Beyond Level-1: Identifiability of a Class of Galled Tree-Child Networks

Abstract

Inference of phylogenetic networks is of increasing interest in the genomic era. However, the extent to which phylogenetic networks are identifiable from various types of data remains poorly understood, despite its crucial role in justifying methods. This work obtains strong identifiability results for large sub-classes of galled tree-child semidirected networks. Some of the conditions our proofs require, such as the identifiability of a network’s tree of blobs or the circular order of 4 taxa around a cycle in a level-1 network, are already known to hold for many data types. We show that all these conditions hold for quartet concordance factor data under various gene tree models, yielding the strongest results from 2 or more samples per taxon. Although the network classes we consider have topological restrictions, they include non-planar networks of any level and are substantially more general than level-1 networks — the only class previously known to enjoy identifiability from many data types. Our work establishes a route for proving future identifiability results for tree-child galled networks from data types other than quartet concordance factors, by checking that explicit conditions are met.

Introduction

The analysis of genomic data sets in recent years has led to the discovery of numerous instances of hybrid speciation or gene flow, with phylogenetic networks and admixture graphs increasingly used to describe evolutionary relationships (e.g., Linder and Rieseberg 2004; Mallet 2005; Noor and Feder 2006; DeRaad et al. 2022; Lopes et al. 2023; Yang et al. 2023; Nielsen et al. 2023; Maier et al. 2023; Ciezarek et al. 2024). Several inference methods used in these analyses have focused on the simplest class of networks, those of level 1, in which reticulations are sufficiently isolated from one another that the network shows only disjoint cycles joined by tree-like edges. While computational difficulties have been partially responsible for this focus, a lack of theoretical understanding of the extent to which more complex network classes are identifiable is also a barrier.

Network identifiability, which is the property that sufficiently large data sets produced in accord with a model allows for the network’s recovery in principle, is essential for valid statistical inference. For level-1 networks, identifiability has been proved (sometimes requiring mild restrictions) under various models and data types (Solís-Lemus and Ané 2016; Gross and Long 2018; Baños 2019; Allman et al. 2019; Gross et al. 2021; Allman et al. 2022; Xu and Ané 2023; Allman et al. 2024).

In this work, we extend our understanding of network identifiability substantially, to a large class of networks, those whose blobs are galled and tree-child. Informally, galled networks are those for which every reticulation lies in a cycle with no others, and a tree-child network is one in which every node has at least one child node that is not a reticulation. Despite the simple structure of these networks, they can be of arbitrary level, and need not be planar. The theoretical results here suggest that these networks should be a good “next step" class of networks on which developers of inference methods might focus.

In establishing our results, we consider quartet concordance factors (CFs) as input data. These are the proportions of gene trees displaying the various unrooted 4-taxon tree relationships for each subset of 4 taxa. Such CFs have formed the basis of a number of inference methods under the multispecies coalescent model, notably ASTRAL (Zhang et al. 2018) for the inference of species trees, and SNaQ (Solís-Lemus and Ané 2016) and NANUQ (Allman et al. 2019, 2025) for the inference of level-1 networks. Quartet information also underlies PhyNEST (Kong et al. 2024), although with site pattern frequencies as input. Quartet CFs are attractive for inference for several reasons, including providing a computational speedup over methods that use full gene trees. In addition, since quartet CFs capture only topological gene tree information, they offer robustness to variability of substitution rates across genes and lineages, to departures from a molecular clock, and to edge length estimation error in gene trees.

Figure 1 provides an informal example of our results which, together with previously established identifiability theorems (Allman et al. 2023, 2024; Rhodes et al. 2025), illustrate that many features of complicated networks are in fact identifiable from CFs. Under commonly-used models of gene tree generation on the species network $N_0$, with one sample per taxon, the tree of blobs T, in which each blob (a maximal connected subgraph with no cut edges) is contracted to a node is first identified. We then analyze each blob individually and are able to identify the full structure of the large (rightmost) blob of $N_0$, since it is galled, tree-child, and has no small cycles. The other blobs of $N_0$, left unresolved and depicted as colored circles in $G_1$, are either not fully identifiable or require additional assumptions to be so. For the two 3-blobs in T, shown with orange circles in $G_1$, by taking two samples per taxon, we can identify that the rightmost 3-blob is trivial and the central 3-blob is not. If a non-trivial 3-blob is assumed to be level-1, as in this case, its hybrid node is identifiable only for certain edge lengths. For the leftmost level-2 blob of $N_0$, since it is outer-labelled planar, the circular order of taxa around it can be identified as well as that taxon $a_3$ descends from a hybrid node. However, that blob’s internal structure cannot be identified with a single sample per taxon. Indeed, edge parameters on $N_1$ can be set (as in (Rhodes et al. 2025, Section 7.2)) such that $N_1$ is undistinguishable from $N_0$, both generating the same quartet concordance factors.

Fig. 1.

Example of a network $N_0$ (top, left) and some of its features identifiable from quartet concordance factors. T (bottom, left) is $N_0$’s tree of blobs. Since the largest (rightmost) blob of $N_0$ is ${\mathfrak{C}}_5$ (see Definition 7), that blob’s full topology is identifiable using one sample per taxon. This blob appears in $G_1$ (top, right), which shows the features of $N_0$ proved to be identifiable in this article, with large circles representing unresolved blobs of uncertain topology. As discussed in the main text, the central 3-blob can be detected as non-trivial and its hybrid node can sometimes be identified. In contrast, the left-most blob is not identifiable: $N_1$ (bottom, right) cannot be distinguished from $N_0$ using quartet concordance factors

After introducing terminology and requisite background results (Section 2) and introducing our new class of networks (Section 3), our arguments are structured into two parts. Following the statements of a few general assumptions on a network class, we give combinatorial arguments for network identifiability under these (or subsets of these) assumptions in Section 4. Then, in Section 5, we show that for quartet CF datas these basic assumptions hold under various models, leading to our main practical result in Theorem 5.7. A discussion in Section 6 concludes this work.

Phylogenetic Networks and Blobs

We use standard terminology for phylogenetic networks, as in (Steel 2016; Solís-Lemus and Ané 2016; Baños 2019; Ané et al. 2024), recalling it briefly in the next two subsections. We then define more specialized terms central to the work here, and establish some basic properties in the remaining subsections.

Rooted Networks

A rooted topological phylogenetic network $N^{+}$ on a set of taxa X is a finite connected rooted directed acyclic graph with vertices comprising a root, hybrid nodes, internal tree nodes and leaves, and edges which are either hybrid or tree edges. The root has in-degree 0. Leaves are of in-degree 1 and out-degree 0 and bijectively labeled by elements of X. Hybrid nodes have in-degree at least 2 and out-degree at least 1. The internal tree nodes make up the remaining nodes. By tree nodes, we mean all non-hybrid nodes. If a non-root node in $N^{+}$ has degree 3, then this node is binary. A hybrid node is bicombining if its in-degree is 2. An edge is hybrid or tree in accord with its child node. Any hybrid edge shares its child node with at least one other partner hybrid edge. An edge incident to a leaf is called pendant.

A network is binary if its root (if it has one) has degree 2 and all other non-leaf nodes are binary. A network $N^{+}$ is metric if each edge e is assigned a pair of parameters $(\ell (e),\gamma (e))$, where $\ell (e)\ge 0$ is an edge length with $\ell (e)>0$ for tree edges, and $\gamma (e)\in (0,1]$ is a hybridization or inheritance parameter, with the sum over partner edges equal to 1. Note that $\gamma (e)=1$ if e is a tree edge, and we require that $\gamma (e)<1$ if e is a hybrid edge.

We say a node or edge s is above or ancestral to another node or edge $s^{\prime }$ in $N^{+}$ (and $s^{\prime }$ is below s or a descendant of s) if there is a (possibly empty) directed path from s to $s^{\prime }$. An up-down path or trek in $N^{+}$ is an undirected path of edges joining nodes $u_1,u_2,\cdots ,u_n$ such that for some $i\in [n]$, the subpaths from $u_i$ to $u_1$ and from $u_i$ to $u_n$ are directed paths in $N^{+}$.

Let $N^{+}$ be a network on X and let $Y\subseteq X$. The least stable ancestor of Y on $N^{+}$, denoted $LSA(Y)$, is the lowest node through which all directed paths from the root to any taxon in Y must pass. The LSA network of $N^{+}$, is the network obtained from $N^{+}$ by deleting all edges and nodes strictly ancestral to $v=LSA(X)$, and rerooting at v. We say $N^{+}$ is an LSA network if $r=LSA(X)$.

