arXiv subjects

Armaan Ahmed

Publications and source records attributed to Armaan Ahmed.

2 recordsLinked to original sources

arXiv2026

The Site Frequency Spectrum in an Exponentially-Growing Population with Selection

We consider a supercritical two-type continuous-time linear birth-death process with mutation and selection, in which wild-type individuals give rise to mutant offspring with a larger net growth rate. In this setting, we investigate the site frequency spectrum (SFS) of driver mutations, describing the number of driver mutations present at any given frequency in the population. First, we derive exact moments for the SFS and establish asymptotic power laws at large times and frequencies. Then, strong laws of large numbers for the driver SFS are proven by constructing suitable $L^2$-approximations. These results apply both to the case when all driver clones have the same selective advantage and when the selective advantage is random. Finally, we allow the frequency to vary with time to examine the number of "intermediate" and "large" driver clones, identifying a cutoff frequency at which there are order 1 number of mutant clones. Overall, our results provide quantitative insights into how selection shapes the site frequency spectrum both at small and large frequencies, which can in principle be leveraged to construct estimators of relevant evolutionary parameters, including the selective advantage of driver mutations.

math.PR↗

arXiv2025

Expected and minimal values of a universal tree balance index

Although the analysis of rooted tree shape has wide-ranging applications, notions of tree balance have developed independently in different domains. In computer science, a balanced tree is one that enables efficient updating and retrieval of data, whereas in biology tree balance quantifies bias in evolutionary processes. The lack of a precise connection between these concepts has stymied the development of universal indices and general results. We recently introduced a new tree balance index, $J^1$, that, unlike prior indices popular among biologists, permits meaningful comparison of trees with arbitrary degree distributions and node sizes. Here we explain how our new index generalizes a concept that underlies the definition of the weight-balanced tree, an important type of self-balancing binary search tree. Our index thus unifies the tree balance concepts of biology and computer science. We provide new analytical results to support applications of this universal index. First, we quantify the accuracy of approximations to the expected values of $J^1$ under two important null models: the Yule process and the uniform model. Second, we investigate minimal values of our index. These results help establish $J^1$ as a universal, cross-disciplinary index of tree balance that generalizes and supersedes prior approaches.

q-bio.QM↗