Those readers active in the field of evolutionary phylogenetic networks will know that there are many different definitions of “rooted phylogenetic network” in circulation. While certain features are almost universal (e.g. rooted, acyclicity), many are not. Why do these differences arise? There are multiple answers to this. Some arise because of differing opinions on what biologically realistic is, and (relatedly) what the correct balance is between biological detail and mathematical abstraction. Others arise because they make optimization problems on networks easier to solve. This should not automatically be viewed as mathematics prescribing reality to biology, but rather as the observation that

*if*evolution looks like this

*then*certain optimization problems can be solved well. Finally, some differences are superficial; they do not lead to any intrinsic differences in the model or its underlying mathematical structure. Of course, “superficial” is highly context dependent, as some of the examples below will show.

Here we list some well-known and less-well known properties that have surfaced in definitions of rooted phylogenetic network. We will take as given that evolution is directed i.e. that the arcs in the network have an explicit orientation (representing the flow of time). Below most properties we show a figure of a network violating it.

The question to our readers, particularly those from the biological side of the spectrum: which of the following properties make sense? And which other restrictions would make more sense biologically?

**Rooted**

A root is a node with indegree 0, meaning that it does not have any ancestors. If evolution is assumed to be acyclic (see below) then there will always be at least one root. Most articles writing about rooted phylogenetic networks assume a

*single*root (which is necessarily an ancestor of every node in the network). Some time ago on this blog David raised the question of whether it would not sometimes be better to allow*multiple*roots. This is an interesting point both from the perspective of interpretation (what does it mean?) and its impact on algorithmic efficiency.**Acyclic**

Most models assume acyclicity: it is not possible to walk along the arcs of the network (respecting their orientation) such that you end up back where you started. The most intuitive argument for this is the passing of time: if arcs represent forward motion through evolutionary time, how can you end up back at an earlier point in time? Recently someone pointed out to me that in the reconciliation literature – where one shows how to reconcile a given gene tree with a given species tree – acyclicity is actually not sacred at all. The reason for this is that, without the acyclicity constraint, the problem becomes computationally tractable. See this recent RECOMB 2013 article [3] for an example of this.

**Time-consistent**

This is an interesting property. It was introduced to prevent reticulation events between non-contemporaneous taxa. That is, to avoid absurd situations such as an organism hybridizing with its ancestor. The most common mathematical articulation of time-consistency is this: it should be possible to put a “time-stamp” on each node of the network such that (a) time always moves strictly forward along tree-edges , and (b) all the nodes involved in a reticulation event have the same time-stamp. The figure here shows a network that is

*not*time-consistent. For many contextualisations of “reticulation event” time-consistency seems to make a lot of sense. But there is a catch. A network might fail to be time-consistent only for the rather artificial reason that we failed to sample a taxon that was, in fact, part of the network (incomplete taxon sampling). Given the reality of incomplete data, demanding time-consistency might be too prohibitive. However, as with all restrictions in this blog, it is perhaps useful as a selection criterion for preferring one network over another.Figure 1: A network that is not time-consistent. The red and blue node are the two parents of a reticulation node but cannot have coexisted in time. |

**Degree-restricted**

The indegree of a node is the number of parents of it, and the outdegree of a node is the number of children of it. In a bifurcating, rooted phylogenetic tree all nodes have indegree-1 (except the root: indegree-0) and outdegree-2. Polytomies have outdegree-3 or higher. In a rooted phylogenetic network we also have reticulation nodes i.e. nodes with indegree-2 or higher. Articles differ in the degree-restrictions they place on nodes; there is an entire zoo of different permutations possible. Many articles agree that nodes with indegree-1 and outdegree-1 should not be allowed, because they are phylogenetically uninformative (and indeed such nodes are also rarely encountered in the phylogenetic tree literature). But what about polytomies? And what about reticulation nodes: should they be permitted to have indegree-3 or higher, and if so how should such “reticulate polytomies” be interpreted? From a parsimony perspective it is usual to argue that reticulate polytomies should be more “expensive” than indegree-2 reticulations, because a single reticulation polytomy can explain far more discordance than a single indegree-2 reticulation. Interestingly, some optimization problems are not really affected at all by degree-restrictions on phylogenetic networks (e.g. the hybridization number problem) while others are highly sensitive to degree-restrictions (e.g. the small parsimony problem).

