Phylogenetic networks represent the evolution of organisms that have undergone reticulate events, such as recombination, hybrid speciation or lateral gene transfer. An important way to interpret a phylogenetic network is in terms of the trees it displays, which represent all the possible histories of the characters carried by the organisms in the network. Interestingly, however, different networks may display exactly the same set of trees, an observation that poses a problem for network reconstruction: from the perspective of many inference methods such networks are indistinguishable. This is true for all methods that evaluate a phylogenetic network solely on the basis of how well the displayed trees fit the available data, including all methods based on input data consisting of clades, triples, quartets, or trees with any number of taxa, and also sequence-based approaches such as popular formalisations of maximum parsimony and maximum likelihood for networks. This identifiability problem is partially solved by accounting for branch lengths, although this merely reduces the frequency of the problem. Here we propose that network inference methods should only attempt to reconstruct what they can uniquely identify. To this end, we introduce a novel definition of what constitutes a uniquely reconstructible network. For any given set of indistinguishable networks, we define a canonical network that, under mild assumptions, is unique and thus representative of the entire set. Given data that underwent reticulate evolution, only the canonical form of the underlying phylogenetic network can be uniquely reconstructed. While on the methodological side this will imply a drastic reduction of the solution space in network inference, for the study of reticulate evolution this is a fundamental limitation that will require an important change of perspective when interpreting phylogenetic networks.
As an example, consider again networks N1 and N2 in Fig. 1, which display the same trees 𝓣 (N) = {T1, T2, T3}. (In the following, 𝓣 (N) denotes the set of trees displayed by N.) By displaying the same trees, these networks display the same clades, the same triples, the same quartets (triples and quartets are rooted subtrees with 3 leaves and unrooted subtrees with 4 leaves, respectively) and in general the same subtrees with an arbitrary number of leaves. Therefore, any method that reconstructs a network based on its consistency with collections of such data will not be able to distinguish between networks N1 and N2. This includes all the methods whose data consists of clusters of taxa (e. g. , [34]), triples (e. g. , [35]), quartets (e. g. , [36]), or any trees (e. g. , [37]). The same holds for many, sequence-based, maximum parsimony and maximum likelihood approaches proposed in recent papers. For maximum parsimony, a practical approach [2,29–31] is to consider that the input is partitioned in a number of alignments A1, A2, …, Am, each from a different non-recombining genomic region (possibly consisting of just one site each), and then take, for each of these alignments, the best parsimony score Ps (T∣Ai) among all those of the trees displayed by a network N. The parsimony score of N is then the sum of all the parsimony scores thus obtained. Formally, we have Ps (N | A 1, A 2, …, A m) = ∑ i = 1 m min T ∈ 𝓣 (N) Ps (T | A i). It is clear that if two networks display the same set of trees (as in Fig. 1), then their parsimony score with respect to any input alignments will be the same—because they take the minimum value over the same set 𝓣 (N) —and thus they are indistinguishable to any method based on the maximum parsimony principle above. As for maximum likelihood (ML), Nakhleh and collaborators [2,32,33,38] have proposed an elegant framework whereby a phylogenetic network N is not only described by a network topology, but also edge lengths and inheritance probabilities associated to the reticulations of N. As a result, any tree T displayed by N has edge lengths—allowing the calculation of its likelihood Pr (A∣T) with respect to any alignment A—and an associated probability of being observed Pr (T∣N). The likelihood function with respect to a set of alignments A1, A2, …, Am, each from a different non-recombining genomic region, is then given by: Pr (A 1, A 2, …, A m | N) = ∏ i = 1 m Pr (A i | N) = ∏ i = 1 m ∑ T ∈ 𝓣 (N) Pr (A i | T) Pr (T | N). Note that an important difference with the consistency-based and parsimony methods described above is that any tree T displayed by a network has now edge lengths and an associated probability Pr (T∣N). Unfortunately, this ML framework is also subject to identifiability problems. For example, it does not allow us to distinguish between networks with topologies N1 and N2 in Fig. 1: for every assignment of edge lengths and inheritance probabilities to N1, there exist corresponding assignments to N2 that make the resulting networks indistinguishable, that is, displaying the same trees, with the same edge lengths and the same probabilities of being observed (see the last section in the Supporting Information, S1 Text). As a result, the likelihoods of these two networks will