Some limit results for Markov chains indexed by trees

We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_\sigma)_{\sigma\in\mathcal T}$, indexed by the full binary tree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcal T_k$ is the $k$th generation of $\mathcal T$. In addition, let $(\Sigma_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $\Sigma_k \in \mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ with $\widetilde R_t^n := X_{\Sigma_{[tn]}}$, arising by observing the Markov chain $\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empirical measure process $\widetilde{\mathcal Z}^n = (\widetilde Z_t^n)_{t\geq 0}$ where $\widetilde{Z}_t^n = \sum_{\sigma\in\mathcal T_{[tn]}} \delta_{X_\sigma^n}$ as $n\to\infty$. Precisely, we show that if $\widetilde{\mathcal R}^n \to \mathcal R$ for some Feller process $\mathcal R = (R_t)_{t\geq 0}$ with deterministic initial condition, then $\widetilde{\mathcal Z}^n \to \mathcal Z$ with $Z_t = \delta_{\mathcal L(R_t)}$.Comment: 12 page

A spatial model for selection and cooperation

We study the evolution of cooperation in an interacting particle system with two types. The model we investigate is an extension of a two-type biased voter model. One type (called defector) has a (positive) bias $\alpha$ with respect to the other type (called cooperator). However, a cooperator helps a neighbor (either defector or cooperator) to reproduce at rate $\gamma$. We prove that the one-dimensional nearest-neighbor interacting dynamical system exhibits a phase transition at $\alpha=\gamma$. For $\alpha>\gamma$ cooperators always die out, but if $\gamma>\alpha$, cooperation is the winning strategy.Comment: 19 pages, 1 figur

Large-scale behavior of the partial duplication random graph

The following random graph model was introduced for the evolution of protein-protein interaction networks: Let $\mathcal G = (G_n)_{n=n_0, n_0+1,...}$ be a sequence of random graphs, where $G_n = (V_n, E_n)$ is a graph with $|V_n|=n$ vertices, $n=n_0,n_0+1,...$ In state $G_n = (V_n, E_n)$, a vertex $v\in V_n$ is chosen from $V_n$ uniformly at random and is partially duplicated. Upon such an event, a new vertex $v'\notin V_n$ is created and every edge $\{v,w\} \in E_n$ is copied with probability~$p$, i.e.\ $E_{n+1}$ has an edge $\{v',w\}$ with probability~$p$, independently of all other edges. Within this graph, we study several aspects for large~$n$. (i) The frequency of isolated vertices converges to~1 if $p\leq p^* \approx 0.567143$, the unique solution of $pe^p=1$. (ii) The number $C_k$ of $k$-cliques behaves like $n^{kp^{k-1}}$ in the sense that $n^{-kp^{k-1}}C_k$ converges against a non-trivial limit, if the starting graph has at least one $k$-clique. In particular, the average degree of a vertex (which equals the number of edges -- or 2-cliques -- divided by the size of the graph) converges to $0$ iff $p<0.5$ and we obtain that the transitivity ratio of the random graph is of the order $n^{-2p(1-p)}$. (iii) The evolution of the degrees of the vertices in the initial graph can be described explicitly. Here, we obtain the full distribution as well as convergence results.Comment: 27 pages, 1 figur

Scaling limits of spatial compartment models for chemical reaction networks

We study the effects of fast spatial movement of molecules on the dynamics of chemical species in a spatially heterogeneous chemical reaction network using a compartment model. The reaction networks we consider are either single- or multi-scale. When reaction dynamics is on a single-scale, fast spatial movement has a simple effect of averaging reactions over the distribution of all the species. When reaction dynamics is on multiple scales, we show that spatial movement of molecules has different effects depending on whether the movement of each type of species is faster or slower than the effective reaction dynamics on this molecular type. We obtain results for both when the system is without and with conserved quantities, which are linear combinations of species evolving only on the slower time scale.Comment: Published at http://dx.doi.org/10.1214/14-AAP1070 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The stationary distribution of a Markov jump process glued together from two state spaces at two vertices

We compute the stationary distribution of a continuous-time Markov chain which is constructed by gluing together two finite, irreducible Markov chains by identifying a pair of states of one chain with a pair of states of the other and keeping all transition rates from either chain (the rates between the two shared states are summed). The result expresses the stationary distribution of the glued chain in terms of quantities of the two original chains. Some of the required terms are nonstandard but can be computed by solving systems of linear equations using the transition rate matrices of the two original chains. Special emphasis is given to the cases when the stationary distribution of the glued chain is a multiple of the equilibria of the original chains, and when not, for which bounds are derived.Comment: 28 pages. Proposition 4, Theorem 7 and their proofs have been changed to correct errors. The notation has been modified. This is a revision of our Author's Original Manuscript submitted for publication to Stochastic Models, Taylor & Francis LL

The Aldous-Shields model revisited (with application to cellular ageing)

In Aldous and Shields (1988), a model for a rooted, growing random binary tree was presented. For some c>0, an external vertex splits at rate c^(-i) (and becomes internal) if its distance from the root (depth) is i. For c>1, we reanalyse the tree profile, i.e. the numbers of external vertices in depth i=1,2,.... Our main result are concrete formulas for the expectation and covariance-structure of the profile. In addition, we present the application of the model to cellular ageing. Here, we assume that nodes in depth h+1 are senescent, i.e. do not split. We obtain a limit result for the proportion of non-senescent vertices for large h.Comment: 13 pages, 2 figure

The infinitely many genes model with horizontal gene transfer

The genome of bacterial species is much more flexible than that of eukaryotes. Moreover, the distributed genome hypothesis for bacteria states that the total number of genes present in a bacterial population is greater than the genome of every single individual. The pangenome, i.e. the set of all genes of a bacterial species (or a sample), comprises the core genes which are present in all living individuals, and accessory genes, which are carried only by some individuals. In order to use accessory genes for adaptation to environmental forces, genes can be transferred horizontally between individuals. Here, we extend the infinitely many genes model from Baumdicker, Hess and Pfaffelhuber (2010) for horizontal gene transfer. We take a genealogical view and give a construction -- called the Ancestral Gene Transfer Graph -- of the joint genealogy of all genes in the pangenome. As application, we compute moments of several statistics (e.g. the number of differences between two individuals and the gene frequency spectrum) under the infinitely many genes model with horizontal gene transfer.Comment: 31 pages, 3 figure

Tree-valued Fleming-Viot dynamics with mutation and selection

The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions. The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanov-type theorem on marked metric measure spaces. To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming-Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming-Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case.Comment: Published in at http://dx.doi.org/10.1214/11-AAP831 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The partial duplication random graph with edge deletion

We study a random graph model in continuous time. Each vertex is partially copied with the same rate, i.e.\ an existing vertex is copied and every edge leading to the copied vertex is copied with independent probability $p$. In addition, every edge is deleted at constant rate, a mechanism which extends previous partial duplication models. In this model, we obtain results on the degree distribution, which shows a phase transition such that either -- if $p$ is small enough -- the frequency of isolated vertices converges to 1, or there is a positive fraction of vertices with unbounded degree. We derive results on the degrees of the initial vertices as well as on the sub-graph of non-isolated vertices. In particular, we obtain expressions for the number of star-like subgraphs and cliques