**T**

**ree-child**

Every

**node has at least one non-reticulate child. This restriction makes sure that there are no "invisible nodes", i.e. nodes from which all paths end up in reticulations. As a result, this restriction makes many computational problems more tractable and mathematical reasoning easier. For example, consider the basic Tree Containment problem, i.e. given a phylogenetic network and a phylogenetic tree, decide if the network displays the tree, see [8]. This problem was shown to be tractable for tree-child networks, while it is NP-hard for most other classes of networks (time-consistent, tree-sibling, regular). This seems important because if it is already hard to tell if a given tree is in a given network, then it seems a daunting task to try to build networks of that class from any kind of data.**

It should be noted that there is of course no guarantee that "real" evolutionary histories are tree-child. In fact, simulation studies show that only under very low recombination rates one can expect tree-child networks [1][2].

**Tree-sibling**

Every reticulate node has at least one non-reticulate sibling. Originally introduced by Cardona, LlabrÃ©s, RossellÃ³ and Valiente who write "Biologically, this condition means that for each of the reticulation events, at least one of the species involved in it has also some descendant through mutation" and showed that networks can efficiently be compared (with a polynomial-time computable metric) if they are both tree-sibling and time-consistent.

An advantage of the class of tree-sibling networks is that it is much larger than the class of tree-child networks. If a reticulation has no non-reticulate siblings, then its parents have no non-reticulate children. Hence, every tree-child network is tree-sibling. However, it can easily be seen that there are many tree-sibling networks that are not tree-child. Computationally, the tree-sibling restriction does not seem to help as much as the tree-child restriction. Again, there is no guarantee that the "real" network is tree-sibling, but it might be more likely, see [1][2].

**R**

**eticulation-visible**

From every reticulation node there is a path to some leaf or cut-arc (a branch whose removal disconnects the network) such that this path does not pass through any reticulations. This is another attempt to weaken the tree-child restriction, thus obtaining a larger class of networks for which many computational problems are still tractable. It is incomparable with tree-sibling (a reticulation-visible network does not have to be tree-sibling and vice versa) but this class clearly contains the class of tree-child networks, and in fact is much larger. It does not forbid all invisible nodes, but just invisible reticulation nodes. It has been shown in the book by Huson, Rupp and Scornavacca that the so-called Cluster Containment problem becomes tractable for reticulation-visible networks. If the same is the case for the Tree Containment problem is still an open question.

**Galled**

**trees**

All reticulation cycles are disjoint. Introduced by Gusfield, Eddhu and Langley [6] although studied before under different names. Makes computational problems much easier but seems biologically unrealistic. On the other hand, galled trees could have a future in the context of data-display networks, by using the galls to show where the reticulate activity is in the network, rather than claiming that each gall represents exactly one reticulation event.

**Galled**

**networks**

Each arc leaving a reticulation node is a cut-arc. Introduced by Huson and KlÃ¶pper [7] (who gave a different but equivalent definition) as a generalization of galled trees, giving a fast algorithm for constructing galled networks from clusters. Hasn't been studied much since. (Be aware that there is also an article using "galled network" as an alternative term for galled tree.)

Figure 5: A network that is not a galled network. The red arc leaves a reticulation node but is not a cut-arc. |

**Level-**

**k**

**(not**

**a**

**restriction,**

**but**

**a**

**measurement)**

Also a generalization of galled trees (which are basically level-1 networks). Every network is a level-

*k*network for some*k*. Hence, “level-*k*” should not really be viewed as a topological restriction, but rather as a measure of how “tangled” (intensely concentrated) the islands of reticulation are in the network. The higher*k*is, the more tangled the network is; level-0 networks are simply trees. For more information, see a previous blog. Other proposed measurements of tangledness include*k*-nested,*r*-reticulation and depth-*k*.**Regular**

**and**

**normal**

If you see a phylogenetic network as a representation of a set of clusters, then it makes sense to consider regular networks (introduced by Baroni, Semple and Steel). A network is regular if it is the so-called "cover digraph" (Hesse diagram) of its set of clusters. Hence, for each set of clusters, there exists a unique regular network with precisely that set of (hardwired) clusters. Normal networks have the additional requirement of being tree-child, thus forming a very restricted class of networks. For example, the network in Figure 1 is not regular and hence also not normal.