be identical, regardless of the data, and no method based on this definition of likelihood will be able to favour one of them over the other. We refer to S1 Text for a more detailed discussion about networks with inheritance probabilities and likelihood-based reconstruction. In general, we believe that these identifiability problems affect all network inference methods which seek consistency with unordered collections of sequence alignments or pre-inferred attributes such as clusters, triples, quartets or trees. In this paper, as in the ML framework above, we adopt networks and trees with edge lengths as the primary objects of our study. The primary motivation for this is that this choice makes our results directly relevant to the statistical approaches for network inference, all of which need edge lengths to measure the fit of a phylogeny with the available data. In addition to ML, these approaches include distance-based and Bayesian methods [39], which are also promising for future work. However, there is another motivation for our choice: accounting for edge lengths solves some of the identifiability problems outlined above, as in some cases it allows to distinguish between networks with different topologies, which would be otherwise impossible to tell apart. For example, consider the three network topologies in Fig. 2 (top), where taxon o is an outgroup used to identify the root of the phylogeny for a, b and c. These networks show three very different evolutionary histories: in N1 taxon b is the only one issued of a reticulation event—in other words the genome of b is recombinant—whereas in N2 and N3, it is a and c, respectively, that are recombinant. However, N1, N2 and N3 display the same tree topologies—those of T1 and T2—and thus would be indistinguishable to any approach that does not model edge lengths. If instead edge lengths are accounted for (e. g. in a ML context) and the data supports T1 and T2 with the edge lengths in Fig. 2, then the only network fitting perfectly the data is N2, with the edge lengths in Fig. 2 (bottom right). It is easy to check that N2 now displays T1 and T2 with the shown edge lengths, whereas no edge length assignment to N1 or N3 can make these networks display T1 and T2. We note that, throughout this paper, as in classical likelihood approaches, edge lengths measure evolutionary divergence, for example in terms of expected number of substitutions per site. No molecular clock is assumed, meaning that we do not expect edge lengths to be proportional to time. Unfortunately, accounting for edge lengths only solves some of the identifiability problems for phylogenetic networks. Consider networks N1 and N2 in Fig. 3: for any set of edge lengths for N1, there exist an infinity of edge length assignments for N2 that make these two networks display exactly the same set of trees with the same edge lengths. In the following, we say that networks such as N1 and N2 are indistinguishable. In fact it is not difficult to construct other examples of indistinguishable networks: each time a network has a reticulation v giving birth to only one edge (i. e. with outdegree 1), then we can reduce by Δλ the length of this edge and correspondingly increase by Δλ the lengths of the edges ending in v, without altering the set of trees displayed by the network. Note that this operation, which we refer to as “unzipping” reticulation v, can result in v coinciding with a speciation node or a leaf when Δλ is taken to equal the length of the edge going out of v. For example in Fig. 3, one may fully unzip the two reticulation nodes in N1, thus obtaining the network N′ of Fig. 4. As expected, N1 and N′ display the same set of trees ({T1, T2, T3}) and are thus indistinguishable. What is most interesting in this example is that, if we fully unzip the two reticulations in N2 (the other network in Fig. 3, also displaying {T1, T2, T3}), then we eventually end up obtaining N′ again. As we shall see in the following, this is not a coincidence: the unzipping transformations described above lead to what we call the canonical form of a network; under mild assumptions, two networks are indistinguishable if and only if they have the same canonical form (e. g. N1, N2 in Fig. 3 have the same canonical form N′; formal definitions and statements in the Results section). Here, we propose to deal with the identifiability issues for phylogenetic networks in the following way: since no data will ever enable any of the standard inference methods described above to prefer a network over all of its indistinguishable equivalents, we propose that these methods should only attempt to reconstruct what they can uniquely identify, that is, networks in canonical form. This is a radical shift, not only for the developers of phylogenetic inference methods, who will see a drastic reduction of the solution space of their algorithms, but also for evolutionary biologists, who should abandon their hopes of seeing a network such as N1 or N2 in Fig. 3 being reconstructed by these inference methods. Limiting the scope of network reconstruction to topologically-constrained classes of networks has been a recurring theme and