Progress on Constructing Phylogenetic Networks for Languages

Abstract

In 2006, Warnow, Evans, Ringe, and Nakhleh proposed a stochastic model (hereafter, the WERN 2006 model) of multi-state linguistic character evolution that allowed for homoplasy and borrowing. They proved that if there is no borrowing between languages and homoplastic states are known in advance, then the phylogenetic tree of a set of languages is statistically identifiable under this model, and they presented statistically consistent methods for estimating these phylogenetic trees. However, they left open the question of whether a phylogenetic network—which would explicitly model borrowing between languages that are in contact—can be estimated under the model of character evolution. Here, we establish that under some mild additional constraints on the WERN 2006 model, the phylogenetic network topology is statistically identifiable and we present algorithms to infer the phylogenetic network. We discuss the ramifications for linguistic phylogenetic network estimation in practice and suggest directions for future research.

Appendix

We restate and then sketch proofs for Theorems 1–3.

Theorem 1

Let N be a rooted phylogenetic network, and let characters evolve down N under the WERN 2023 model, and let Q be the output of QTC. Then every quartet tree in Q will be in Q(N_r). Furthermore, as the number of characters increase, with probability converging to 1, every quartet tree in Q(N_r) will appear in Q. Thus, QTC is a statistically consistent estimator of Q(N_r).

Proof

We begin by showing that every quartet tree placed in Q is also in Q(N_r). Recall that quartet tree uv|xy is included in Q if and only if some character α is found such that α(u) = α(v) ≠ α(x) = α(y) and the states α(u), α(x) are non-homoplastic. This character evolves down some tree T contained inside the network. Moreover, since the states exhibited at u, v, x, y are non-homoplastic, there is a path in T connecting u and v and another path connecting x and y and these two paths do not share any vertices. Hence, the quartet tree uv|xy is in Q(N_r).

We now show that in the limit, every quartet tree in Q(N_r) is also in Q. Let ab|cd be a quartet tree in Q(N_r). Hence, there is a rooted tree T contained in N that induces this quartet tree (when T is considered as an unrooted tree). With positive probability, a character will evolve down T. Without loss of generality, assume a and b are siblings in the rooted version of T, so that their least common ancestor, lca_T (a, b), lies strictly below the root of the tree T.

Since a and b are siblings, there is an edge e above lca_T (a, b) within T. It follows that the probability that a random character evolves down T, selecting a non-homoplastic state at the root, and then changing on e, but on no other edge in T, is strictly positive. Note that for any such character α, we have α(a) = α(b) and α(c) = α(d) where α(a) and α(b) are different and both are non-homoplastic states. In such a case, Q will include quartet tree ab|cd. Thus, in the limit as the number of characters increases, with probability converging to 1, Q will contain every quartet tree in Q(N_r).

Since in the limit Q ⊆ Q(N_r) and Q(N_r) ⊆ Q, it follows that Q = Q(N_r) with probability converging to 1.

Theorem 2

The QBTE (Quartet-Based Topology Estimation) method is statistically consistent for estimating the unrooted topology of the network N under the WERN 2023 model when the rooted network N is a level-1 network where all cycles have length at least six; furthermore, QBTE runs in polynomial time.

Proof

By Theorem 1, we have shown that as the number of characters increases, we can construct Q(N_r). Due to space constraints, we direct the reader to Warnow et al. (2023) https://arxiv.org/abs/2306.06298 for the rest of the proof.

Theorem 3

Let N be the true unrooted level-1 network and let C₀ denote the set of homoplasy-free phonological characters that exhibit both 0 and 1 at the leaves of N. Rooting N on any edge returned by Root-Network will produce a rooted network on which all characters in C₀ can evolve without homoplasy, and the edge containing the true location of the root will be in the output returned by Root-Network. Furthermore, when given the unrooted topology of the true phylogenetic network as input, Root-Network is a statistically consistent estimator of the root location under the assumption that the probability of homoplasy-free phonological characters is positive.

Proof

We sketch the proof due to space constraints. It is straightforward to verify that an edge is colored red for a character α if and only if subdividing the edge and labeling the introduced node by 0 for α makes α homoplastic on every tree contained within the network. Furthermore, it is not hard to see that if we root the network on any edge that remains green throughout Root-Network, then all characters in C₀ will be homoplasy-free. As a result, the first part of the theorem is established.

For the second part of the theorem, if the probability of homoplasy-free phonological characters is positive, then with probability converging to 1, for every edge in the true network, there is a character α that changes on the edge, but on no other edge; hence, α will be non-constant and homoplasy-free. Let e₁ and e₂ be the two edges incident to the root, and suppose the input set of characters contains α₁ and α₂ homoplasy-free characters that change on e₁ and e₂, respectively, then these two characters will mark as red every edge below e₁ and e₂. In the unrooted topology for N, the root is suppressed and edges e₁ and e₂ are merged into the same single edge, e. Hence, when Root-Network is applied to the unrooted topology for N, if characters α₁ and α₂ are in the input, then the only edge that is not colored red will be the edge e containing the suppressed root. In conclusion, since the probability of homoplasy-free phonological characters is strictly positive, as the number of such characters increase, the probability that every edge other than the root edge will be red will converge to 1. Thus, Root-Network will uniquely leave the single edge containing the suppressed root green, establishing that it is statistically consistent for locating the root in the network.