozano , Elizabeth Gross , Max Hill , Colby Long , Devon Olds , Kawika O'Connor
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Udani Ranasinghe Christin Sum
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Pith reviewed 2026-06-26 02:18 UTC · model grok-4.3

classification 🧬 q-bio.PE
keywords phylogenetic networksJukes-Cantor modelsemialgebraic setsidentifiabilityhybridizationtrianglessite-pattern probabilitiesevolutionary models
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The pith

Three Jukes-Cantor network models with embedded triangles produce overlapping but distinct full-dimensional sets of site-pattern probabilities.

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

The paper supplies a complete semialgebraic description of the probability distributions generated by three distinct 3-leaf phylogenetic network models under the Jukes-Cantor substitution process, each containing an embedded triangle. It establishes that any two models intersect in a full-dimensional region of the space of site-pattern probability distributions and that their set differences are likewise full-dimensional. This shows the models are algebraically indistinguishable by polynomial invariants yet are not the same set and therefore not identifiable from data. The result supplies a direct biological reading that the signal of a hybridization event can be detected at first but fades until the orientation of the triangle edges becomes impossible to recover.

Core claim

The three 3-leaf Jukes-Cantor phylogenetic network models with embedded triangles admit a complete semialgebraic description. For any pair of these models, both the intersection and the set differences consist of full-dimensional regions in the space of site-pattern probability distributions. Consequently the models are algebraically indistinguishable, not identical, and not identifiable or generically identifiable.

What carries the argument

The semialgebraic sets (regions cut out by polynomial equalities and inequalities) that exactly describe the site-pattern probability distributions for each of the three network models.

Load-bearing premise

Resolving the identifiability question for the three 3-leaf base cases is enough to settle identifiability for arbitrary networks that merely contain embedded triangles.

What would settle it

An explicit probability vector that satisfies the semialgebraic inequalities for exactly one of the three models, or a dimension calculation showing that any intersection or difference has dimension strictly less than the ambient space.

Figures

Figures reproduced from arXiv: 2606.26673 by Aviva K. Englander, Bryan Currie, Christin Sum, Colby Long, Devon Olds, Elizabeth Gross, Jose A. Esparza-Lozano, Kawika O'Connor, Max Hill, Udani Ranasinghe.

Figure 1
Figure 1. Figure 1: The three 3-leaf level-one triangle networks, each rooted on one of the non-hybrid leaf edges. reticulation edges are length ϵ > 0 and represent a near instantaneous hybridization event. For this example, we assume half of the genetic information is inherited along each edge. If ℓ is sufficiently large, then the sites observed at leaves 2 and 3 are nearly independent. If it is also the case that s is very … view at source ↗
Figure 2
Figure 2. Figure 2: Left: The network N1 from [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The three semi-directed networks corresponding to models M1,M2, and M3, with associated Fourier edge parameters. We begin by noting that because the Fourier transform gives a linear change of coordinates, for the network model, each qω is parameterized by a convex combination of the parameter￾izations for the displayed trees in Fourier coordinates. For example, if we delete the edge labeled a6 with probabi… view at source ↗
Figure 4
Figure 4. Figure 4: Networks (A)-(D) show four different choices of branch lengths for N1 that all give rise to the same site-pattern probability distribution. The networks are rooted along edge e5 for display, but as noted above, the location of the root is not identifiable. Branch lengths, measured in expected number of mutations per site, are displayed in the figure as horizontal distances. we can construct a new set of pa… view at source ↗
Figure 5
Figure 5. Figure 5: Venn diagram showing the percentage of points in ∆4 which belong to the regions of intersections of the models M1,M2, and M3. The vast majority of simplex points (96.2%) do not correspond to any of the three network models [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The proportion of networks in which the hybrid node is distin￾guishable as a function of δ, for N1 with branch lengths t1, . . . , t6 iid∼ unif(0, m). 3.5. Implications for Network Inference in Practice. The previous section shows that the three models overlap substantially, especially when considering biologically relevant pa￾rameters. Here we elaborate on some of the implications of the distinguishabilit… view at source ↗
Figure 7
Figure 7. Figure 7: A 5-leaf network with branch lengths in expected number of mu￾tations per site [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The network in [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Left: The 3-leaf phylogenetic network N1 with root placed on the edge corresponding to the parameter a5. Right: The proportion of such networks for which the hybrid node is distinguishable, assuming the intervals h1, h2−h1, h3−h2 and h4−h3 are drawn uniformly at random from the interval (0, .5). See Appendix C. Theorem 3.7. Under the ghost-lineage scenario shown in [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A 3-leaf phylogenetic network rooted on edge 3 at time h4. The network exhibits a 3-cycle arising from ghost lineages that diverged from species 2 at times h3 and h2 and later hybridized at time h1. Note that a3 is the Fourier parameter of leaf 3 when the root is suppressed. 3.6. Applications to Biological Data. In this section, we present two examples from the literature where a phylogenetic network is i… view at source ↗
Figure 11
Figure 11. Figure 11: Network from van der Heijden et al. (2025) on three species of neotropical butterflies with branch lengths converted to expected number of mutations per site. thresholding, while constraining the number of hybridizations to one. Our code is available in the supplemental materials in src/rhizoplaca/. The top-scoring network is shown in [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: ; branch lengths were converted from coalescent units to expected number of muta￾tions per site using the population mutation parameter θ = 2 × 10−3 , which roughly accords with estimates found by Leavitt et al. (2013). This network appears to closely match the network obtained in (Keuler et al., 2020, [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
read the original abstract