Throughout this work we assume all rooted networks are LSA networks.

Semidirected Networks

The semidirected phylogenetic network $N^{-}$ of a (LSA) network $N^{+}$ is the graph obtained by undirecting all tree edges and suppressing its root if it has degree 2. The network $N^{+}$ is an example of a rooted partner of $N^{-}$ (Linz and Wicke 2023). A semidirected phylogenetic network may have more than one rooted partner, since many rootings of $N^{-}$ might be consistent with hybrid edge directions in $N^{-}$.

We use uv to denote an undirected edge between nodes u and v. When relevant, we describe directed edges as ‘from u to v’, as the notation (u, v) fails to distinguish parallel edges. An up-down path or trek in $N^{-}$ is a path in the fully undirected graph without any pair of consecutive edges directed from $p_1$ to h and from $p_2$ to h in $N^{-}$. As shown by Xu and Ané (2023), up-down paths in $N^{+}$ are in bijection with up-down paths in $N^{-}$. A semidirected path in $N^{-}$ is a path joining nodes $u_1,u_2,\cdots ,u_n$ such that each edge is either undirected, or directed from $u_i$ to $u_{i+1}$. An up-down cycle or trek cycle is a non-empty up-down path starting and ending at the same node, which is necessarily hybrid.

Removing all hybrid edges from $N^{-}$ results in connected components (called skeleton trees below), with exactly one of these, the root component, containing possible root locations (Maxfield et al. 2024). The set of edges where $N^{-}$ might be rooted consists of those edges in the root component and the hybrid edges incident to the root component. Note that, by definition, any edge not in the root component has the same direction in all rooted partners $N^{+}$.

We say that a node or edge s is above or ancestral to another node or edge $s^{\prime }$ in $N^{-}$ (and $s^{\prime }$ is below or a descendant of s) if s is above $s^{\prime }$ in every rooted partner of $N^{-}$. In particular, tree edges in non-root components of $N^{-}$ are all below at least one hybrid node.

For any rooted or semidirected graph G, the reduced graph of G is obtained from G by suppressing all degree-2 nodes (other than the root). In this work the reduced version of a graph G may be denoted by $\overline{G}$ or simply specified in words. For a rooted or semidirected network N, the induced network $N_Y$ of a network N is the network obtained from N by retaining only the up-down paths between all pairs of taxa in Y. Baños (2019) showed that the operations of inducing and semidirecting commute, that is, as reduced graphs ${(N_Y^{+})}^{-}={(N^{-})}_Y$.

For the remainder of this work, we focus on semidirected phylogenetic networks $N^{-}$, denoted for simplicity by N.

Skeletons and Blobs

Our arguments will require a number of special subgraphs of phylogenetic networks that we now define.

Definition 1

The unreduced skeleton forest of N is the graph obtained by removing all hybrid edges. The unreduced skeleton trees of N are the connected components of its unreduced skeleton forest. The unreduced root skeleton tree is the root component. The skeleton forest, skeleton trees and root skeleton tree of N are the corresponding reduced graphs. Finally, a skeleton tree is trivial if it only contains a single node, or a single edge.

Note that, by definition, each edge in a skeleton tree T is composed of one or more tree edges in N.

Skeleton trees of a phylogenetic network are not necessarily phylogenetic trees, as they may have unlabeled leaves. See, for example, the root skeleton tree of the network $N^{\prime }$ in Figure 2. Related concepts like tree-node forest and tree-node components were introduced by Gunawan et al. (2017), but differ from the definition here in that hybrid nodes are removed, and hence the child edges of those nodes are also deleted.

Fig. 2.

The semidirected network $N^{\prime }$ (top left) is unreduced with a degree-2 node marked with a dot. Its unreduced skeleton forest, obtained by removing hybrid edges (blue), has 3 non-trivial unreduced skeleton trees (orange). $N^{\prime }$ has 3 non-trivial blobs: $B_1$, $B_2$, and $B_3$. The blob $B_1$ is a level-1 3-blob with 2 boundary nodes, and $B_3$ is a 7-blob with 6 boundary nodes. $N^{\prime }$ is a 2-blob extension of N (center left). T (bottom left) is the tree of blobs for both N and $N^{\prime }$. The subnetwork of $N^{\prime }$ induced by $\{a_1,f,c_1,c_2,g_2,g_1,d\}$ is an extended bloblet with $B_3$ its internal blob. In the language of Section 3, $N_1$ and $N_3$ (right) are bloblets generated by $B_1$ and $B_3$ respectively. Both are non-binary, but in $N_3$ hybrid nodes are binary. $N_1$ is a galled, weakly (but not strongly) tree-child bloblet. $N_3$ is ${\mathfrak{C}}_k$ for $k=3,4,5$, with its unreduced root skeleton tree shown in brown

Lemma 2.1

If N is a network with k hybrid nodes, then the skeleton forest of N has $k+1$ skeleton trees.

Proof

Letting $N^{+}$ be a rooted partner of N with root $\rho$, pick a lowest hybrid node v in $N^{+}$ (and N). By deleting all v’s partner hybrid edges we obtain a graph with 2 connected components since all nodes below v remain connected to v, and all other nodes are connected to $\rho$ by some path. The first component has no hybrid nodes and is thus a tree, while the component containing $\rho$ has $k-1$ hybrid nodes. Repeating this process on the component containing $\rho$ until no hybrid nodes remain, gives $k+1$ components which are the unreduced skeleton trees. $\square$

Let G be an arbitrary graph. A blob of G is a maximal connected subgraph with no cut edges (that is, a 2-edge-connected component). A trivial blob is one consisting of a single node. Note that a non-trivial blob is a biconnected component (or block) if the network is binary, but otherwise may contain one or more blocks (Xu and Ané 2023).

A node in a blob is a boundary node if it is incident to one or more cut edges. A blob incident to exactly m cut edges is an m-blob. If a network is binary, a non-trivial m-blob has exactly m boundary nodes. In general, an m-blob has m or fewer boundary nodes, but any network with an m-blob must have at least m taxa. For examples, see Figure 2.

For a network N the unreduced tree of blobs is the tree obtained from N by contracting each of its blobs to a vertex. Since blobs are 2-edge connected components, this contraction is well-defined, and distinct blobs are contracted to distinct vertices in the unreduced tree of blobs. The tree of blobs T(N) of N is obtained by reducing its unreduced tree of blobs. See Figure 2 for an example.

Definition 2

A semidirected phylogenetic network $N^{\prime }$ is a 2-blob extension of a network N if N is obtained from $N^{\prime }$ by contracting some of its 2-blobs and suppressing the resulting degree-2 nodes.

If $N^{\prime }$ is a 2-blob extension of N, then $N^{\prime }$ has the same leaf set and the same tree of blobs T as N, and any node in T corresponds to the same blob in N and $N^{\prime }$ (see Figure 2).

Definition 3

A semidirected network N is a bloblet if, in addition to the trivial blobs of its leaves, it has a single additional blob that is an m-blob with $m\ge 2$. A network $N^{\prime }$ is an extended bloblet if it is a 2-blob extension of a bloblet N with $m\ge 3$. The internal blob of an extended bloblet is its unique m-blob with $m\ge 3$.

Figures 2 and 3 show examples of bloblets. The name “bloblet" was suggested by the term sunlet, which has been used in many works for graphs with a single blob that is a cycle in a binary network (a binary level-1 bloblet). Note that if N is a bloblet on n taxa, then its internal blob is an n-blob with $k\le n$ boundary nodes, and its tree of blobs is a star tree. The same holds when $N^{\prime }$ is an extended bloblet on $n\ge 3$ taxa: it has an internal n-blob and star tree of blobs. Finally, if a bloblet N’s internal blob is trivial, then N itself is a star tree.

Fig. 3.

Examples of bloblets. Left: A 4-blob B on a network that is not tree-child and not galled, with a below B’s lowest node $h_2$. The root skeleton tree, on the taxa $\{b,c,d\}$, is the star tree, so $h_2$ is augmenting. Right: A 5-blob B on a network that is tree-child but not galled. B’s unique lowest node $h_2$ is augmenting

Definition 4

A node v in a blob B of a semidirected N is a lowest node if, in every rooted partner $N^{+}$ of N, v has no proper descendants in B.

Note that every non-trivial blob has at least one lowest node, and that all lowest nodes are necessarily hybrid. This definition extends that of Rhodes et al. (2025) for rooted networks, where it was proved that any node without descendants in a rooted non-trivial blob must be hybrid. If v is a lowest node of a non-trivial blob B in N, then its descendant nodes and edges are the same across all of N’s rooted partners, and we denote v’s descendant leaves by L(v).