an important goal in the literature on phylogenetic networks. Examples of such classes include galled trees [40,41], galled networks [42], level-k networks [43], tree-child networks [44], tree-sibling networks [45], networks with visible reticulations [1]. Although the ultimate goal should be to establish what can be inferred from biological data, most of the proposed definitions are computationally-motivated: in general the rationale behind these classes is the possibility of devising an efficient algorithm to solve some formalization of the reconstruction problem. None of these definitions claims to have biological significance. Our goals are more basic: starting from the observation that not all phylogenetic networks are identifiable, since many of them are mutually indistinguishable with most inference approaches, we aim to define a class of networks that is (existence goal) large enough that every phylogenetic network has an equivalent (i. e. indistinguishable) network within this class and (distinguishability goal) small enough that no two networks within this class are indistinguishable. From our standpoint, the computationally-motivated definitions above are at the same time too broad and too restrictive. Too broad, because they determine a set of networks that includes many pairs of indistinguishable networks: for example the three indistinguishable networks in Fig. 2 are all galled trees—and thus belong to every single one of the classes mentioned above (which are all generalizations of galled trees). Too restrictive, because these classes of networks do not include simple networks that it should be possible to reconstruct from real data. For example, Fig. 5a shows a network N with edge lengths that is not tree-sibling, nor has the visible property, and thus is not galled, nor tree-child (for definitions, see [1]), but which in practice should be reconstructible: apart from the lengths of three edges (x, y, z), N is uniquely determined by the trees that it displays (a consequence of the formal results that we will show in the following), meaning that, given large amounts of data strongly supporting each of these (seven) trees with their correct edge lengths, any method for network inference properly accounting for edge lengths (e. g. based on ML) should be able to reconstruct N, or its canonical form N′. To the best of our knowledge, only three classes of networks have claims of unique identifiability: reduced networks [46,47], regular networks [48] and binary galled trees with no gall containing exactly 4 nodes [49]. These approaches bear some resemblances to ours, but do not include edge lengths in the definition of a network. Moreover, we argue that these classes of networks are still too narrow to be biologically relevant. We briefly describe and comment these previous works below. Moret et al. [46] defined notions of reconstructible, indistinguishable and reduced networks that resemble concepts that we will introduce here. Although some of their results were flawed [47,50], some of the arguments in this introduction are inspired by their paper. Particularly relevant to the current paper is a reduction algorithm to transform a network into its reduced version. (However, the exact definition of the reduced version is unclear: as one of the authors later pointed out [47], “the reduction procedure of Moret et al. [46] is, in fact, inaccurate” and “in this paper we do not attempt to fix the procedure”.) The concept of reduced version is analogous to that of canonical form here, as the authors claim that networks displaying the same tree topologies have the same reduced version (up to isomorphism; Theorem 2 in [46]). This is somehow a weaker analogue of one of our results (Corollary 1); weaker, because it does not claim that, conversely, networks with the same reduced version display the same tree topologies. To have an idea of the difference between our canonical form and the reduced version of Moret and colleagues, in Fig. 6 we compare the canonical form and the reduced version of the same network N1. (N1 and its reduced version are taken from Fig. 15 of [46] to avoid possible issues with the reduction algorithm.) As one can see, the canonical form retains more of the complexity of the original network. Another reduction procedure on network topologies has been studied by Gambette and Huber [49], who prove that if two network topologies reduce to the same topology, then they must display the same tree topologies. Again, this is analogous to, but somehow weaker than our results, since it only provides a sufficient condition for networks to be indistinguishable (which in their context means to display the same tree topologies). This means that there can be irreducible networks that are indistinguishable (e. g. those in Fig. 2) thus failing to achieve the distinguishability goal. Moreover, Gambette and Huber [49] show that a particular class of network topologies (binary galled trees with no gall containing exactly 4 nodes) are uniquely identified by the tree topologies they display. It is clear that this class is too small to achieve the existence goal. Finally, a regular network