arXiv subjects

Poly H. da Silva

Publications and source records attributed to Poly H. da Silva.

6 recordsLinked to original sources

arXiv2025

Continuous approximations for the fixation probability of the Moran processes on star graphs

We consider a generalized version of the birth-death (BD) and death-birth (DB) processes introduced by Kaveh, Komarova, and Kohandel (2015), in which two constant fitnesses, one for birth and the other for death, describe the selection mechanism of the population. Rather than constant fitnesses, in this paper we consider more general frequency-dependent fitness functions (allowing any smooth functions) under the weak-selection regime. A particular case arises in evolutionary games on graphs, where the fitness functions are linear combinations of the frequencies of types. For a large population structured as a star graph, we provide approximations for the fixation probability which are solutions of certain ODEs (or systems of ODEs). For the DB case, we prove that our approximation has an error of order $1/N$, where $N$ is the size of the population. The general BD and DB processes contain, as special cases, the BD-* and DB-* (where * can be either B or D) processes described in Hadjichrysanthou, Broom, and Rychtář (2011) -- this class includes many examples of update rules used in the literature. Our analysis shows how the star graph may act as an amplifier, suppressor, or remains isothermal depending on the scaling of the initial mutant placement. We identify an analytical threshold for this transition and illustrate it through applications to evolutionary games, which further highlight asymmetric structural effects across different game types. Numerical examples show that our fixation probability approximations remain accurate even for moderate population sizes and across a wide range of frequency-dependent fitness functions, extending well beyond previously studied linear cases derived from evolutionary games, or constant fitness scenarios.

math.PR↗

arXiv2022

Another view of sequential sampling in the birth process with immigration

Models of counts-of-counts data have been extensively used in the biological sciences, for example in cancer, population genetics, sampling theory and ecology. In this paper we explore properties of one model that is embedded into a continuous-time process and can describe the appearance of certain biological data such as covid DNA sequences in a database. More specifically, we consider an evolving model of counts-of-counts data that arises as the family size counts of samples taken sequentially from a Birth process with Immigration (BI). Here, each family represents a type or species, and the family size counts represent the type or species frequency spectrum in the population. We study the correlation of $S(a,b)$ and $S(c,d)$, the number of families observed in two disjoint time intervals $(a,b)$ and $(c,d)$. We find the expected sample variance and its asymptotics for $p$ consecutive sequential samples $\mathbf{S}_p:=(S(t_0,t_1),\dots, S(t_{p-1},t_p))$, for any given $0=t_0<t_1<\dots<t_p$. By conditioning on the sizes of the samples, we provide a connection between $\mathbf{S}_p$ and $p$ sequential samples of sizes $n_1,n_2,\dots,n_p$, drawn from a single run of a Chinese Restaurant Process. The properties of the latter were studied in da Silva et al. (2022). We show how the continuous-time framework helps to make asymptotic calculations easier than its discrete-time counterpart. As an application, for a specific choice of $t_1,t_2,\dots, t_p$, we revisit Fisher's 1943 multi-sampling problem and give another explanation of what Fisher's model could have meant in the world of sequential samples drawn from a BI process.

math.PR↗

arXiv2022

Markov chains arising from biased random derangements

We explore the cycle types of a class of biased random derangements, described as a random game played by some children labeled $1,\cdots,n$. Children join the game one by one, in a random order, and randomly form some circles of size at least $2$, so that no child is left alone. The game gives rise to the cyclic decomposition of a random derangement, inducing an exchangeable random partition. The rate at which the circles are closed varies in time, and at each time $t$, depends on the number of individuals who have not played until t. A $\{0,1\}$-valued Markov chain $ X^n$ records the cycle type of the corresponding random derangement in that any $1$ represents a hand-grasping event that closes a circle. Using this, we study the cycle counts and sizes of the random derangements and their asymptotic behavior. We approximate the total variation distance between the reversed chain of $X^n$ and its weak limit $X^\infty$, as $n\to\infty$. We establish conditional (and push-forward) relations between $X^n$ and a generalization of the Feller coupling, given that no $11$-pattern ($1$-cycle) appears in the latter. We extend these relations to $X^\infty$ and apply them to investigate some asymptotic behaviors of $X^n$.

math.PR↗

arXiv2020

Random derangements and the Ewens Sampling Formula

We study derangements of $\{1,2,\ldots,n\}$ under the Ewens distribution with parameter $θ$. We give the moments and marginal distributions of the cycle counts, the number of cycles, and asymptotic distributions for large $n$. We develop a $\{0,1\}$-valued non-homogeneous Markov chain with the property that the counts of lengths of spacings between the 1s have the derangement distribution. This chain, an analog of the so-called Feller Coupling, provides a simple way to simulate derangements in time independent of $θ$ for a given $n$ and linear in the size of the derangement.

math.PR↗

arXiv2018

Partial geodesics on symmetric groups endowed with breakpoint distance

The notion of partial geodesic was introduced by Jamshidpey et al. in "Sets of medians in the non-geodesic pseudometric space of unsigned genomes with breakpoints", 2014. In this paper, we study the density of points on non-trivial partial geodesics between two permutations $ξ_1^{(n)}$ and $ξ_2^{(n)}$ chosen uniformly and independently at random from the symmetric group $S_n$, where $S_n$ is endowed with the breakpoint distance. For a permutation $π:= π_1 \ ... \ π_n$, any unordered pair $\{π_i , π_{i+1}\}$, for $i=1, ..., n-1$, is called an adjacency of $π$. The set of all adjacencies of $π$ is denoted by $\mathcal{A}_π$. Denote by $id^{(n)}$ the identity permutation, and let $I_n$ be an arbitrary subset of $\mathcal A_{id^{(n)}}$. We classify the set of all adjacencies of a permutation $π\in S_n$ into four types, with respect to $I_n$. Then for a permutation $ξ^{(n)}$ chosen uniformly at random from $S_n$, we derive a convergence theorem for the normalized number (after dividing by $n$) of adjacencies of each type in $ξ^{(n)}$ with respect to $I_n$ (for some random or deterministic choices of $I_n$), as $n\rightarrow \infty$. We also see an application of this convergence theorem to find the appropriate choices of $I_n$. A geodesic point of $u$ and $v$ in a pseudometric space $(S,ρ)$ is a point $w$ of the space that $ρ(u,w)+ρ(w,v)=ρ(u,v)$. We find an upper bound for the number of permutations $x\in S_n$ for which there exists at least one non-trivial geodesic point between $id^{(n)}$ and $x$, far from both. This partially verifies the conjecture of Haghighi and Sankoff stated in "Medians seek the corners, and other conjectures", 2012, namely we prove that, with high probability, there is no breakpoint median of two permutations $ξ_1^{(n)}$ and $ξ_2^{(n)}$ chosen uniformly and independently at random from $S_n$, far from both of them.

math.CO↗

arXiv2017

Median inverse problem and approximating the number of $k$-median inverses of a permutation

We introduce the "Median Inverse Problem" for metric spaces. In particular, having a permutation $π$ in the symmetric group $S_n$ (endowed with the breakpoint distance), we study the set of all $k$-subsets $\{x_1,...,x_k\}\subset S_n$ for which $π$ is a breakpoint median. The set of all $k$-tuples $(x_1,...,x_k)$ with this property is called the $k$-median inverse of $π$. Finding an upper bound for the cardinality of this set, we provide an asymptotic upper bound for the probability that $π$ is a breakpoint median of $k$ permutations $ξ_1^{(n)},...,ξ_k^{(n)}$ chosen uniformly and independently at random from $S_n$.

math.CO↗