Basic Blob Structure

Definition 5

Let N be a semidirected network on taxon set X, with T its tree of blobs. Suppose v is a lowest node in a non-trivial blob B of N. Let ${\overline{T}}_{X\backslash L(v)}$ denote the reduced graph of $T_{X\backslash L(v)}$. We say that v links B if $N_{X\backslash L(v)}$’s tree of blobs is more resolved than ${\overline{T}}_{X\backslash L(v)}$. If the tree of blobs of $N_{X\backslash L(v)}$ is equal to ${\overline{T}}_{X\backslash L(v)}$, then v augments B.

Since any cut edge in N remains a cut edge in $N_{X\backslash L(v)}$ (if present in $N_{X\backslash L(v)}$), $N_{X\backslash L(v)}$’s tree of blobs must have all the edges from ${\overline{T}}_{X\backslash L(v)}$, and either be identical to or more resolved than ${\overline{T}}_{X\backslash L(v)}$. Therefore any lowest node in a non-trivial blob is either linking or augmenting. For examples, note that the unique hybrid node in a k-taxon sunlet network ($k\ge 5$) is a linking node, whereas $h_2$ in Figure 3 (right) is an augmenting node. Also, if a blob B has a linking node v, there must be at least 4 taxa in $X\backslash L(v)$, implying that X has $n\ge 5$ taxa. If an m-blob B with $m\ge 5$ has a single lowest node, then this node is not necessarily linking, as shown in Figure 3 (right).

Note that these definitions depart from those in Rhodes et al. (2025), where linking and augmenting lowest nodes are defined in rooted networks with at least two lowest nodes.

Lemma 2.2

Let N be a bloblet on a set X, with internal blob B. Then the tree of blobs of $N_{X\backslash \{x\}}$ is not a star tree if, and only if, x is below a linking node of B, and x is the only leaf below that node.

Proof

By hypothesis, the tree of blobs of N is a star tree. Let $x\in X$ with v its adjacent internal node, which is a boundary node of B, and let $T_{x^{-}}$ denote the tree of blobs of $N_{X\backslash \{x\}}$. If v is also adjacent to a second leaf, then $N_{X\backslash \{x\}}$ contains v and all edges and nodes of B, so $T_{x^{-}}$ is a star. Thus, we may hereafter assume that v is adjacent to a single leaf and that $L(v)=\{x\}$.

Assume first that v is a lowest node of B. Then, by definition, $T_{x^{-}}$ is a star if v is augmenting and $T_{x^{-}}$ is not a star if v is linking.

Now suppose that v is not a lowest node of B. Then in every rooted partner $N^{+}$, v has a descendant edge in B. If v is hybrid, this follows since v is not lowest. If v is a tree node, this follows since v is in B: If all its descendant edges in $N^{+}$ were not in B, then v would be incident to at most 1 edge in B, a contradiction.

To show $T_{x^{-}}$ is the star tree, we will show that (unreduced) $N_{X\backslash \{x\}}$ cannot have an internal cut edge. Suppose for the sake of contradiction that it does, and on a fixed partner $N^{+}$ rooted at some node $\rho$ of N, pick an edge e that becomes an internal cut edge in $N_{-}^{+}=N_{X\backslash \{x\}}^{+}$. Let s and t be the parent and child nodes of e. Removing e from $N_{-}^{+}$ leaves two connected components, $M_s$ containing s and $M_t$ containing t. Let $X_s$ be the taxa on $M_s$ and $X_t$ the taxa on $M_t$, so $X_s\bigsqcup X_t=X\backslash \{x\}$.

For use several times in the argument below we claim ($†$): For any taxon $a\in X_s$, no directed path in $N_{-}^{+}$ ending at a may pass through e. To see this, suppose there were such a path passing through e. Then truncating this path gives a path from t to a in $N^{+}$ that avoids e. But from t there is also a directed path in $M_t$ to some $b\in X_t$. After truncating at a lowest common node, these two paths combine to give an up-down path from a to b, which is therefore in $N_{-}^{+}$. But this shows $M_s$ and $M_t$ are connected in $N_{-}^{+}$ by an up-down path avoiding e, which contradicts that e is a cut edge of $N_{-}^{+}$.

Since e is not a cut edge in $N^{+}$, by Lemma 10 of Ané et al. (2024) there is an up-down cycle C in $N^{+}$ containing e. Pick such a C with lowest and highest nodes $\ell$ and h, respectively. There is some taxon below $\ell$, so first suppose x is. Then since x has v as its only parent, v is also below $\ell$. But since there is a child edge of v in the blob, v must also be above another taxon. Thus whether x is below $\ell$ or not, there is some other taxon b below $\ell$. In fact b must be in $X_t$, since if b were in $X_s$ there would be a directed path from h through the part of C containing e to $\ell$ and then to b, contradicting ($†$). Thus $\ell$ is above some $b\in X_t$.

Next consider a directed path avoiding e from the root $\rho$ to h then through part of C to $\ell$ and on to b. For any $a\in X_s$ choose a directed path from $\rho$ to a. By construction for the first, and by ($†$) for the second, neither of these paths pass through e. Truncating them at their lowest common node yields an up-down path from a to b which is therefore in $N_{-}^{+}$. But since e is not on this path between $M_s$ and $M_t$ this contradicts that e is a cut edge of $N_{-}^{+}$. $\square$

A final structure lemma we need is the following.

Lemma 2.3

Let N be binary bloblet on 3 taxa, with a non-trivial 3-blob. Then N has a level-1 subnetwork with a single 3-cycle and no 2-cycles. In other words, N has a 3-sunlet subnetwork.

Proof

We proceed by induction on the number of hybrid nodes in the 3-blob. For the base case, when the 3-blob has a single hybrid, N is a 3-cycle network and there is nothing to prove.

Assuming now that N’s 3-blob has more than one hybrid node, then there exists a lowest hybrid node with a descendant taxon, say a. Let $M=N_{\{b,c\}}$ be the subnetwork composed of all edges on up-down paths connecting the taxa b and c. Thus M has the form of a chain of 2-blobs (some possibly trivial) joined by cut edges. The funnel of a in N is all edges in up-down paths from a to M, terminating at the funnel’s attachment nodes on M, which in number are at least 2. If any attachment node v of a’s funnel is in a non-trivial 2-blob of M, then we may pick one path in the funnel from v to a; and removing from N all other funnel edges not on this path gives a subnetwork with 1 fewer hybrid node and a non-trivial 3-blob, and possibly some 2-blobs. By choosing some semidirected up-down path through each 2-blob and deleting all edges in 2-blobs not on these paths, we reduce to a network for which the inductive hypothesis applies.

Otherwise all attachment nodes are trivial 2-blobs of M, at which two cut edges of M join. Thus any up-down path in M between b and c contains all attachment nodes of a’s funnel. Picking two of these attachment nodes, v, w and retaining only a path from b to c and one path each from v, w to a in the funnel yields the desired subnetwork. $\square$

Galled and Tree-Child Semidirected Networks

A rooted phylogenetic network $N^{+}$ is tree-child, if every non-leaf node v has a child that is a tree node (Steel 2016). A semidirected network N is strongly tree-child (or simply tree-child) if all its rooted partners are tree-child, and weakly tree-child if at least one of its rooted partners is tree-child (Maxfield et al. 2024). Examples of semidirected networks that are strongly, weakly, or not tree-child are given in Figure 4. A sunlet with 3 or more leaves is strongly tree-child.

Fig. 4.

Examples of galled bloblets N, with root skeleton trees T shown in orange. Left: N is neither strongly nor weakly tree-child, with no taxa on T. Middle: N is weakly but not strongly tree-child, as some rooted partners are tree-child (root at the hybrid’s parent node or along a hybrid edge) and some are not (root along the edge incident to a). T has one taxon, a. Right: N is tree-child, but not ${\mathfrak{C}}_4$. T has taxa $\{a,c,d\}$. The reduced graph of the subnetwork $N_{\{a,c,d,b_1,b_2,g\}}$ (f omitted) is ${\mathfrak{C}}_4$

A (rooted or semidirected) network N is galled if, for every hybrid node h and every pair of partner hybrid edges e and $e^{\prime }$ sharing child h, there exists a cycle in N (considering edges as undirected) that contains e and $e^{\prime }$ and no other hybrid edges. Such a cycle is called a tree cycle. For example, the network $N_3$ in Figure 2 is galled, but neither of the networks in Figure 3 are galled. The term “galled network” should not be confused with “galled trees,” a class of networks now commonly referred to as level-1 networks (when binary).

