Raw Strongly Tree-Child Level-2 Species Networks from Metric Genealogy Laws under the NMSC: Signed pair co-occupancy, rooted graph reconstruction, and demographic fibers

Abstract

We classify the observable rooted graph and parameters of finite strict temporally separated binary strongly tree-child species networks of level at most two under the independent-lineage Network Multispecies Coalescent (NMSC), with a fixed known surjective taxon sampling map. Equality of complete normalized rooted metric-genealogy laws is equivalent, pointwise on the full positive domain, to an explicit quotient generated by eight move types.

The proof recovers current pair co-occupancy from the analytic germ of a finite positive Kingman mixture. Signed jumps reveal every tree event and every pair-exposed reticulation. Descendant clusters, chronology, tree-childness and level-two shortcut completion then reconstruct the retained graph, including arbitrary decorations. Deterministic witnesses and actual three-lineage merger densities recover oriented inheritance weights and the entire finite demographic rate words on exposed populations. Scalar propagation through ancestral reticulations links both incoming arms, including equal rates and inheritance one-half. The indexed collection of all genealogy marginals on at most three sampled lineages suffices for the complete quotient.

A reticulation with one descendant sampled lineage retains its two typed first-exposure outlets and route weights, while its internal clock and singly occupied demographic words are unobservable. Sister-transfer direction is invisible exactly when both sides are sample unary; the corresponding reversal transfers the whole exposed donor rate word. Together with presentation suppression, equal-rate composition, unary-corridor replacement, parent exchange and labelled isomorphism, these responses exhaust the quotient. Exact hidden-clock and rank-one nonbridge examples explain essential obstructions. A separate graph argument derives the five cores and twelve minimum strong repairs.

The companion study transfers the classification to all indexed ordered locus laws with known boundaries, jointly with the stationary distribution and trace-normalized shared irreducible stationary reversible substitution generator. The reconstruction uses observable current-grouping scalars throughout.

The idea in plain English

A species history can contain both branching and gene flow between branches. Different genetic lineages can therefore trace different family trees through the same species network. The question here is what the complete probability distribution of those genealogies could reveal about the underlying history.

For a restricted class of networks, the paper classifies exactly which changes leave that distribution unchanged. It claims recovery of the retained rooted graph, inheritance probabilities and the histories of coalescence rates in populations that can contain two sampled ancestral lineages.

The distinction between visible and invisible history is central. An event carrying only one sampled lineage can retain identifiable route probabilities while its internal timing remains invisible. In a sister-transfer triangle, direction is ambiguous when each side supplies only one sampled lineage; additional exposure can distinguish the directions. These are statements about ideal probability laws under the model, not guarantees for a finite genetic dataset.

The precise result

The main theorem considers finite rooted binary strongly tree-child networks of level at most two under the independent-lineage Network Multispecies Coalescent. Samples are contemporaneous, and the known sampling map is surjective onto the terminal taxa. Finite population durations and piecewise-constant Kingman pair rates are positive, inheritance weights lie strictly between zero and one, and distinct internal biological events have distinct ages. The above-root population eventually has a positive constant coalescence rate.

Strongly tree-child means that the semi-directed reduction is tree-child under every admissible rooting, which is stronger than checking the supplied rooting alone. Level at most two limits each blob to two reticulations; it does not limit the entire network to two reticulations. Raw reticulations have distinct parents.

Two such models have the same complete normalized rooted metric-genealogy law exactly when they agree under eight specified equivalence moves, Q1–Q8. These include harmless presentation changes, parent exchange, replacement of unobservable single-lineage corridors and two-outlet routing fans, and reversal of a bi-unary sister triangle with its entire exposed donor rate history transferred.

The indexed collection of every labelled genealogy marginal on at most three sampled lineages suffices for this quotient. The result is pointwise on the stated positive domain, including equal population rates and inheritance probability one-half. It concerns the observable equivalence class, not unique recovery of every raw event and parameter.

Through the companion paper, the classification also applies to all indexed ordered locus laws with known boundaries under a shared irreducible stationary reversible substitution generator. This additionally identifies the stationary distribution and trace-normalized generator; physical time and rate retain the common scale ambiguity.

The companion’s finite-determination theorem further gives a finite determining locus length on each fixed finite union of compact semianalytic normalized finite-presentation charts in this class. It provides no effective practical sequence-length bound.

Why the result holds

On a short interval with no population boundary, pair survival is a finite sum of positive exponentials plus a constant. The constant corresponds to ancestral lineages in separate populations. Exact local analytic information therefore determines their current co-occupancy probability without observing their routes.

A tree join produces a positive co-occupancy jump; independent splitting at an exposed reticulation produces a negative one. With distinct biological event times, these jumps reveal descendant clusters and chronology. Tree-child witnesses and the level-two graph restrictions then determine the retained graph up to the stated sister-triangle ambiguity.