An important consideration for a model-based method of phylogenetic network inference is the identifiability of the network parameter of the model. A recurring theme in previous works exploring this issue is that it is often difficult to identify the orientation of edges in a triangle of the network. In fact, it has been shown that for some models it is impossible to determine the orientation of triangle edges utilizing the standard algebraic technique of phylogenetic invariants. In this work, we consider one such model with a Jukes-Cantor site-substitution process and no coalescence. We give a complete semialgebraic description of three, 3-leaf Jukes-Cantor phylogenetic network models with embedded triangles. By describing these base cases, we resolve several questions about the identifiability of networks with embedded triangles. We show that for any pair of models, the intersection and set differences of the models are full-dimensional regions of the space of site-pattern probability distributions. Thus, despite being algebraically indistinguishable, these network models are not identical, nor are they identifiable (or generically identifiable). Our results also yield a straightforward biological interpretation--that the signal from a hybridization event may be immediately detectable but decays over time until it is impossible to identify the orientation of edges in the triangle of a network.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript computes complete semialgebraic descriptions of three 3-leaf Jukes-Cantor phylogenetic network models with embedded triangles. It shows that pairwise intersections and set differences of these models are full-dimensional subsets of the site-pattern probability simplex. The authors conclude that the models are algebraically indistinguishable yet distinct and non-identifiable (nor generically identifiable), and that the 3-leaf base cases thereby resolve identifiability questions for arbitrary networks containing embedded triangles.

Significance. If the semialgebraic descriptions are correct and the generalization to larger networks is justified, the work supplies a concrete algebraic distinction between models that standard phylogenetic invariants cannot separate. The full-dimensionality results and the biological interpretation of hybridization-signal decay constitute a useful contribution to the literature on network identifiability.

major comments (1)
  1. [Abstract and final paragraph] Abstract and final paragraph: the claim that describing the three 3-leaf base cases resolves identifiability questions for arbitrary networks with embedded triangles is not accompanied by an explicit reduction argument (e.g., a marginalization lemma, an embedding of the 3-leaf distributions, or a proof that any triangle-orientation ambiguity in a general network projects onto one of the three base cases).
minor comments (1)
  1. The manuscript would benefit from displaying the explicit polynomials or inequalities that define each semialgebraic set, together with the verification steps used to confirm completeness and full-dimensionality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and final paragraph] Abstract and final paragraph: the claim that describing the three 3-leaf base cases resolves identifiability questions for arbitrary networks with embedded triangles is not accompanied by an explicit reduction argument (e.g., a marginalization lemma, an embedding of the 3-leaf distributions, or a proof that any triangle-orientation ambiguity in a general network projects onto one of the three base cases).

    Authors: We agree that an explicit reduction argument would strengthen the presentation of the generalization. Although the manuscript positions the 3-leaf cases as fundamental base cases (with the implication that marginals on any embedded triangle fall into one of the three models), we acknowledge the absence of a formal statement. In the revised manuscript we will add a short marginalization paragraph establishing that, for any larger network containing an embedded triangle, the induced distribution on those three leaves lies in one of the three semialgebraic sets we describe; the full-dimensionality of the intersections and differences then carries over directly to show that orientation ambiguity persists. This addition clarifies the claim without altering the core semialgebraic results or the 3-leaf theorems. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation rests on direct semialgebraic computation of 3-leaf base cases

full rationale

The paper computes complete semialgebraic descriptions of three specific 3-leaf Jukes-Cantor network models and verifies that pairwise intersections and symmetric differences are full-dimensional in the probability simplex. These steps are presented as explicit algebraic results rather than reductions to fitted parameters, self-citations, or definitional equivalences. No equations or load-bearing self-citations appear in the abstract or described claims, and the work is self-contained against external algebraic benchmarks. The assertion that 3-leaf cases resolve identifiability for arbitrary networks is an interpretive extension but does not render the central computations tautological by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all such items are therefore recorded as empty.

pith-pipeline@v0.9.1-grok · 5791 in / 1156 out tokens · 29750 ms · 2026-06-26T02:18:04.142864+00:00 · methodology

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Reference graph

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