In the next section, we prove that galled tree-child bloblets are identifiable — provided additional assumptions hold — and then extend these results to networks with multiple internal blobs. In preparation for this, we establish some key properties and introduce definitions leading to a new subclass of galled tree-child networks. Some of the properties of galled networks we need have already been developed, for example by Huson and Klöpper (2007); Huson et al. (2010); Gunawan et al. (2017), and Gunawan et al. (2020), although we generally give self-contained proofs for the sake of readability.

Let N be a galled network with hybrid node h. Then h is a lowest node of the blob containing it and, if in addition h is the child of exactly two hybrid edges, then h is in a unique (tree) cycle.

If N is a galled bloblet, then each hybrid node and its children are the nodes of one skeleton tree, with all remaining nodes contained in (and connected by) the unreduced root skeleton tree. Figure 4 illustrates the root skeleton tree may or may not be trivial, and can have 0, 1 or more taxa.

Suppose further that a galled bloblet N is weakly tree-child, and consider a tree-child rooted partner $N^{+}$ of N. Since $N^{+}$ is tree-child, there is at least one path of tree edges from its root to some leaf, and the taxon set $X_0$ of the root skeleton tree is not empty. See, for example, Figure 4, middle. The next lemma shows that if N is strongly tree-child, then the root skeleton tree has all leaves labelled, and thus has at least 2 taxa, as in Figure 4 right, where $X_0=\{a,c,d\}$.

Lemma 3.1

Let N be a galled, tree-child bloblet, and $X_h$ its taxa with hybrid parent nodes. Then the unreduced root skeleton tree of N is a phylogenetic tree that is composed of all of N’s tree nodes and tree edges, except for $X_h$ and pendant edges leading to taxa in $X_h$.

Proof

Let $\tilde{T}$ denote the unreduced root skeleton tree of N. If N has k hybrid nodes, by Lemma 2.1 it has k non-root skeleton trees in addition to the root skeleton tree. Because N is a galled bloblet, each non-root skeleton tree consists of a lowest hybrid node in the blob, its descendant leaves, and the pendant edges connecting them. Thus $\tilde{T}$ is composed of all of N’s other tree edges and tree nodes.

To finish the proof, we need to show that $\tilde{T}$ is a phylogenetic tree, that is, it does not have unlabeled leaves. Maxfield et al. (2024) showed N can be rooted at any node in $\tilde{T}$. Let $e=uv$ be a pendant edge in $\tilde{T}$, with leaf node v, and consider the rooted partner $N_u^{+}$ rooted at u. Since $N_u^{+}$ is tree-child, there is a (possibly empty) directed path of tree edges starting from v to some labeled leaf y. Since this path contains only tree edges, it must be in $\tilde{T}$ which implies that $v=y$ is labeled. $\square$

As a corollary to Lemma 3.1, each hybrid edge e in a galled tree-child bloblet N has its parent node u in N’s unreduced root skeleton tree. If u is binary in that tree, then u is suppressed when reducing the unreduced root skeleton tree of N, and we say that e attaches to an edge in the root skeleton tree T. If u is not suppressed in T, then we say that e attaches to a node in T, or that it attaches to multiple edges in T (those incident to u in T).

Definition 6

Let N be a semidirected network with a blob B. The bloblet generated by B is the subgraph of N comprised of the edges in B and the cut-edges incident to B, with new and distinct labels assigned to all unlabeled leaves.

Note that the bloblet generated by a blob B is semidirected, and its topology depends only on B and the number of cut-edges in N incident to B’s boundary nodes. In Figure 2, for example, the network $N_3$ is the bloblet generated by $B_3$ in N. If N is itself a bloblet with internal blob B, then the bloblet generated by B is simply N.

We now define the new class of networks central to this work.

Definition 7

Let N be a semidirected network, B a blob of N, and N(B) the bloblet generated by B. Suppose that the leaf set of N(B) is $Y_0\bigsqcup Y_{\ell }$ where $Y_{\ell }$ are the leaves below the lowest nodes of B.

We say that B is ${\mathfrak{C}}_k$, or in the class ${\mathfrak{C}}_k$, $k\ge 3$, if N(B) is reduced, galled, tree-child, with all hybrid nodes of out-degree 1, and for every $y\in Y_{\ell }$, the internal blob of the reduced graph of $N{(B)}_{Y_0\bigsqcup \{y\}}$ is a tree cycle of size k or more. A network is ${\mathfrak{C}}_k$ if it is reduced and all its blobs are ${\mathfrak{C}}_k$.

To illustrate these ideas, we consider again the networks displayed in Figure 2. The network $N_3$ is a galled tree-child bloblet with $Y_{\ell }=\{g_1,g_2\}$, and in class ${\mathfrak{C}}_k$ for $k=3,4,5$. Since the bloblet generated by $B_2\subset N^{\prime }$ is not tree-child, this bloblet (and therefore $N^{\prime }$) can not be in ${\mathfrak{C}}_k$ for any k. In like manner, since $N_1$ is only weakly tree-child, N is not ${\mathfrak{C}}_k$, for any k.

Trivial blobs (single nodes) have no hybrids so they are ${\mathfrak{C}}_k$ for all k. Using the notation of Definition 7, note that if the hybrid parent of $y\in Y_{\ell }$ were not bicombining, then the internal blob of $N{(B)}_{Y_0\bigsqcup \{y\}}$ would have at least 2 cycles and this bloblet would not be ${\mathfrak{C}}_k$. We state this formally.

Lemma 3.2

If N is a ${\mathfrak{C}}_k$ network, then all of its hybrid nodes are bicombining, hence binary.

In a galled network, if a hybrid creates a tree cycle of size k or more, then the cycle contains $k-2$ or more tree edges. When $k=4$, the criterion of Definition 7 thus simply means that partner hybrid edges do not attach to the same edge of the network’s (reduced) skeleton forest. Similar reasoning for $k=3,5$ yields the following.

Lemma 3.3

A bloblet N is ${\mathfrak{C}}_3$ if it is reduced, galled, tree-child, with all hybrid nodes binary, and partner hybrid edges do not attach to the same node of N’s unreduced root skeleton tree. It is ${\mathfrak{C}}_4$ if in addition the partner hybrid edges do not attach to the same edge in its root skeleton tree, and ${\mathfrak{C}}_5$ if, in addition, partner hybrid edges do not attach to adjacent edges in its root skeleton tree.

For example, the bloblet N in Figure 4 (right) is ${\mathfrak{C}}_3$, but not ${\mathfrak{C}}_4$ because the hybrid edges above f attach to the same skeleton tree edge, creating a tree cycle of length 3. The reduced subnetwork of N on all taxa except f is ${\mathfrak{C}}_4$, but not ${\mathfrak{C}}_5$, since each subnetwork on $\{a,c,d\}$ and any one of the hybrid taxa $b_1$, $b_2$, or g is a level-1 network with a 4-cycle. More generally, ${\mathfrak{C}}_{k+1}$ is a proper subclass of ${\mathfrak{C}}_k$.

Finally, in closing this section, we prove that the ${\mathfrak{C}}_k$ property implies a lower bound on the number of taxa.

Lemma 3.4

A network N in ${\mathfrak{C}}_k$ has no non-trivial m-blob with $m<k$. If N has at least one non-trivial blob, then N has at least k taxa.

Proof

If N is a bloblet in ${\mathfrak{C}}_k$ with a non-trivial m-blob B, then each of its hybrid nodes is in a tree cycle with at least k edges. Thus, its root skeleton tree has at least $k-1$ taxa. Since at least one taxon descends from each hybrid node, the bloblet’s number of taxa satisfies $m\ge k$.

If B is an m-blob in a general network N in ${\mathfrak{C}}_k$, then the bloblet generated by B is in ${\mathfrak{C}}_k$. Therefore the result follows from the bloblet case. The final statement is immediate. $\square$

Identifiability of Galled Tree-Child Networks with Large Cycles

Depending on data type and model, certain specific properties can be established that aid in showing network identifiability. We now explicitly state a number of these as assumptions, so that we may show their role in proving network identifiability through combinatorial arguments. Later, in Section 5, we prove (subsets of) these assumptions hold for specific models with quartet concordance factors as data. All networks are semidirected, without further restriction unless stated explicitly.

A-ToB.

If N is a semidirected network, the topology of its tree of blobs is identifiable.

A-4circ.

If N is known to be a 4-taxon level-1 network, the set of circular orders congruent with this level-1 network is identifiable. Specifically, if N has a non-trivial split, this split is identifiable; and if N has a 4-cycle, then the circular order of taxa around this cycle is identifiable.

A-3blob.