is a network topology N in which, among other requirements, no two distinct nodes have the same set of descendant leaves (see [48] for a formal definition and characterizations). This requirement implies, among other things, that N cannot contain any reticulation v with outdegree 1 (v and its direct descendant would have the same descendant leaves), which in turn implies that regular networks are special cases of our canonical networks (the latter however also specify edge lengths). In fact regular networks satisfy a property that is analogous to the one we prove here for canonical networks: a regular network N is uniquely determined by the tree topologies that it displays [51], meaning that there can be no other regular network N′ displaying exactly the same set of tree topologies. Willson [51] shows this constructively by providing an algorithm that, given the (exponentially large) set of tree topologies displayed by a regular network R, reconstructs R itself. However, unlike for our canonical forms, for a given network there may exist no regular network displaying the same set of trees (e. g. consider the topology of N′′ in Fig. 5b), thus failing to meet the existence goal. Regularity is in fact a very restrictive constraint for a network. For example, none of the networks in Fig. 5 and Fig. 7 is regular, despite the fact that their topologies are uniquely determined by the trees with edge lengths that they display (a consequence of our results further below). Finally, going back to Fig. 6, collapsing the edge above taxa c and d in R (N1) yields the regular network displaying the same tree topologies as N1 and N2. Again, this shows that the canonical form retains more of the complexity of the original network than its regular counterpart.
All the phylogenies considered here—trees or networks—are rooted. This is because we assume that the analysis uses an outgroup (possibly consisting of multiple taxa, and with no reticulations) for rooting. For simplicity, outgroup lineages are not included in our phylogenies (an exception to this is in Fig. 2). Note however that, because our phylogenies have edge lengths, and because omitting the outgroup is just a convention, the omitted lineages must have the same lengths for a network and all the trees it displays. For example, if we wish to omit the outgroup from N2 in Fig. 2 and from the trees that it displays (T1 and T2 in Fig. 2), then what we obtain are N 2 ′, T 1 ′ and T 2 ′ in Fig. 7. This has a notable consequence: the trees displayed by a rooted network with edge lengths may have a root with outdegree 1 (e. g. T 1 ′ in Fig. 7). For flexibility, we also allow a network to have a root with outdegree 1. Moreover, we allow multiple lengths for an edge in a network, but not in a tree. For example, in Fig. 6, network N 2 ′ has an edge with two lengths (λ7 + λ12 + λ14 and λ7 + λ11 + λ13 + λ14). The motivation behind multiple lengths lies in the observation that, whereas each edge in a phylogenetic tree describing the evolution of non-reticulating organisms trivially corresponds to a unique evolutionary path in the underlying real evolutionary history, when reticulate events have occurred this is not necessarily true: Fig. 8 and Fig. 9 show that some evolutionary scenarios can either be represented using multiedges (multiple edges with the same endpoints) or edges with multiple lengths. Although these two options are mathematically equivalent, graphically the second one leads to more compact representations, and this is why we choose to allow multiple lengths rather than multiedges. For our purposes we only need to consider the case where e has a finite set of lengths (Λ (e) = {λ1 (e), …, λk (e) }). Another unconventional aspect of our networks is the possibility of having nodes with in-degree and out-degree both greater than one. (See, e. g. , the last common ancestor of c and d in N 2 ′ in Fig. 6.) Traditionally, the internal nodes in a phylogenetic network are constrained to belong to one of two different categories: reticulate nodes, with more than one incoming edge and just one outgoing edge, and speciation (or coalescence) nodes, with one incoming edge and multiple outgoing edges. Because reticulate and speciation events are clearly distinct, it is reasonable to constrain internal nodes to only fall in the two categories above. In our framework, this requirement is dropped, and some networks, notably those in canonical form, may have nodes that both represent reticulate and speciation events. In this case, it is important to understand that these nodes represent a potentially complex (and unrecoverable) reticulate scenario, followed by one or more speciation events. Compare, for example, network N and its canonical form N′ in Fig. 5, or N2 and N 2 ′ in Fig. 6. (In the latter, it is especially instructive to consider the reticulate history above the direct ancestor of taxon e.) We use network N1 of Fig. 3 to illustrate the NELP property. In N1 there are three distinct weighted paths having as endpoints the root of N1 and the direct ancestor of b. The lengths of these paths are ℓ1 = λ1 + λ6, ℓ2 = λ2 + λ3 + λ5 + λ8 and ℓ3 = λ2 + λ10 + λ9 + λ8. Moreover, there is another pair of paths having the same endpoints: those of lengths ℓ4 = λ3 + λ5 and ℓ5 = λ10 + λ9. Thus N1 has the NELP property if and only if the three numbers ℓ1, ℓ2 and ℓ3 are all different (note that this implies that also ℓ4 and ℓ5 are different). If edge lengths are taken to represent evolutionary change, rather than time, this is a very mild requirement: when edge lengths are drawn at random from a continuous distribution, the probability that two paths get exactly the same length is zero. On the other hand, the NELP property does not hold for phylogenetic networks where edge lengths are taken to represent time. For these networks, canonical forms may not be unique (see Fig. 10 for an example of this). Even in this case, we believe that inference methods should only consider phylogenetic networks in their canonical form, as this allows to reduce the solution space without any loss in “expressive power”: since every network N has (at least one) canonical form that displays exactly the same set of trees—and therefore has the same fit with the data as N—restricting the solution space to canonical forms always leaves at least one optimal network within this space. The real weakness of using canonical forms in a molecular clock context is that if a canonical form is not unique, then it cannot be considered representative of all the networks indistinguishable from it. As an example of this, consider the indistinguishable networks in Fig. 10: none of these is representative of all the others.
Our results are both negative and positive. The bad news is that any method that scores the fit between a network N and the available data—which may be sequences, distances, splits, trees (with or without edge lengths) —based on the set of trees displayed by N must face an important theoretical limitation: regardless of the amount of available data from the taxa under consideration, some parts of the network representing their evolutionary history may be impossible to recover—most notably the relative order of consecutive reticulate events (see, e. g. , Fig. 3). The good news is that, when edge lengths are taken into account, we can set precise limits to what is recoverable: the canonical form of a network N is a simplified version of N that excludes all the unrecoverable aspects of N. In a canonical form, reticulate events are brought as forward in time as possible, causing the collapse of multiple consecutive nodes. (Compare again network N2 and its canonical form N 2 ′ in Fig. 6.) The importance of the canonical form N′ of a network N lies in the fact that, if we restrict our consideration to networks with the NELP property, N′ is the unique canonical network consistent with perfect and unlimited data from the taxa in N. There is an interesting analogy between soft polytomies in classical phylogenetics and collapsed nodes in a canonical network. Both represent lack of knowledge about the order of evolutionary events: speciations or more generally lineage splits in the first case, and reticulate events in the second. However, there is also an important difference between them: whereas in principle polytomies can be resolved by collecting further data from the taxa in the tree (for example, by extensive sequencing of their genomes [52]), the standard network inference methods considered here cannot resolve collapsed nodes in a canonical network, irrespective of the amount of data from the taxa under consideration. This difference is mitigated by the observation that increased taxon sampling may indeed permit to resolve the collapsed nodes, when the new lineages break adjacencies between reticulate nodes. However, such lineages may not always exist or they may be difficult to sample. The present work has several consequences that should be of interest both to the biologists concerned by the use of methods for phylogenetic network inference, and to the researchers interested in the development of these methods. We illustrate these consequences starting from a well-known problem of network inference methods, that of multiple optima. It has been noted before that many of the inference methods that have been recently proposed—especially those solely based on topological features—often return multiple optimal networks: Huson and Scornavacca show a striking example of this (Fig. 2 in [53]), where the problem of finding the simplest network displaying two given tree topologies admits at least 486 optimal solutions. The existence of multiple optimal networks for a given data set is essentially due to two reasons: insufficient data and non-identifiability. For the example of 486 optimal solutions, this large number may be partly due to the fact that the goal was to achieve consistency with only two tree topologies. More data may enable to discriminate among the 486 returned networks. Non-identifiability, which occurs when none of the allowed data can discriminate between two or more networks, is a more serious problem than insufficient data, as it cannot be solved by simply increasing the size of the input sample. Another interesting example appears in a paper by Albrecht et al. [54], which