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Identifiability of phylogenetic networks and quintet concordance factors

This paper presents an algorithm and Macaulay2 implementation for computing $n$ -tet Concordance Factors to demonstrate that quintet (5-taxon) data resolves identifiability issues for level-1 phylogenetic networks under the Network Multispecies Coalescent model that remain unsolvable using traditional quartet (4-taxon) methods.

Original authors: Joseph Cummings, Maize Curiel, Bryan Currie, Bryson Kagy, Udani Ranasinghe, John A. Rhodes

Published 2026-08-05

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CC BY 4.0

Original authors: Joseph Cummings, Maize Curiel, Bryan Currie, Bryson Kagy, Udani Ranasinghe, John A. Rhodes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are trying to solve a massive family mystery, but instead of just a straight line of parents and children, your family tree has secret shortcuts. Maybe two cousins had a baby together, or a great-grandparent had a child with someone from a completely different branch of the family. In the world of biology, this is called a "phylogenetic network." It's a map of how species are related, but unlike a simple tree, it has loops and crossings because nature loves to mix things up through hybridization and gene swapping. Scientists use these maps to understand evolution, but there's a catch: the data they have is often fuzzy. They can't see the exact history; they can only see the "echoes" of that history in the DNA of living species.

To make sense of these echoes, scientists look at small groups of four or five species at a time. They ask: "If I pick these four, what does their family tree look like?" These tiny snapshots are called "quartets" (four species) and "quintets" (five species). For a long time, scientists relied mostly on quartets. It was like trying to solve a jigsaw puzzle by only looking at pieces with four corners. It worked well for many things, but it left some big holes. Specifically, quartets couldn't tell you where the "root" of the tree was (which species is the oldest ancestor) or exactly how the loops in the network were formed. It was like looking at a shadow and not knowing if the object casting it was a cat or a dog. The big question was: if we look at quintets—groups of five instead of four—do we get enough extra information to finally solve the mystery of the root and the loops?

This paper is the story of a team of mathematicians and biologists who decided to upgrade the puzzle pieces from four to five. They built a powerful new computer tool (an algorithm) that can calculate the exact mathematical "fingerprints" of these five-species groups for any kind of network. Think of it as a super-calculator that can predict what a five-species family tree should look like if the family history involved a secret hybrid event. They then used this tool to test every possible shape of a five-species network that doesn't have the weirdest, most confusing loops (called 2-cycles).

What they found is that looking at five species at a time is a game-changer. Just like adding one more piece to a puzzle can suddenly reveal the whole picture, quintets provide enough information to identify the root of the network in almost every case. They also found that quintets can spot certain small loops (3-cycles) that quartets completely miss. However, the paper also shows that the mystery isn't fully solved yet. While they can pinpoint the root in many situations, there are still some tricky spots—specifically when the root is hidden inside a small loop—where even five species aren't enough to tell the difference between two different family histories. The authors are very clear that they haven't solved the problem for every possible network (especially those with the trickiest loops), but they have proven that moving from four to five species unlocks a whole new level of clarity. They didn't just guess; they used heavy-duty algebra and computer simulations to prove that these new "fingerprints" are mathematically distinct for different network shapes. So, while the door isn't fully open, they've definitely kicked it wide open, showing us that the key to understanding our evolutionary past might just be counting one more relative.

Technical Summary: Identifiability of Phylogenetic Networks and Quintet Concordance Factors

Problem Statement
Phylogenetic networks are increasingly necessary to model evolutionary histories involving hybridization, admixture, and lateral gene flow, which violate the assumptions of standard tree models. However, inferring these networks is computationally and statistically challenging. A fundamental prerequisite for any inference method is identifiability: the ability to uniquely determine network features (such as root location, cycle structures, and hybrid nodes) from the distribution of data generated by the model.

While the use of quartet Concordance Factors (CFs)—frequencies of 4-taxon topological relationships on gene trees—has facilitated inference under the Network Multispecies Coalescent (NMSC) model, it suffers from known non-identifiability issues. Specifically, quartet CFs cannot identify the root location of a network or certain features of small cycles (e.g., the presence of 2-cycles or the specific hybrid node within 3- or 4-cycles). The authors posit that quintet CFs (frequencies of 5-taxon relationships) may resolve these ambiguities, as they have been shown to identify root locations in phylogenetic trees. However, the algebraic complexity of computing quintet CFs for networks has hindered a systematic study of their identifiability properties.

Methodology
The paper employs a combination of algorithmic development, symbolic computation, and algebraic geometry to investigate the identifiability of level-1 phylogenetic networks using quintet CFs.