If N is an extended bloblet with an internal 3-blob B, then whether B is trivial or non-trivial is identifiable.

A-4len.

If N is known to be a 4-taxon extended bloblet with an internal 4-cycle whose hybrid node is known, then the lengths of tree edges in the cycle are identifiable. If N is known to be a 4-taxon tree, possibly extended by 2-blobs on its pendant edges, then the length of the internal tree edge is identifiable.

A-hyb$(\mathfrak{C})$.

If N is known to be an extended bloblet with internal blob B in some class $\mathfrak{C}$ of networks, then the set of taxa below the hybrid nodes in B is identifiable.

We refer to a class $\mathfrak{C}$ of networks in this last assumption since for applicability to the common models of data generation considered in the next section we must impose restrictions on network structure such as those for ${\mathfrak{C}}_4$ or ${\mathfrak{C}}_5$ . In contrast, A-ToB is known to hold quite generally, and the other assumptions concern only specific network structures.

Remark 1

For any class $\mathfrak{C}$ that includes level-1 networks, where conditions A-4circ and A-hyb$(\mathfrak{C})$ hold, these assumptions together give full topological identifiability of 4-cycles in level-1 networks in $\mathfrak{C}$.

We first prove a result on identifying hybrid nodes.

Lemma 4.1

Let ${\mathfrak{C}}_5^e$ be the class of extended bloblets whose internal blob B is ${\mathfrak{C}}_5$. If A-ToB holds, then A-hyb(${\mathfrak{C}}_5^e$) holds. More specifically, taxon x is below a hybrid node of B if and only if there is a subset Y of 4 taxa such that the tree of blobs $T(N_{Y\cup \{x\}})$ is a star tree but the tree of blobs $T(N_Y)$ is resolved.

Proof

Let N be a network in ${\mathfrak{C}}_5^e$ with internal blob B, and taxon set $X=X_0\bigsqcup X_h$ where $X_h$ is the subset of taxa with a hybrid ancestor in B.

Let $x\in X_h$. By Definition 7, the reduced induced subnetwork $\overline{N_{X_0\cup \{x\}}}$ is a level-1 network with a single k-cycle with $k\ge 5$. Therefore there exists a subset of 4 taxa $Y\subseteq X_0$ such that the reduced graph $\overline{N_{Y\cup \{x\}}}$ is level-1 with a single 5-cycle, and $T(N_{Y\cup \{x\}})$ is a star tree. Since all taxa in Y are on the root skeleton tree of N and $\overline{N_{Y\cup \{x\}}}$ has a 5-cycle, $T(N_Y)$ is resolved.

Conversely let Y be a 4-taxon set such that $T(N_{Y\cup \{x\}})$ is a star tree but $T(N_Y)$ is resolved. Note that $N_{Y\cup \{x\}}$ must be an extended bloblet with a non-trivial internal 5-blob B for this to occur. By Lemma 2.2, x is below a linking lowest node of B, which is hybrid, and $x\in X_h$. $\square$

Another useful result implying identifiability of hybrids is the following.

Lemma 4.2

Consider the class ${\mathfrak{C}}_4^e$ of extended bloblets whose internal blob is ${\mathfrak{C}}_4$. If A-ToB and A-3blob hold, then so does A-hyb(${\mathfrak{C}}_4^e$).

Proof

Let N be a ${\mathfrak{C}}_4$ extended bloblet, with internal blob B. We identify the descendants $X_h$ of hybrid nodes in B by identifying the complementary set of taxa $X_0=X\backslash X_h$.

Consider all subsets Y of taxa on N with $|Y|\ge 3$, and the tree of blobs $T(N_Y)$ (identifiable by A-ToB) of the induced network $N_Y$. For each internal node in each $T(N_Y)$, use A-3blob on all subtrees determined by picking 3 edges emanating from the node to determine and discard those subsets $Y\subseteq X$ with $N_Y$ containing a non-trivial k-blob, $k\ge 3$, as such Y must contain a taxon in $X_h$.

Some remaining subsets Y may still contain a taxon a in $X_h$ because the cycle in N formed by the hybrid edges above a in B and edges in an unreduced skeleton tree of N has been collapsed to a degree-2 node, which was then suppressed in $T(N_Y)$. However, for such sets arising from ${\mathfrak{C}}_4$ networks we can remove a and include at least 2 additional taxa from $X_0$ which are in distinct groups off of the cycle. This produces a larger set $Y^{\prime }$ which has an additional degree-3 node in its tree of blobs, which A-3blob identifies as trivial. Moreover, $Y^{\prime }$ has one less taxon from $X_h$.

Thus taking the largest set Y which produces a tree of blobs, all of whose internal nodes arise from trivial blobs as tested by A-3blob, gives precisely $X_0$. $\square$

We now prove our main combinatorial results on bloblet identifiability.

Theorem 4.3

Consider the class ${\mathfrak{C}}_4^e$ of extended bloblets whose internal blob is ${\mathfrak{C}}_4$, and suppose A-ToB, A-4circ, A-4len, and A-hyb$({\mathfrak{C}}_4^e)$ hold. For a network $N^e$ in ${\mathfrak{C}}_4^e$, let N be the reduced bloblet that $N^e$ extends. Then the semidirected topology of N and the length of its internal tree edges are identifiable.

Combining with Lemma 4.1 and using ${\mathfrak{C}}_5\subseteq {\mathfrak{C}}_4$, this implies the following.

Corollary 4.4

Consider the class ${\mathfrak{C}}_5^e$ of extended bloblets whose internal blob is ${\mathfrak{C}}_5$, and suppose A-ToB, A-4circ and A-4len hold. For a network $N^e$ in ${\mathfrak{C}}_5^e$, let N be the bloblet that $N^e$ extends. Then the semidirected topology of N and the length of its internal tree edges are identifiable.

Proof of Theorem 4.3

For $N^e$ and N as in the statement, let B be their common internal blob. By assumption A-hyb(${\mathfrak{C}}_4^e$) we can identify the partition $X=X_0\bigsqcup X_h$ where $X_h$ is the set of taxa that are below hybrid nodes of B. $X_h$ may be empty, in which case B is trivial and there is nothing to prove.

Otherwise, the taxa $X_0$ are those on the root skeleton tree $T_{\rho }$ of N, so by assumption A-ToB we can identify the topology of ${(N^e)}_{X_0}$’s tree of blobs, which is $T_{\rho }=\overline{N_{X_0}}$.

Now take $x\in X_h$. As N is ${\mathfrak{C}}_4$, the reduced graph $\overline{N_{X_0\cup \{x\}}}$ is level-1, with a single cycle, of at least 4 edges. By considering induced 4-taxon networks on x, a, b, c for all choices of 3 taxa $a,b,c\in X_0$ and using A-4circ, we may determine those taxa which attach to the single cycle of $\overline{N_{X_0\cup \{x\}}}$ by paths to each of the nodes in the cycle. This is enough to determine the edges (and possibly nodes) of $T_{\rho }$ onto which the hybrid edges above x attach.

From this information across all taxa in $X_h$, we can identify which edges in $T_{\rho }$ arise from a path of multiple edges in the unreduced $N_{X_0}$, and which edges in $T_{\rho }$ match a single edge in $N_{X_0}$.

Next we need to identify for each taxon x in $X_h$, the precise locations at which the two hybrid edges above x originate on $T_{\rho }$, and the length of those internal tree edges in B which are internal tree edges of $N_{X_0}$. A key aspect of this is determining the placement of attachment nodes for hybrid edges in different cycles along the same edge of the skeleton tree, as illustrated by Figure 5.

Fig. 5.

Top: Network $N\in {\mathfrak{C}}_4$ has $X_h=\{x,y,z_1,z_2\}$ highlighted in blue, and root skeleton tree $T_{\rho }=\overline{N_{X_0}}$. Bottom: In the induced network $\overline{N_{X_0\cup \{x\}}}$, $a(x)=a_2$ is an attachment node for x. In $\overline{N_{X_0\cup \{y\}}}$, y has attachment node $a(y)=a_1$. By A-4len, the edge lengths highlighted in orange can be identified, from which we can identify that a(y) is closer to u than a(x) is to u

Let uv be an edge in $T_{\rho }$. If uv arises from a single internal edge in $N_{X_0}$ (no hybrid edge attaches to uv except possibly at its ends), then uv arises from an internal edge in ${(N^e)}_{X_0}$ and its edge length is identifiable by A-4len.