**DAG**

**or**

**tree-with-edges-added?**

Figure 6: A network that cannot be obtained by adding edges to a tree (for common edge-adding rules). |

It can be shown that any tree-sibling network can be constructed by adding edges to a tree, but the network in Figure 3 shows that the converse is not true (the shown network can be constructed by adding edges to a tree, but is not tree-sibling).

This restriction touches on the fundamental question whether there exists something like a species tree, and if it might be possible to reconstruct this species tree before starting network analysis.

A related but stronger (?) restriction was recently used by Wu [9]. In his RECOMB 2013 article, he writes “(R1) For a network N, when only one of the incoming edges of each reticulation node is kept and the other is deleted, we always derive a tree T'.”

**Conclusion**

We see from this list that there are already some quite different topological properties and restrictions in circulation for rooted phylogenetic networks. To biologists these discussions might appear to be a strange side-show to keep computer scientists in work. But it runs deeper than that, because it touches on three fundamental issues. Firstly, what are we trying to model exactly? Secondly, the importance of understanding the networks that your favourite software for constructing rooted phylogenetic networks will not build, however biologically relevant, due to the fact that they are a priori excluded from its search space. Finally, since the total number of networks is huge, it could be inevitable to focus on certain restricted classes of networks when one wants to search through network-space efficiently.

**Note:**There is a follow-up post Topological restrictions: some comments.
REFERENCES

[1] Miguel Arenas, Mateus Patricio, David Posada and Gabriel Valiente. Characterization of Phylogenetic Networks with NetTest. In BMCB, Vol. 11:268, 2010.

[2] Miguel Arenas, Gabriel Valiente and David Posada, Characterization of Reticulate Networks Based on the Coalescent with Recombination, Mol Biol Evol (2008) 25 (12):2517-2520.

[3] Mukul S. Bansal, Eric J. Alm, Manolis Kellis, Reconciliation Revisited: Handling Multiple Optima when Reconciling with Duplication, Transfer, and Loss, RECOMB 2013.

[4] Gabriel Cardona, Merce Llabres, Francesc Rossello, Gabriel Valiente, The comparison of tree-sibling time consistent phylogenetic networks is graph isomorphism-complete, arXiv:0902.4640 [q-bio.PE], 2009.

[4] Gabriel Cardona, Merce Llabres, Francesc Rossello, Gabriel Valiente, The comparison of tree-sibling time consistent phylogenetic networks is graph isomorphism-complete, arXiv:0902.4640 [q-bio.PE], 2009.

[5] Gabriel Cardona, MercÃ¨ LlabrÃ©s, Francesc RossellÃ³ and Gabriel Valiente. A Distance Metric for a Class of Tree-Sibling Phylogenetic Networks. In BIO, Vol. 24(13):1481-1488, 2008.

[6] Dan Gusfield, Satish Eddhu and Charles Langley. Efficient reconstruction of phylogenetic networks with constrained recombination. In CSB03, Pages 363-374, 2003.

[7] Daniel H. Huson and Tobias KlÃ¶pper. Beyond Galled Trees - Decomposition and Computation of Galled Networks. In RECOMB 2007, Vol. 4453:211-225 of LNCS.

[8] Leo van Iersel, Charles Semple and Mike Steel, Locating a Tree in a Phylogenetic Network,

*Information**Processing**Letters*, 110 (23), pp. 1037-1043 (2010).
[9] Yufeng Wu. An Algorithm for Constructing Parsimonious Hybridization Networks with Multiple Phylogenetic Trees. RECOMB 2013.

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