Symbolic Algorithm for CF Computation:
The authors develop a new recursive algorithm to compute symbolic formulas for the probability of any unrooted gene tree (or $m$ -taxon subtree) under the NMSC model.
- Unlike previous methods that enumerate all coalescent histories (which is computationally infeasible for symbolic derivation), this algorithm reduces the network and gene tree by considering partial coalescent histories within single edges.
- It handles both tree edges and hybrid edges. For hybrid edges, it recursively "splits" the hybrid node based on the partition of descendant lineages entering each parental edge, weighted by hybridization parameters ( $\lambda$ ).
- The algorithm transforms edge lengths into edge probabilities ( $x = e^{-\ell}$ ), ensuring that the resulting CFs are polynomial functions of the parameters.
- This algorithm is implemented in Macaulay2, allowing for the generation of symbolic parameterizations for arbitrary networks.
Algebraic Framework for Identifiability:
The authors treat the map from network parameters to CFs as a polynomial map. They analyze the vanishing ideals ( $I_N$ ) and algebraic varieties ( $V_N$ ) associated with these maps.
- Two networks are generically distinguishable if their associated varieties differ.
- Since computing full Gröbner bases for high-degree polynomials is often infeasible, the authors utilize algebraic matroids to distinguish varieties when full ideals cannot be computed.
- They employ homogenization techniques and multigraded implicitization to compute low-degree invariants (up to degree 4) and verify primality and dimension to confirm when full ideals have been found.
Scope of Analysis:
The study focuses on 5-taxon, rooted, binary, level-1 networks without 2-cycles. The exclusion of 2-cycles is necessary because their inclusion would result in an infinite number of network topologies for 5 taxa. The analysis covers all possible "trees of blobs" (the underlying tree structure of the network) for 5 taxa.

Key Contributions

New Algorithm: A recursive, symbolic algorithm for computing $m$ -tet Concordance Factors on arbitrary phylogenetic networks under the NMSC, implemented in Macaulay2.
Systematic Classification: An exhaustive classification of the algebraic varieties generated by quintet CFs for all 190 rooted level-1 networks on 5 taxa (excluding 2-cycles).
Identifiability Results: A detailed characterization of which network features become identifiable when moving from quartet to quintet CFs.

Results
The computational results yield specific identifiability statements for different network topologies:

Tree of Blobs (Fully Resolved):
- Quintet CFs can determine the presence of a 3-cycle at the central vertex of the tree of blobs.
- If a central 3-cycle exists, the presence of additional 3-cycles at the "cherry" positions (adjacent to specific taxon pairs) cannot be determined.
- The root location is generally identifiable, except when the root lies within a "cluster" (a 3-cycle and its pendant edges) where the root's exact position within that cluster cannot be distinguished.
- Hybrid nodes within 3-cycles are generally not identifiable, except for constraints imposed by the root location.
Tree of Blobs (4-multifurcation):
- The hybrid node of a 4-cycle is identifiable.
- Root location is identifiable in many cases, though some ambiguity remains when the root lies in a cluster containing a 3-cycle.
- The presence of a 3-cycle in a specific cluster is only detectable if the root lies within that cluster.
- Note: For certain root locations on the 4-cycle, the authors found numerical evidence (via pseudo-witness sets and homotopy continuation) suggesting distinguishability, but could not provide a rigorous algebraic proof due to the high degree of missing invariants (degrees 13, 29, or 62).
Tree of Blobs (5-multifurcation):
- Both the network root and hybrid node locations are identifiable.

Significance and Claims
The paper claims that quintet CFs provide strictly more information than quartet CFs regarding the root location and small-cycle structures (specifically 3-cycles) in level-1 networks.

Foundation for Inference: The authors emphasize that identifiability is a necessary prerequisite for statistical consistency. By establishing which features are identifiable, this work lays the theoretical foundation for future inference methods that utilize quintet CFs.
Limitations: The authors are modest about their claims, noting that:
- The results are specific to 5-taxon, level-1 networks without 2-cycles.
- Some non-identifiability results are based on low-degree invariants and matroids rather than full ideal computations, meaning they reflect indistinguishability with respect to currently known algebraic relations rather than definitive proofs of equivalence.
- The study does not address 2-cycles, which will be the subject of future work, as their inclusion creates an infinite class of networks.
- The analysis focuses on algebraic distinguishability; semialgebraic constraints (inequalities) required for real-parameter identifiability are not fully explored.

In summary, the work demonstrates that while quintet CFs resolve several ambiguities inherent in quartet-based methods (particularly regarding root placement), they do not eliminate all identifiability issues, especially concerning the precise location of roots within complex cycle clusters.

Technical Summary: Identifiability of Phylogenetic Networks and Quintet Concordance Factors

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