If one or more hybrid edges attach to uv, let p be the corresponding path in N of tree edges joining tree nodes $u,a_1\dots ,a_k,v$ ($k\ge 1$) that reduces to uv in $T_{\rho }$. It is possible for u or v to be a leaf, but not both (because $T_{\rho }$ has 3 or more taxa by Lemma 3.4). We thus assume that u is internal. At this point, we have already identified the set of taxa $X_{\mathit{uv}}\subseteq X_h$ below a hybrid edge that attaches to uv. For $x\in X_{\mathit{uv}}$, let a(x) be the associated attachment node. It suffices to show that we can identify the distance between u and a(x) for each $x\in X_{\mathit{uv}}$.

To this end, fix $x\in X_{\mathit{uv}}$. In $\overline{N_{X_0\cup \{x\}}}$, u and a(x) are internal nodes (of degree $\ge 3$), and v is either a leaf or internal. The cycle in $\overline{N_{X_0\cup \{x\}}}$ contains 4 or more nodes, two of which are a(x) and the hybrid node above x.

If v is a leaf the cycle also includes u since N is in ${\mathfrak{C}}_4$. By A-4len, we can identify the length of tree edges in this cycle on $N^e$ which includes the length of ua(x). If v is internal, we can similarly identify the length of either ua(x) or of va(x), whichever is part of the cycle. But since uv is an internal edge of the tree $\overline{N_{X_0}}$, using A-4len we can identify its length on $N^e$. Subtracting the length of va(x) from it gives the length of ua(x) (see Figure 5).

Finally, note that since all hybrid nodes are binary, no taxa $x,x^{\prime }\in X_h$ who share one or two attachment nodes may be descended from the same hybrid edge(s) (e.g., see $z_1$ and $z_2$ in Figure 5). $\square$

Remark 2

In the proof of Theorem 4.3, edge lengths, identified by A-4len, are used to precisely locate the attachment points of hybrid nodes on the root skeleton tree. In fact identifying the relative edge lengths is sufficient to identify the semidirected topology of the blob. That is, if A-4len is weakened to identifying only whether whether $\ell (e)<\ell (e^{\prime })$, $\ell (e)=\ell (e^{\prime })$ or $\ell (e)>\ell (e^{\prime })$ when e and $e^{\prime }$ are composite edges formed from a path and subpath of tree edges in the network, then a weakened theorem can be established.

In particular, edge lengths may be considered in any units (consistently across 4-taxon sets), such as years, number of generations, coalescent units, or substitutions per sites. If only an ordering by magnitude of edge lengths can be identified, then edge lengths in the full network are not identifiable, but the claim of topological identifiability stated in Theorem 4.3 remains valid. This suggests robustness of network topology estimation to edge length inference error.

We extend the two previous results from bloblets to general networks.

Theorem 4.5

Suppose A-ToB, A-4circ, A-4len, and A-hyb(${\mathfrak{C}}_4$) hold. For a semidirected network N in ${\mathfrak{C}}_4$, the topology of N and the lengths of internal tree edges in its blobs are identifiable.

Proof

By A-ToB, the tree of blobs of N is identifiable. To identify the structure of an individual blob B, we pass to an extended bloblet as follows: For each cut edge of N incident to B choose one taxon which is separated from B by that edge, forming a taxon subset Y Then consider $\overline{N_Y}$, an extended bloblet with blob B. By Theorem 4.3 the topology of B and its internal tree edge lengths are identified. This gives the full semidirected network and the lengths of all tree edges inside blobs, as claimed. $\square$

A similar argument, using the weaker hypotheses of Corollary 4.4, yields the following.

Theorem 4.6

Suppose A-ToB, A-4circ, and A-4len hold. For a semidirected network in ${\mathfrak{C}}_5$, the topology and the lengths of internal tree edges in non-trivial blobs are identifiable.

Note that the theorems do not claim identifiability of the lengths of cut edges of N. However, A-4len can be applied to any cut edge uv whose endpoints are not incident to a blob, by choosing four taxa that define uv.

Remark 3

Identifying lengths of hybrid edges and tree edges incident to non-trivial blobs seems to depend on more than the general assumptions made above. Even in the level-1 case studied by Allman et al. (2024), lengths of edges near 3-cycles and leaves can be nonidentifiable.

Remark 4

Arguing as in the proof of Theorem 4.5 or 4.6, the topology of ${\mathfrak{C}}_4$ or ${\mathfrak{C}}_5$ blobs of a general network N may be identifiable, even if the full network N is not of those classes. The topologies of its ${\mathfrak{C}}_4$ blobs are identifiable if A-ToB, A-4circ, A-4len, and A-hyb(${\mathfrak{C}}_4$) hold, and topologies of ${\mathfrak{C}}_5$ blobs if only the first three assumptions hold. The network $N_0$ of Figure 1, discussed in the introduction, provides an example of this since only one blob is ${\mathfrak{C}}_5$.

Remark 5

Without assuming A-4len, the proof of Theorem 4.3 shows that, for each hybrid in a blob, we can identify the 2 edges onto which its parent edges attach. The proof could thus be modified to show topological identifiability of N for a smaller class of networks where no two hybrid edges attach to the same edge of the root skeleton tree. For example, the rightmost blob of $N_0$ in Figure 1 is ${\mathfrak{C}}_5$ and in this more restrictive class.

Even if multiple hybrid edges with different children attach to the same edge in the root skeleton tree, without A-4len we can still determine a finite list of networks, one of which is the true network.

Models and Quartet Concordance Factor Data

We now present several models of gene tree formation on a network, as well as the quartet concordance factor data type. Then we establish that the assumptions needed to apply Theorems 4.5 and 4.6 hold, and conclude with our main result in Theorem 5.7.

Models for Gene Trees

We describe three models of gene trees forming within species networks.

Definition 8

Let $N^{+}$ be a rooted metric phylogenetic network with edge lengths $\ell (e)$ in coalescent units (generations/population size). Then the following models determine distributions of topological or metric unrooted gene trees.

1. Displayed tree (DT) model: For each edge e of $N^{+}$, we also specify the effective population size $\eta (e)>0$ and mutation rate $\mu (e)>0$ in substitutions per site per generation. Only gene trees whose topology is displayed in $N^{+}$ have positive probability, equal to the product of inheritance probabilities of all edges in $N^{+}$ forming T:

   $$\mathbb{P}(T)=\gamma (T)=\prod _{e\in T}\gamma (e).$$

   Each edge e of T is assigned length $s(e)=\ell (e)\eta (e)\mu (e)$, giving a distribution of metric unrooted gene trees.

2. Network multispecies coalescent model with independent inheritance (NMSCind): Gene trees form according to the coalescent model within each population (edge) of the network. At a hybrid node with parental edges $e_1,\cdots ,e_m$, each lineage is inherited from population $e_k$ ($1\le k\le m$) with probability $\gamma (e_k)$, independently of the other lineages (Degnan et al. 2012). This gives a distribution of topological unrooted gene trees.

3. Network multispecies coalescent model with common inheritance (NMSCcom): Gene trees form according to the coalescent model within each population. At a hybrid node with parental edges $e_1,\cdots ,e_m$, all lineages of a given gene are inherited from the same population $e_k$ ($1\le k\le m$), chosen with probability $\gamma (e_k)$. Equivalently, a displayed tree is chosen with probability as in item 1, and a gene tree forms within it according to the coalescent process as in item 2 restricted to a tree (Gerard et al. 2011). This gives a distribution of topological unrooted gene trees.

The NMSCind and NMSCcom models (items 2 and 3) are the two extreme cases of a model with correlated inheritance of lineages at reticulations (Fogg et al. 2023), which for simplicity we do not consider in full generality here.

We need not explicitly consider models of sequence evolution on gene trees, as we treat the gene trees themselves as data. In practice, one of course needs to assume these gene trees can be robustly inferred from sequences, as all “2-stage” inference methods utilizing inferred gene trees must do.

Quartet Concordance Factors

A number of network identifiability results and practical network inference methods are based on quartet concordance factors (CFs) (Solís-Lemus and Ané 2016; Baños 2019; Allman et al. 2019, 2023, 2024, 2025). Although many of these results are limited to level-1 networks, we draw on them to obtain results for the more general classes of networks in this work.

For a network with n taxa, quartet CFs are the $3·\left(\begin{array}{c}n\\ 4\end{array}\right)$ probabilities of the unrooted gene quartet topologies that might relate a subset of four taxa. These probabilities can be calculated under any model M of gene tree formation on a network N. If M generates metric gene trees on the full set of n taxa, CFs are obtained by pruning taxa except those in a quartet, and then marginalizing over edge lengths and root location. For the NMSCind and NMSCcom models only resolved quartet trees have positive probability, so CFs for each 4-taxon set have the form

$$\begin{array}{c}\hfill {\text{CF}}_{\mathit{abcd}}=({\text{CF}}_{ab|cd},{\text{CF}}_{ac|bd},{\text{CF}}_{ad|bc})=({\mathbb{P}}_M(ab|cd),{\mathbb{P}}_M(ac|bd),{\mathbb{P}}_M(ad|bc)).\end{array}$$

For the DT model on a binary network we also have probability 0 of an unresolved quartet topology.