we reproduce here in Fig. 11. Here, there are only three optimal networks, essentially differing for which of the three clades {A. bicornis, A. longissima, A. sharonensis}, {A. uniaristata, A. comosa} and {A. tauschii} is considered as a hybrid (in this example reticulations represent hybridizations). This pattern is entirely analogous to that of the three networks in Fig. 2 (with a, b and c replaced by the three clades above), meaning that these three networks are indistinguishable to methods not accounting for edge lengths. Therefore, in this example, the existence of multiple optimal solutions is entirely due to non-identifiability. All this motivates three recommendations: Correspondingly, we recommend that edge lengths should be accounted for in the analyses (point 1) and, for each of the indistinguishable classes resulting from this choice, we identify a canonical network that, for all practical purposes, can be considered to be unique. Most important to the end users, we propose that a canonical network N ̂ is what should be given as the result of the inference, with the caveat that N ̂ is a way to represent a class of networks that are all equally supported (point 2). In a canonical form N ̂, the aspects that are not common to all networks in this class are collapsed, as described above. This will help the evolutionary biologist to locate the uncertainties in the phylogeny, and possibly to choose further taxa to resolve them. Finally, we propose that inference methods only attempt to search among—or construct—phylogenetic networks in their canonical form (point 3). We note that accounting for yet more characteristics of the data may reduce (or eliminate altogether) the identifiability issues for phylogenetic networks. In the case of sequence-based methods, one may take into account the natural order of sites within a sequence [11–13,55,56]. Similarly, for reconstruction methods based on collections of subtrees, one could observe and use the relative position of the different genomic regions supporting the input trees. However, these relative positions must be conserved across the genomes being analyzed, a condition which may hold for recombining organisms (e. g. individuals within a population or different viral strains), but which is not obvious when studying a group of taxa that have undergone reticulate events (e. g. , hybridization) at some point in a distant past. The main conclusion of the present study is the following: unless one abandons any optimization criterion that scores a network solely based on the trees it displays, the reconstruction should be carried out in a reduced space of networks: that of the canonical forms defined here. The motivation for this lies in the fact that canonical networks are guaranteed to be uniquely determined, if sufficient data are available. Once a canonical form N ̂ is inferred, it must be kept in mind that even assuming that the inference is free of statistical error, the true phylogenetic network is just one of the many networks having N ̂ as canonical form. Compared to what biologists are used to for phylogenetic trees—where in principle it is always possible to resolve uncertainties—it is clear that this requires an important change of perspective.
In order to prove that any network N has a canonical form, we describe an algorithm to transform N into a canonical network indistinguishable from N. The algorithm simply consists of repeatedly applying to N = (V, E, φ, Λ) one of the following two reduction rules, until neither can be executed (see Fig. 12): Funnel suppression (R1). Given a funnel v with k ≥ 1 in-edges (u1, v), (u2, v), …, (uk, v) and out-edge (v, w), remove v and all these edges from N and introduce k new edges (u1, w), (u2, w), …, (uk, w). For all i ∈ {1,2, …, k} assign to (ui, w) the lengths Λ ( (ui, w) ): = Λ ( (ui, v) ) + Λ ( (v, w) ), where the sum of two sets of numbers A and B is defined as A + B = {a + b: a ∈ A, b ∈ B}. Multiedge merging (R2). Given a collection of multi-edges (u, w) with multiplicity k and lengths Λ 1 ′, Λ 2 ′, …, Λ k ′, replace these edges with a single edge with lengths ⋃ i = 1 k Λ i ′. An example of the reduction of a network to its canonical form is shown in Fig. 13. Note that, even if the algorithm may temporarily produce multi-edges, the network produced in the end obviously does not have any multi-edge (otherwise we could still apply rule R2). Proof of part (i) of Theorem 1. We must prove that any network N = (V, E, φ, Λ) has a canonical form. For this, we apply the reduction algorithm described above, thus obtaining a sequence N0 = N, N1, …, Nm, where each Ni+1 is obtained from Ni by applying either R1 or R2. Neither R1 nor R2 can be applied to Nm. We prove that Nm is a canonical form of N. Although, strictly speaking, Ni may not be a network (as it potentially contains multi-edges), the notion of trees displayed by Ni, and thus that of indistinguishability, trivially extends to these multigraphs. First, note that the algorithm terminates after a finite number of iterations (m). This is true because at each iteration the size of E is reduced by at least one. Moreover, the resulting network Nm is funnel-free, since no reduction of type R1 can be applied to it. What is left to prove is that Nm is indistinguishable from N = N0. To this end we prove that, at each iteration, Ni and Ni+1 are indistinguishable, i. e. 