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On the Variance of Internode Distance Under the Multispecies Coalescent

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Book cover Comparative Genomics (RECOMB-CG 2018)

Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 11183))

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Abstract

We consider the problem of estimating species trees from unrooted gene tree topologies in the presence of incomplete lineage sorting, a common phenomenon that creates gene tree heterogeneity in multilocus datasets. One popular class of reconstruction methods in this setting is based on internode distances, i.e. the average graph distance between pairs of species across gene trees. While statistical consistency in the limit of large numbers of loci has been established in some cases, little is known about the sample complexity of such methods. Here we make progress on this question by deriving a lower bound on the worst-case variance of internode distance which depends linearly on the corresponding graph distance in the species tree. We also discuss some algorithmic implications.

This work was supported by funding from the U.S. National Science Foundation DMS-1149312 (CAREER), DMS-1614242 and CCF-1740707 (TRIPODS). We thank Tandy Warnow for suggesting the problem and for helpful discussions.

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Notes

  1. 1.

    In keeping with much of the literature on the MSC, we use the generic term gene to refer to any genomic region experiencing low rates of internal recombination, not necessarily a protein-coding region.

  2. 2.

    Note however that the associated branch lengths may differ from \(\varGamma \).

  3. 3.

    Note that it is trivial that \((d^{\mathcal {T}_j}_\mathrm {g}(x,y))_{x,y}\) is an additive metric associated to gene tree \(\mathcal {T}_j\). On the other hand it is far from trivial that averaging over the MSC leads to an additive metric associated to the species tree.

  4. 4.

    A straightforward modification of the argument also works for odd n.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin–Madison, Madison, WI, 53706, USA

    Sébastien Roch

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  1. Sébastien Roch
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Correspondence to Sébastien Roch .

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  1. McGill University, Montréal, QC, Canada

    Mathieu Blanchette

  2. Université de Sherbrooke, Sherbrooke, QC, Canada

    Aïda Ouangraoua

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Roch, S. (2018). On the Variance of Internode Distance Under the Multispecies Coalescent. In: Blanchette, M., Ouangraoua, A. (eds) Comparative Genomics. RECOMB-CG 2018. Lecture Notes in Computer Science(), vol 11183. Springer, Cham. https://doi.org/10.1007/978-3-030-00834-5_11

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  • DOI: https://doi.org/10.1007/978-3-030-00834-5_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-00833-8

  • Online ISBN: 978-3-030-00834-5

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