Weak Hierarchies: A Central Clustering Structure

Bertrand, Patrice; Diatta, Jean

doi:10.1007/978-1-4939-0742-7_14

Cited by 5 publications

(6 citation statements)

References 14 publications

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“…From the point of view of clustering systems, phylogenetic networks suggest properties that may also be of relevance in practical data analysis beyond applications in phylogenetics. A clustering systems between hierarchies and weak hierarchies [11], (L)-clustering systems appear as an attractive alternative e.g. to pyramidal clustering [10] for data that are not naturally linearly ordered.…”

Section: Discussionmentioning

confidence: 99%

Clustering Systems of Phylogenetic Networks

Hellmuth¹,

Schaller²,

Stadler³

2022

Preprint

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Rooted acyclic graphs appear naturally when the phylogenetic relationship of a set X of taxa involves not only speciations but also recombination, horizontal transfer, or hybridization, that cannot be captured by trees. A variety of classes of such networks have been discussed in the literature, including phylogenetic, level-1, tree-child, tree-based, galled tree, regular, or normal networks as models of different types of evolutionary processes. Clusters arise in models of phylogeny as the sets C(v) of descendant taxa of a vertex v. The clustering system C N comprising the clusters of a network N conveys key information on N itself. In the special case of rooted phylogenetic trees, T is uniquely determined by its clustering system C T . Although this is no longer true for networks in general, it is of interest to relate properties of N and C N . Here, we systematically investigate the relationships of several well-studied classes of networks and their clustering systems. The main results are correspondences of classes of networks and clustering system of the following form: If N is a network of type X, then C N satisfies Y, and conversely if C is a clustering system satisfying Y then there is network N of type X such that C ⊆ C N .This, in turn, allows us to investigate the mutual dependencies between the distinct types of networks in much detail.

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Section: Discussionmentioning

confidence: 99%

Clustering Systems of Phylogenetic Networks

Hellmuth¹,

Schaller²,

Stadler³

2022

Preprint

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“…Weak hierarchies were introduced in Bandelt and Dress ( 1989 ) and subsequently have been studied in detail in the context of clustering systems, e.g., in Brucker and Gély ( 2009 ); Bertrand and Diatta ( 2014 ). Binary clustering system are considered systematically in Barthélemy and Brucker ( 2008 ).…”

Section: Least Common Ancestors and Lca-networkmentioning

confidence: 99%

Clustering systems of phylogenetic networks

Hellmuth,

Schaller,

Stadler

2023

Theory Biosci.

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Rooted acyclic graphs appear naturally when the phylogenetic relationship of a set X of taxa involves not only speciations but also recombination, horizontal transfer, or hybridization that cannot be captured by trees. A variety of classes of such networks have been discussed in the literature, including phylogenetic, level-1, tree-child, tree-based, galled tree, regular, or normal networks as models of different types of evolutionary processes. Clusters arise in models of phylogeny as the sets $${{\,\mathrm{\texttt{C}}\,}}(v)$$ C ( v ) of descendant taxa of a vertex v. The clustering system $$\mathscr {C}_N$$ C N comprising the clusters of a network N conveys key information on N itself. In the special case of rooted phylogenetic trees, T is uniquely determined by its clustering system $$\mathscr {C}_T$$ C T . Although this is no longer true for networks in general, it is of interest to relate properties of N and $$\mathscr {C}_N$$ C N . Here, we systematically investigate the relationships of several well-studied classes of networks and their clustering systems. The main results are correspondences of classes of networks and clustering systems of the following form: If N is a network of type $$\mathbb {X}$$ X , then $$\mathscr {C}_N$$ C N satisfies $$\mathbb {Y}$$ Y , and conversely if $$\mathscr {C}$$ C is a clustering system satisfying $$\mathbb {Y},$$ Y , then there is network N of type $$\mathbb {X}$$ X such that $$\mathscr {C}\subseteq \mathscr {C}_N$$ C ⊆ C N .This, in turn, allows us to investigate the mutual dependencies between the distinct types of networks in much detail.

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“…Moreover, they admit a convenient graphical representation (Brucker and Préa, 2015). Finally totally balanced hypergraphs are weak-hierarchies Gély, 2010 -see Bandelt andDress (1989) for a definition of weak-hierarchies), they have a lot of good practical properties (see Bertrand and Diatta, 2014 for instance) like admitting a relatively small number of overlapping clusters (at most the square of the number of elements) or being equivalent to an underlying metric model.…”

Section: Introductionmentioning

confidence: 99%

Totally Balanced Dissimilarities

2019

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We show in this paper a bijection between totally balanced hypergraphs and so called totally balanced dissimilarities. We give an efficient way (O(n 3 ) where n is the number of elements) to (i) recognize if a given dissimilarity is totally balanced and (ii) approximate it if it is not the case. We also introduce a new kind of dissimilarity which generalize chordal graphs and allows a polynomial number of cluster that can be easily computed and interpreted.

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Weak Hierarchies: A Central Clustering Structure

Cited by 5 publications

References 14 publications

Clustering Systems of Phylogenetic Networks

Clustering Systems of Phylogenetic Networks

Clustering systems of phylogenetic networks

Totally Balanced Dissimilarities

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