arXiv subjects

Stephen J. Willson

Publications and source records attributed to Stephen J. Willson.

4 recordsLinked to original sources

arXiv2016

Comparing and simplifying distinct-cluster phylogenetic networks

Phylogenetic networks are rooted acyclic directed graphs in which the leaves are identified with members of a set X of species. The cluster of a vertex is the set of leaves that are descendants of the vertex. A network is "distinct-cluster" if distinct vertices have distinct clusters. This paper focuses on the set DC(X) of distinct-cluster networks whose leaves are identified with the members of X. For a fixed X, a metric on DC(X) is defined. There is a "cluster-preserving" simplification process by which vertices or certain arcs may be removed without changing the clusters of any remaining vertices. Many of the resulting networks may be uniquely determined without regard to the order of the simplifying operations.

math.CO↗

arXiv2016

CSD Homomorphisms Between Phylogenetic Networks

Since Darwin, species trees have been used as a simplified description of the relationships which summarize the complicated network $N$ of reality. Recent evidence of hybridization and lateral gene transfer, however, suggest that there are situations where trees are inadequate. Consequently it is important to determine properties that characterize networks closely related to $N$ and possibly more complicated than trees but lacking the full complexity of $N$. A connected surjective digraph map (CSD) is a map $f$ from one network $N$ to another network $M$ such that every arc is either collapsed to a single vertex or is taken to an arc, such that $f$ is surjective, and such that the inverse image of a vertex is always connected. CSD maps are shown to behave well under composition. It is proved that if there is a CSD map from $N$ to $M$, then there is a way to lift an undirected version of $M$ into $N$, often with added resolution. A CSD map from $N$ to $M$ puts strong constraints on $N$. In general, it may be useful to study classes of networks such that, for any $N$, there exists a CSD map from $N$ to some standard member of that class.

q-bio.PE↗

arXiv2010

Restricted trees: simplifying networks with bottlenecks

Suppose N is a phylogenetic network indicating a complicated relationship among individuals and taxa. Often of interest is a much simpler network, for example, a species tree T, that summarizes the most fundamental relationships. The meaning of a species tree is made more complicated by the recent discovery of the importance of hybridizations and lateral gene transfers. Hence it is desirable to describe uniform well-defined procedures that yield a tree given a network N. A useful tool toward this end is a connected surjective digraph (CSD) map f from N to N' where N' is generally a much simpler network than N. A set W of vertices in N is "restricted" if there is at most one vertex from which there is an arc into W, thus yielding a bottleneck in N. A CSD map f from N to N' is "restricted" if the inverse image of each vertex in N' is restricted in N. This paper describes a uniform procedure that, given a network N, yields a well-defined tree called the "restricted tree" of N. There is a restricted CSD map from N to the restricted tree. Many relationships in the tree can be proved to appear also in N.

q-bio.PE↗

arXiv2009

Regular networks are determined by their trees

A rooted acyclic digraph N with labelled leaves displays a tree T when there exists a way to select a unique parent of each hybrid vertex resulting in the tree T. Let Tr(N) denote the set of all trees displayed by the network N. In general, there may be many other networks M such that Tr(M) = Tr(N). A network is regular if it is isomorphic with its cover digraph. This paper shows that if N is regular, there is a procedure to reconstruct N given Tr(N). Hence if N and M are regular networks and Tr(N) = Tr(M), it follows that N = M, proving that a regular network is uniquely determined by its displayed trees.

q-bio.PE↗