Ordinary population rates come from a merger density divided by co-occupancy. Three-lineage merger densities distinguish incoming reticulation arms; forward propagation of already recovered parameters recovers the remaining arm histories. These are observable scalar calculations, not measurements of historical route labels.

Single-lineage regions cannot contain a coalescent merger. Their observable effect reduces to how they deliver a lineage to its first exposed outlets. Keeping those weighted outlets while discarding invisible internal clocks explains why the theorem identifies an equivalence class.

Scope and limitations

This is an unrefereed preprint developed with extensive generative-AI assistance. The author is responsible for its claims and welcomes independent expert review.

The theorem requires the strong tree-child, binary, level-at-most-two, strict positivity, temporal-separation and sampling hypotheses. It does not classify arbitrary reticulate species histories, simultaneous biological events or limiting zero-duration or zero-rate models.

The observations are exact rooted metric-genealogy probability laws, including merger times. Recovering analytic germs and their continuations is a theoretical identifiability argument. No sample-size guarantee, noise-stability bound or efficient practical estimator is supplied.

Some clocks and rate histories are intrinsically unobservable when only one sampled lineage traverses a population. Sister-triangle reversal can also preserve the law when both sides are sample unary. All claims about recovered demographics are modulo the stated equivalence moves.

The small-marginal statement requires all relevant labelled pairs and triples. It does not say that one triple, three nucleotide sites or three observed genealogies determine a network.

The sequence-law corollary relies on the companion preprint and its additional model assumptions. Unknown locus boundaries, recombination, nonreversible substitution and arbitrary demographic mixtures are outside this result. Finite computational checks support the arguments within their declared domains and do not prove the universal theorem.

Status and AI use

This is an unrefereed preprint developed with extensive generative-AI assistance. The author is responsible for its claims and welcomes independent expert review.

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Collaborate on this result

I welcome substantive collaboration on an independent review of the probability and graph arguments, particularly the recovery of latent co-occupancy from observable genealogy laws and the reconstruction of demographic histories, or on developing a journal submission.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Raw Strongly Tree-Child Level-2 Species Networks from Metric Genealogy Laws under the NMSC: Signed pair co-occupancy, rooted graph reconstruction, and demographic fibers (Version 1.0.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.22729545

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Background and prior work

• Alec Kriebel, Direct Sequence Laws under the Network Multispecies Coalescent (2026): The companion preprint supplies the sequence-law to normalized metric-genealogy equivalence used by this paper. The graph classification and its sequence corollary therefore have different dependency chains.
• Marc Hellmuth, David Schaller and Peter F. Stadler, Clustering systems of phylogenetic networks (2023): Develops the relationship between descendant clusters and network structure. This is background for the tree-child cluster arguments; clusters alone do not identify arbitrary networks.
• Ziheng Yang and Tomáš Flouri, Estimation of cross-species introgression rates using genomic data despite model unidentifiability (2022): Earlier work analyzes parameter ambiguities and sampling in coalescent introgression models. The present paper should be read as an exact quotient for its larger specified network class, not as the first observation of introgression nonidentifiability.
• Yuttapong Thawornwattana and colleagues, Inferring the Direction of Introgression Using Genomic Sequence Data (2023): Already explains the sister-species direction ambiguity with one sequence per species and the value of additional sampling. Here that phenomenon is incorporated into a graph-and-demographic classification with finite rate histories.
• Elizabeth S. Allman, Cécile Ané, Hector Baños and John A. Rhodes, Beyond Level-1: Identifiability of a Class of Galled Tree-Child Networks (2025): Establishes topology identifiability for restricted galled tree-child semi-directed networks, including networks of arbitrary level, from suitable data such as quartet concordance factors. Its network class and recovered information differ from this metric-genealogy quotient.
• Niels Holtgrefe and colleagues, Distinguishing Phylogenetic Level-2 Networks with Quartets and Inter-Taxon Quartet Distances (2025): Characterizes a canonical form for outer-labelled planar galled semi-directed level-2 networks using displayed quartets. The present result uses richer metric-genealogy laws and retains rooted numerical information within its own restrictions.
• Aviva K. Englander and colleagues, Identifiability of Phylogenetic Level-2 Networks under the Jukes-Cantor Model (2025): Gives generic identifiability of triangle-free strongly tree-child semi-directed level-2 topology under a Markov DNA-evolution model. This is relevant prior level-2 identifiability work; its model and topology target differ from the NMSC demographic classification here.
• Jari Brits, Niels Holtgrefe, Leo van Iersel and Samuel Martin, On Tree-Network Distinguishability and Full Identifiability of Phylogenetic Networks (2026): Studies pointwise level-1 topology identifiability, modulo triangle redirection, under JC, K2P and K3P Markov models. It is further context for avoiding a broad claim to the first pointwise network-identifiability result.