Identifiability Assumptions

We next investigate the validity of the assumptions laid out in Section 4 in the context of the three models and quartet CFs just introduced. This requires additional assumptions — including that networks are binary and numerical parameters are generic (lie outside some subset of measure zero) — to utilize already published identifiability results. These restrictions do not alter the arguments of Section 4, and the main topological identifiability results there still apply.

A-ToB

The identifiability of the tree of blobs of a binary network from quartet CFs under the NMSCind model was proved by Allman et al. (2023), with Allman et al. (2024) providing a practical inference algorithm. Rhodes et al. (2025, Corollary 6.6) extended that work to the DT and NMSCcom models, but made additional assumptions of no “anomalous quartets” on the network in order to study circular orders of blobs.

The proof of tree of blobs identifiability referenced here is primarily combinatorial, with the key exception the fundamental case for 4-taxon networks (Allman et al. 2023, Theorem 1). Noting that the proofs of those results did not require assuming no anomalous quartets, straightforward modifications extend those arguments to the DT and NMSCcom models. We state this formally as the next proposition.

Proposition 5.1

Under the DT, NMSCind, and NMSCcom models with quartet CFs from a binary network with generic numerical parameters, the network’s tree of blobs is identifiable, so assumption A-ToB holds.

A-4circ

The identifiability of circular orders for outer-labelled planar networks with blobs of any size was studied by Rhodes et al. (2025), but the A-4circ assumption only concerns level-1 4-taxon networks, where the result follows immediately from (Baños 2019, Theorem 4) for the NMSCind model. For the other models on a level-1 4-taxon network N with generic parameters, one can directly compute that if N displays the split ab|cd, then ${\text{CF}}_{\mathit{abcd}}$ has the form (1, 0, 0) under DT, and (p, q, q) under NMSCcom and NMSCind, while if N has a 4-cycle with circular order (a, b, c, d) the CFs have the form (p, 0, q), $p,q>0$ for DT and (p, r, q) with $p,q>r>0$, $p\ne q$ under NMSCcom and NMSCind. If the 4-taxon network has a 4-polytomy the CF is (1/3, 1/3, 1/3) for the NMSC models, while for DT gene trees are unresolved with probability 1. Thus we obtain the following.

Proposition 5.2

Under the DT, NMSCcom, and NMSCind models with generic parameters using quartet CFs, assumption A-4circ holds.

A-4len

Let N be a 4-taxon level-1 network with a 4-cycle with circular order a, b, c, d whose hybrid node is ancestral to taxon a. First consider the DT model. As CFs capture only topological information on gene trees, ${\text{CF}}_{\mathit{abcd}}=(p,0,q)$ is independent of all branch lengths. Therefore A-4len does not hold from CFs for this model. However, from one metric unrooted gene tree of each topology, one can identify the lengths (in substitutions per site) of the tree edges in the cycle, as these are simply internal branch lengths on one of the unrooted trees displayed by N. Similarly if N is a tree, possibly extended with 2-cycles on its pendant edges, the CFs alone are not enough to determine the internal edge length under DT, but one metric unrooted gene tree is. Note that under the DT model, A-4len is then satisfied using edge lengths in substitutions per site: the units that can be identified on gene trees from (arbitrary length) sequence data.

For the NMSCind model, the identifiability of the tree edge lengths in a 4-cycle from CFs is dependent on having multiple samples (either from different taxa or within the same taxon) from specific cut edges attached to the 4-cycle, as characterized by Allman et al. (2024, Proposition 29). Specifically, under NMSCind, if the circular order of taxon groups around the cycle is (A, B, C, D), where each group is the subset of taxa separated from the blob by a common boundary node, and if A is below the hybrid node, one needs either two samples from B or two from D, or 2 from both A and C. For NMSCcom, two samples from A is sufficient. (See Appendix 7.3 for details.) Rather than use these facts with such exactness, we simply say that these lengths are identifiable if we have 2 samples per taxon group.

Proposition 5.3

Using quartet CFs, assumption A-4len holds

1. under the DT model if one has a metric gene tree of each possible topology from the 4-taxon network, and

2. under the NMSCcom and NMSCind models if there are 2-samples per taxon.

For the strongest statements on identifiability of tree edge lengths in 4-cycles from CFs under the NMSCind and NMSCcom models, see (Allman et al. 2024) and Appendix 7.3 of this work.

A-3blob

We establish the validity of assumption A-3blob for the two models with a coalescent process.

Proposition 5.4

Under the NMSCind or NMSCcom model, with at least 2 samples per taxon and quartet CF data, assumption A-3blob holds for binary semidirected networks with generic parameters.

Proof

Let a, b, c denote the 3 taxa on the network, and denote the 2 samples per taxon by $a_1$, $a_2$ from a etc.

Let $G_{\mathit{abc}}$, $G_{\mathit{bca}}$, and $G_{\mathit{cab}}$ be the polynomials in quartet CFs given by

$$G_{\mathit{abc}}={\text{CF}}_{a_1{c}_1|a_2{c}_2}{\text{CF}}_{a_1{b}_1|b_2{c}_2}-2{\text{CF}}_{b_1{c}_1|b_2{c}_2}{\text{CF}}_{a_1{b}_1|a_2{c}_2}+{\text{CF}}_{a_1{b}_1|a_2{b}_2}{\text{CF}}_{a_1{c}_1|b_2{c}_2}$$

and similarly for $G_{\mathit{bca}}$ and $G_{\mathit{cab}}$ by permuting taxon labels. Then for level-1 networks and the NMSCind and NMSCcom models Allman et al. (2024, Proposition 9 and Remark 1) established that N has a non-trivial 3-blob if and only if at least one of the G polynomials does not vanish: $G_{\mathit{abc}}\ne 0$ or $G_{\mathit{bca}}\ne 0$ or $G_{\mathit{cab}}\ne 0$.

If the network has an arbitrary non-trivial 3-blob, by specializing some of the hybridization parameters to 0 or 1 all 2-blobs can be effectively replaced by edges and the 3-blob can be effectively reduced to a 3-cycle by Lemma 2.3. This implies that at least one $G_{\mathit{xyz}}$, viewed as a function of the network parameters, does not vanish for one specialization of the parameters. As the CFs are analytic functions of the parameters, so is $G_{\mathit{xyz}}$, and the non-vanishing of an analytic function at a single point implies its non-vanishing at generic points. Thus for generic parameter values, this $G_{\mathit{xyz}}$ is non-zero. $\square$

The argument in this proof fails for the DT model, since, as pointed out by Allman et al. (2024), in the level-1 case the CFs appearing in the G polynomials are all for quartets not displayed on the network, and thus are 0 under that model. It remains an open question whether other approaches might imply A-3blob holds under DT.

We note that the 2-sample assumption is necessary for the leaves of a binary 3-bloblet, but can be weakened for larger networks, as in the similar discussion for A-4len above. Concretely, for a 3-blob, each boundary node corresponds to a taxon group and the argument for Proposition 5.4 is valid when each taxon group has at least 2 samples. In Figure 1, for example, the middle 3-blob in $N_0$ can be identified as non-trivial even with a single sample per taxon.

A-hyb($\mathfrak{C}$)

We consider the question of identifying descendants of hybrid nodes of a blob for classes ${\mathfrak{C}}_4$ and ${\mathfrak{C}}_5$. Although A-hyb(${\mathfrak{C}}_4)$ implies A-hyb(${\mathfrak{C}}_5$) since ${\mathfrak{C}}_5⊊{\mathfrak{C}}_4$, we first state a general identifiability result for ${\mathfrak{C}}_5$, since it requires few restrictions. For class ${\mathfrak{C}}_4$, we give a different argument, valid under the coalescent models but requiring multiple samples per taxon.

Proposition 5.5

If N is a binary 2-blob extension of a bloblet B in ${\mathfrak{C}}_5$ with generic numerical parameters, then under the models DT, NMSCcom, and NMSCind the set of taxa descended from a hybrid node in B is identifiable from quartet CFs. Thus A-hyb(${\mathfrak{C}}_5$) holds.