𝓣 (Ni) = 𝓣 (Ni+1). In other words any tree T is displayed by Ni if and only if T is displayed by Ni+1. Let T be displayed by Ni. Then T can be obtained by suppressing all suppressible nodes from a tree Ti contained in Ni. We consider three cases. (1) If none of the edges in Ti is involved in the reduction transforming Ni into Ni+1, then clearly Ti is still contained in Ni+1 and thus T is still displayed by Ni+1. (2) If Ti is involved in a R1 reduction, then it contains a funnel v and it contains one of the in-edges of the funnel, say (uj, v), with length λj ∈ Λj = Λ ( (uj, v) ), along with the out-edge (v, w), with length λ0 ∈ Λ0 = Λ ( (v, w) ). Now, let Ti+1 be the tree obtained from Ti by suppressing the suppressible node v and thus creating a new edge (uj, w) with length λj + λ0. Because the R1 reduction creates a new edge (uj, w) with length set Λj + Λ0, containing the value λj + λ0, then Ti+1 is contained in Ni+1. Moreover, it easy to see that T can still be obtained by suppressing all suppressible nodes from Ti+1. Thus T is still displayed by Ni+1. (3) If Ti is involved in a R2 reduction, then it contains one of the edges of a multi-edge (u, w), with a length λ belonging to one of the length sets Λ 1 ′, Λ 2 ′, …, Λ k ′ associated to the k copies of (u, w). Thus we have that λ ∈ ⋃ i = 1 k Λ i ′, which implies that Ti is still contained in Ni+1 and thus T is still displayed by Ni+1. This concludes the proof of 𝓣 (Ni) ⊆ 𝓣 (Ni+1). In order to prove that, conversely, 𝓣 (Ni+1) ⊆ 𝓣 (Ni), one can proceed in a similar way as above: if T is displayed by Ni+1, then T can be obtained by suppressing all suppressible nodes from a tree Ti+1 contained in Ni. By considering three cases analogous to the ones above regarding the involvement of Ti+1 in the reduction transforming Ni into Ni+1, we can prove that in all these cases T is already displayed by Ni. Thus Ni and Ni+1 are indistinguishable, which concludes our proof. □ We note informally that the order of application of the possible reductions in the algorithm above is irrelevant to the end result. To see this, it suffices to show that if two different reductions are applicable to a network, then the result of applying them is the same irrespective of the order of application. As we do not need this remark for the other results in this paper, we do not give a formal proof of it. Lemma 1. Let N be a network and N′ a canonical form of N obtained by applying the reduction algorithm. If N satisfies the NELP property, then N′ satisfies the NELP property. Proof: We prove that for each basic step of the reduction algorithm—transforming Ni into Ni+1 via a reduction rule R1/R2—if Ni satisfies the NELP property, then Ni+1 also satisfies it. Suppose the contrary; then, Ni+1 contains two distinct weighted paths ρ1, ρ2 with the same endpoints u and v and same lengths. Because R1/R2 cannot create new nodes, u and v are also nodes in Ni. Moreover, it is easy to see that each weighted path ρ in Ni from u to v gives rise to exactly one weighted path f (ρ) in Ni+1 from u to v, with exactly the same length as ρ. Now take two weighted paths in Ni, one in the preimage f−1 (ρ1) and the other in the preimage f−1 (ρ2). These two weighted paths in Ni are distinct (as ρ1 ≠ ρ2), have the same endpoints (u and v) and the same length. But then Ni violates the NELP property, leading to a contradiction. We thus have that if Ni satisfies the NELP property, then Ni+1 also satisfies it. By iterating the argument above for each step in the reduction algorithm, the lemma follows. □ The proof of Theorem 1, part (ii), is rather technical. In this section, we introduce a number of new concepts and state the main intermediate results that are necessary to obtain this result. We leave their detailed proofs to S1 Text, together with the obvious definitions of basic concepts such as that of isomorphic networks, sub-network and union of two networks. Definition 3. (Root-leaf path, prefix, postfix, wishbone, crack.) Let N be a network on 𝓧 and (π, λ) be a weighted path in N from the root of N to a leaf labelled by x ∈ 𝓧. Now consider the sub-network P = (V (π), E (π), φ∣{x}, λ) on {x} consisting of all the nodes and edges in π and associated labels. Any sub-network of N such as P is called a root-leaf path of N. Given a root-leaf path P and a node v belonging to it, any weighted path formed by all the ancestors [descendants] of v in P is a prefix [suffix] of P. Note that a prefix [suffix] only consists of one node when v is the root [leaf] of P. A wishbone of N is any sub-network of N formed by taking the union of two root-leaf paths that have in common only a prefix. A crack of N is any sub-network of N formed by taking the union of two root-leaf paths that have in common only a prefix and a suffix. Fig. 14 illustrates the definitions above. Note that any root-leaf path P is both a wishbone and a crack, as P is the result of the union of P with itself, and P has