2024/05/28 by Sean S. Svihla, Svihla, Sean P., Manuel E. Lladser +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #60C05 #65F05 #65F15 #65F55 #68R05 #92-08 #92C70 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Biological sciences #FOS: Computer and information sciences #FOS: Mathematics #Fractal and DNA sequence analysis #G.2.1 #G.2.2 #G.3 #Graph theory and applications #J.2 #J.3 #Populations and Evolution (q-bio.PE) #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2405.17847
openalex publication_date 2024/05/28 · openalex created_date 2024/05/30 · openalex updated_date 2026/07/28
Consider a tree T=(V,E) with root ∘ and edge length function ℓ:E→ℝ+. The phylogenetic covariance matrix of T is the matrix C with rows and columns indexed by L, the leaf set of T, with entries C(i,j):=∑e∈[i\wedge j,o]ℓ(e), for each i,j∈ L. Recent work [15] has shown that the phylogenetic covariance matrix of a large, random binary tree T is significantly sparsified with overwhelmingly high probability under a change-of-basis with respect to the so-called Haar-like wavelets of T. This finding notably enables manipulating the spectrum of covariance matrices of large binary trees without the necessity to store them in computer memory but instead performing two post-order traversals of the tree. Building on the methods of [15], this manuscript further advances their sparsification result to encompass the broader class of k-regular trees, for any given k≥2. This extension is achieved by refining existing asymptotic formulas for the mean and variance of the internal path length of random k-regular trees, utilizing hypergeometric function properties and identities.