Abstract
In recent years, several different versions of the Shapley value have been introduced in phylogenetics for the purpose of ranking biodiversity data in order to decide whether to preserve the data or not. Two of these Shapley values are the rooted and unrooted Shapley value which have been compared with the fair proportion index since this index is easier to compute. In particular, it was proved for the former that it is identical with the fair proportion index and numerical data was presented by several authors that the latter is strongly correlated with the fair proportion index. In this paper, we will prove a theoretical result which supports this observation. More precisely, we will prove that in random phylogenetic trees under the \(\beta \)-splitting model, the correlation coefficient between the unrooted Shapley value and the fair proportion index indeed tends to one for all \(\beta \) with \(\beta >-\,1\). We also present data which suggests that the convergence worsens as \(\beta \) is approaching \(-\,1\).
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References
Aldous D (1996) Probability distributions on cladograms. In: Random discrete structures (Minneapolis, MN, 1993), vol 76, pp 1–18
Blum MGB, François O (2006) Which random processes describe the tree of life? A large-scale study of phylogenetic tree imbalance. Syst Biol 55:685–691
Flajolet P, Sedgewick R (2009) Analytic combinatorics. Cambridge University Press, Cambridge
Fuchs M, Jin EY (2015) Equality of Shapley value and fair proportion index in phylogenetic trees. J Math Biol 71(5):1133–1147
Haake C-J, Kashiwada A, Su FE (2008) The Shapley value of phylogenetic trees. J Math Biol 56(4):479–497
Hartmann K (2013) The equivalence of two phylogenetic biodiversity measures: the Shapley value and fair proportion index. J Math Biol 67(5):1163–1170
Roura S (2001) Improved master theorems for divide-and-conquer recurrences. J ACM 48(2):170–205
Sokal RR, Rohlf FJ (1962) The comparison of dendrograms by objective methods. Taxon 11(2):33–40
Stahn H (2019) Biodiversity, Shapley value and phylogenetic trees: some remarks. J Math Biol (to appear)
Steel M (2016) Phylogeny-discrete and random processes in evolution. In: CBMS-NSF regional conference series in applied mathematics. Society for industrial and applied mathematics (SIAM), vol 89, Philadelphia, PA
Wicke K, Fischer M (2017) Comparing the rankings obtained from two biodiversity indices: the fair proportion index and the Shapley value. J Theor Biol 430:207–214
Acknowledgements
We thank both reviewers for a careful reading and many insightful comments which led to an improvement of the paper. We also acknowledge support by the Ministry of Science, Taiwan under the Grants MOST-104-2923-M-009-006-MY3 and MOST-107-2115-M-009-010-MY2.
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A. R. Paningbatan: On leave from the Institute of Mathematics, University of the Philippines, Diliman, Quezon City 1101, Philippines.
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Fuchs, M., Paningbatan, A.R. Correlation between Shapley values of rooted phylogenetic trees under the beta-splitting model. J. Math. Biol. 80, 627–653 (2020). https://doi.org/10.1007/s00285-019-01435-3
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DOI: https://doi.org/10.1007/s00285-019-01435-3