a common prefix and a common suffix with P. Moreover, any sub-network R that can be obtained from a root-leaf path by attributing two lengths to one of its edges e is a crack. Finally, note that wishbones and cracks are networks, and thus the notion of isomorphism (Definition 5 in S1 Text) can be applied to them. The proof of part (ii) in Theorem 1 depends on two important results (Propositions 1 and 2 below), whose proofs can be found in S1 Text. The first states that a network with the NELP property is uniquely determined by the wishbones and cracks it contains. Proposition 1. Two networks N1 and N2 with the NELP property are isomorphic if and only if they contain the same wishbones and cracks (up to isomorphism). Proposition 1 is interesting on its own as it suggests an enumerative algorithm to verify whether two networks with the NELP property are isomorphic. Unfortunately this algorithm would be impractical, as the number of wishbones (or cracks) in a network is not polynomial in the size of the network. Also note that we require N1 and N2 to satisfy the NELP property because there exist non-isomorphic networks containing the same wishbones and cracks: for example the networks in the bottom line of Fig. 10. The second result that we need is the following: Proposition 2. Let N1 and N2 be two indistinguishable funnel-free networks, satisfying the NELP property. Then they contain the same wishbones and cracks (up to isomorphism). Proof of part (ii) of Theorem 1. Let N be a network with the NELP property and N′ a canonical form of N obtained by applying the reduction algorithm. By Lemma 1, N′ satisfies the NELP property. Now suppose that there exists another canonical form of N, called N′′, satisfying the NELP property. By transitivity, N′ and N′′ are indistinguishable. Because N′ and N′′ are indistinguishable, funnel-free and with the NELP property, N′ and N′′ must contain the same wishbones and cracks (because of Proposition 2). But then, because of Proposition 1, N′ and N′′ are isomorphic. □ We note that some of our arguments in S1 Text lead us to conjecture that a funnel-free network satisfying the NELP property cannot be indistinguishable from a funnel-free network violating the NELP property. This claim would allow us to simplify the statement of Theorem 1: networks with the NELP property would be guaranteed to have a unique canonical form (not just among networks with the NELP property, but among all networks). Unfortunately, to this date, we were unable to prove this conjecture. Nonetheless, note that the reduction algorithm returns, for any network with the NELP property, its unique canonical form with the NELP property (by Lemma 1). It remains to prove the two corollaries at the end of the Results section. The first one states that two networks N1 and N2 satisfying the NELP property are indistinguishable if and only if their unique canonical forms with the NELP property, N 1 ′ and N 2 ′ respectively, are isomorphic. By Lemma 1, N 1 ′ and N 2 ′ can be obtained by applying the reduction algorithm to N1 and N2. Proof of Corollary 1. The if part trivially follows from the transitivity of indistinguishability. As for the only if part, note that (again by transitivity) N 1 ′ is indistinguishable from N2. As it is also funnel-free, N 1 ′ is a canonical form of N2. Because N2 can only have one canonical form satisfying the NELP property (by Theorem 1 (ii) ), N 1 ′ and N 2 ′ must be the same network (up to isomorphism). □ As for Corollary 2, we recall that it states that a canonical network N with the NELP property is uniquely determined by the trees it displays. Proof of Corollary 2. Let N and N′ be indistinguishable canonical networks satisfying the NELP property. Then, N and N′ are both canonical forms of N satisfying the NELP. But then, by Theorem 1 (ii), N and N′ must be the same network (up to isomorphism). □
We consider here an elementary question for the inference of phylogenetic networks: what networks can be reconstructed. Indeed, whereas in theory it is always possible to reconstruct a phylogenetic tree, given sufficient data for this task, the same does not hold for phylogenetic networks: most notably, the relative order of consecutive reticulate events cannot be determined by standard network inference methods. This problem has been described before, but no solutions to deal with it have been put forward. Here we propose limiting the space of reconstructible phylogenetic networks to what we call “canonical networks”. We formally prove that each network has a (usually unique) canonical form—where a number of nodes and branches are merged—representing all that can be uniquely reconstructed about the original network. Once a canonical network N ̂ is inferred, it must be kept in mind that—even with perfect and unlimited data—the true phylogenetic network is just one of the potentially many networks having N ̂ as canonical form. This is an important difference to what biologists are used to for phylogenetic trees, where in principle it is always possible to resolve uncertainties, given enough data.
Reconstructible Phylogenetic Networks: Do Not Distinguish the Indistinguishable