Proof

This follows directly from Proposition 5.1 and Lemma 4.1. $\square$

Proposition 5.6

If N is a binary 2-blob extension of a bloblet B in ${\mathfrak{C}}_4$ with generic numerical parameters and 2 samples per taxon, then under the models NMSCcom and NMSCind the set of taxa descended from a hybrid node in B is identifiable from quartet CFs. Thus A-hyb(${\mathfrak{C}}_4$) holds.

Proof

For such a network and sampling, A-ToB and A-3blob hold by Propositions 5.1 and 5.4. Lemma 4.2 then yields the claim. $\square$

Main Results

We now state and prove our main theorem.

Theorem 5.7

Let N be a binary semidirected phylogenetic network in ${\mathfrak{C}}_4$. Then under the NMSCind and NMSCcom models with generic numerical parameters and 2 samples per taxon, the semidirected topology of N and the lengths of internal tree edges in blobs are identifiable from quartet CFs.

Under the DT model with CFs and metric gene trees, or the coalescent models with a single sample per taxon, the same result holds for binary networks in ${\mathfrak{C}}_5$.

Proof

For the NMSCind and NMSCcom models with 2 samples per taxon, A-ToB holds by Proposition 5.1, A-4circ by Proposition 5.2, A-4len by Proposition 5.3, and A-hyb(${\mathfrak{C}}_4$) by Proposition 5.6. Thus Theorem 4.5 yields the first claim.

Under the DT model, the same propositions show A-ToB and A-4circ hold, as does A-4len since we have metric gene trees. While we do not have A-hyb(${\mathfrak{C}}_4$), applying Theorem 4.6 establishes the second claim. The coalescent models on ${\mathfrak{C}}_5$ networks with 1 sample per taxon are handled similarly. $\square$

For the NMSC models we obtain a weaker, yet still interesting result, that does not depend on A-4len or A-3blob (as motivated by Remark 5). We omit the proof, since it closely follows previous arguments.

Proposition 5.8

Let N be a binary semidirected phylogenetic network in ${\mathfrak{C}}_5$ with the additional requirement that no two hybrid edges attach to the same edge in the skeleton forest. Then, under the DT, NMSCcom, and NMSCind models with 1 sample per taxon and generic numerical parameters, the topology of N is identifiable from quartet CFs.

While the network family in this proposition is a proper subset of ${\mathfrak{C}}_5$, it still includes networks of arbitrary level, and many that are not outer-labeled planar, including for example a network containing the rightmost non-trivial blob of $N_0$ in Figure 1.

Discussion

The classes of networks that we have shown to be identifiable from quartet concordance factors are essentially those that are binary, galled, with all blobs tree-child, and with no “small” cycles. These include networks of arbitrary level, well-beyond the level-1 structure currently assumed by most practical inference methods (Solís-Lemus and Ané 2016; Allman et al. 2019; Kong et al. 2024; Allman et al. 2025; Holtgrefe et al. 2025a), and ones without the outer-labelled planar embeddings that previously have been shown to lead to additional identifiability results beyond level-1 (Rhodes et al. 2025). Since we build our work on first identifying a network’s tree of blobs, and then analyze each blob separately, it also allows for the structure of some blobs to be identified while others may be too complex to do so (with current understanding).

As this work was in review, Holtgrefe et al. (2025b, Section 5) established results implying that in some circumstances every blob of a network is strongly tree-child exactly when the network is strongly tree child. Specifically, consider a reduced semidirected network without parallel edges in which each hybrid node has a single descendant. Then if no non-leaf tree node is incident to a single tree edge and no hybrid node has a hybrid child, the network is strongly tree-child. Since the class ${\mathfrak{C}}_k$ assumes hybrid nodes have a single child, and $k\ge 3$ implies no parallel edges, this simplifies verifying that a blob or full network is strongly tree-child for applicability of our results.

While the network restrictions considered here are not biologically motivated, they are natural ones for identifiability results. For instance galled networks are ones in which the various reticulations in a blob have some independence of one another, with each determining a unique cycle and none ancestral to each other. While this is much weaker than requiring the non-intersecting cycles of level-1 networks, it allows, to some degree, for an analysis one cycle at a time. Tree-child networks are those in which every node is ancestral to a leaf by a path with no reticulations. This can be viewed as giving some ‘direct’ data on all nodes in the network, not obscured by the genetic interchange the reticulations model. Finally, while biologically one might expect small cycles, representing gene flow between closely related species, to be most common, it is also plausible that they will be the most difficult to infer correctly due to the similarity of the intermixed genomes. In particular 3-cycles represent gene flow between sister species, and if this occurred soon after species divergence it might only be detectable through more detailed data than that we consider here.

By largely focusing on quartet CFs, which are determined by topological gene tree information alone, our results are likely to concern what can be most robustly inferred. Identifiability results from metric gene trees that are applicable to empirical data require a detailed model of the substitution process along gene trees. Variation in this process across the genome may be substantial (both in across-site rate variation and in the substitution process itself). Rate variation across lineages, violating a molecular clock, is known to reduce network accuracy and increase the detection of spurious reticulations for methods using metric gene trees that assume no rate variation (Ogilvie et al. 2017; Flouri et al. 2022; Frankel and Ané 2023; Cao et al. 2024; Koppetsch et al. 2024). Thus while metric gene trees could provide more information on network structure, how to extract that information accurately needs further development.

One final aspect of our results worth highlighting, for both empiricists and developers of methods, is that multiple samples per taxon can provide more information on network structure than single samples do. While already inherent in other theoretical works, and especially prominent in (Allman et al. 2024), many empirical data sets currently include only single samples. While the cost, in time, effort, and funds for routinely collecting multiple samples cannot be dismissed, doing so would increase what can be inferred. With a coalescent process part of the assumed model, one may view each sample as a ‘probe’ into the past, with multiple probes from the same source allowing their differing histories to provide more insight than does any one alone.

Appendix

Proposition 5.3, part 2, gives sufficient conditions for Assumption A-4len to hold for quartet CF data under NMSCcom. However, assuming two samples from each of the four taxon groups is stricter than is needed. Assume that N is a network with a 4-cycle, where the circular order of taxon groups around the cycle is (A, B, C, D) with A the hybrid group. Assume further that the tree edge lengths in the cycle between groups B and C and between C and D are $t_1$ $t_2$, respectively.

Note first that if only a single sample is available from among the hybrid descendants then the CFs from models NMSCcom and NMSCind are equal and the necessary condition follows from (Allman et al. 2024). Specifically, with a single hybrid sample a necessary and sufficient condition for identifiability of the lengths of the tree edges in the 4-cycle is that at least two samples are taken from either group that is neighbor to the hybrid group.

In contrast, Allman et al. (2024) showed that if 2 samples are taken from the hybrid group and only 1 from the other groups, under NMSCind $t_1,t_2$ are not identifiable from quartet CFs. However, under NMSCcom these branch lengths are identifiable. To establish this, we used the computational algebra software Singular 4.4.0 (Decker et al. 2024) to show that the edge probabilities $X_i=exp(-t_i)$ for the tree edges in the 4-cycle can be computed from CFs under NMSCcom. For example, adopting a short notation with $C_{\mathit{acab}}={\text{CF}}_{a_{1}c|a_{2}b}$, etc.,

$$\begin{array}{c}\hfill X_1=\frac{2C_{\mathit{acab}}{C}_{\mathit{acbd}}+C_{\mathit{acab}}{C}_{\mathit{adbc}}-2C_{\mathit{acbd}}{C}_{\mathit{adab}}-C_{\mathit{acbd}}{C}_{\mathit{adac}}-C_{\mathit{adab}}{C}_{\mathit{adbc}}+C_{\mathit{adac}}{C}_{\mathit{adbc}}}{-C_{\mathit{acbd}}{C}_{\mathit{adab}}+C_{\mathit{adab}}{C}_{\mathit{adbc}}+C_{\mathit{acab}}-C_{\mathit{adab}}}.\end{array}$$

A formula for $X_2$ is obtained from that for $X_1$ by exchanging the taxon labels b and d.

Funding

This work was supported in part by the National Science Foundation through grants DMS-2051760 (ESA and JAR), DMS-2331660 (HB), DMS-2023239 (CA), and by grant DMS-1929284 while all authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Theory, Methods, and Applications of Quantitative Phylogenomics" program.

Data Availability

No data was used in this work.

Declarations

Competing interests

The authors have no relevant financial or non-financial interests to disclose. HB serves as a Guest Editor to the Bulletin of Mathematical Biology for the topical collection associated with the ICERM semester program “Theory, Methods and Applications of Quantitative Phylogenomics”, but was not involved in the review or editorial decision-making for this manuscript.