On the push&pull protocol for rumour spreading

The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumour in a graph $G$, works as follows. Independent Poisson clocks of rate 1 are associated with the vertices of $G$. Initially, one vertex of $G$ knows the rumour. Whenever the clock of a vertex $x$ rings, it calls a random neighbour $y$: if $x$ knows the rumour and $y$ does not, then $x$ tells $y$ the rumour (a push operation), and if $x$ does not know the rumour and $y$ knows it, $y$ tells $x$ the rumour (a pull operation). The average spread time of $G$ is the expected time it takes for all vertices to know the rumour, and the guaranteed spread time of $G$ is the smallest time $t$ such that with probability at least $1-1/n$, after time $t$ all vertices know the rumour. The synchronous variant of this protocol, in which each clock rings precisely at times $1,2,\dots$, has been studied extensively. We prove the following results for any $n$-vertex graph: In either version, the average spread time is at most linear even if only the pull operation is used, and the guaranteed spread time is within a logarithmic factor of the average spread time, so it is $O(n\log n)$. In the asynchronous version, both the average and guaranteed spread times are $\Omega(\log n)$. We give examples of graphs illustrating that these bounds are best possible up to constant factors. We also prove theoretical relationships between the guaranteed spread times in the two versions. Firstly, in all graphs the guaranteed spread time in the asynchronous version is within an $O(\log n)$ factor of that in the synchronous version, and this is tight. Next, we find examples of graphs whose asynchronous spread times are logarithmic, but the synchronous versions are polynomially large. Finally, we show for any graph that the ratio of the synchronous spread time to the asynchronous spread time is $O(n^{2/3})$.Comment: 25 page

Rare events in population genetics: Stochastic tunneling in a two-locus model with recombination

We study the evolution of a population in a two-locus genotype space, in which the negative effects of two single mutations are overcompensated in a high fitness double mutant. We discuss how the interplay of finite population size, $N$, and sexual recombination at rate $r$ affects the escape times $t_\mathrm{esc}$ to the double mutant. For small populations demographic noise generates massive fluctuations in $t_\mathrm{esc}$. The mean escape time varies non-monotonically with $r$, and grows exponentially as $\ln t_{\mathrm{esc}} \sim N(r - r^\ast)^{3/2}$ beyond a critical value $r^\ast$.Comment: 4 pages, 3 figure

Self-organized patterns of coexistence out of a predator-prey cellular automaton

We present a stochastic approach to modeling the dynamics of coexistence of prey and predator populations. It is assumed that the space of coexistence is explicitly subdivided in a grid of cells. Each cell can be occupied by only one individual of each species or can be empty. The system evolves in time according to a probabilistic cellular automaton composed by a set of local rules which describe interactions between species individuals and mimic the process of birth, death and predation. By performing computational simulations, we found that, depending on the values of the parameters of the model, the following states can be reached: a prey absorbing state and active states of two types. In one of them both species coexist in a stationary regime with population densities constant in time. The other kind of active state is characterized by local coupled time oscillations of prey and predator populations. We focus on the self-organized structures arising from spatio-temporal dynamics of the coexistence. We identify distinct spatial patterns of prey and predators and verify that they are intimally connected to the time coexistence behavior of the species. The occurrence of a prey percolating cluster on the spatial patterns of the active states is also examined.Comment: 19 pages, 11 figure

Simple models of genomic variation in human SNP density

<p>Abstract</p> <p>Background</p> <p>Descriptive hierarchical Poisson models and population-genetic coalescent mixture models are used to describe the observed variation in single-nucleotide polymorphism (SNP) density from samples of size two across the human genome.</p> <p>Results</p> <p>Using empirical estimates of recombination rate across the human genome and the observed SNP density distribution, we produce a maximum likelihood estimate of the genomic heterogeneity in the scaled mutation rate <it>θ</it>. Such models produce significantly better fits to the observed SNP density distribution than those that ignore the empirically observed recombinational heterogeneities.</p> <p>Conclusion</p> <p>Accounting for mutational and recombinational heterogeneities can allow for empirically sound null distributions in genome scans for "outliers", when the alternative hypotheses include fundamentally historical and unobserved phenomena.</p

Population Dynamics in Spatially Heterogeneous Systems with Drift: the generalized contact process

We investigate the time evolution and stationary states of a stochastic, spatially discrete, population model (contact process) with spatial heterogeneity and imposed drift (wind) in one- and two-dimensions. We consider in particular a situation in which space is divided into two regions: an oasis and a desert (low and high death rates). Carrying out computer simulations we find that the population in the (quasi) stationary state will be zero, localized, or delocalized, depending on the values of the drift and other parameters. The phase diagram is similar to that obtained by Nelson and coworkers from a deterministic, spatially continuous model of a bacterial population undergoing convection in a heterogeneous medium.Comment: 8 papes, 12 figure

Limit Theorem for Continuous-Time Quantum Walk on the Line

Concerning a discrete-time quantum walk X^{(d)}_t with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: X^{(d)}_t /t \to dx / \pi (1-x^2) \sqrt{1 - 2 x^2} as t \to \infty. The present paper shows that a similar type of weak limit theorems is satisfied for a {\it continuous-time} quantum walk X^{(c)}_t on the line as follows: X^{(c)}_t /t \to dx / \pi \sqrt{1 - x^2} as t \to \infty. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: Y_{t}/ \sqrt{t} \to e^{-x^2/2} dx / \sqrt{2 \pi} as t \to \infty. The work deals also with issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.Comment: 15 pages, title correcte

Stationarity of SLE

A new method to study a stopped hull of SLE(kappa,rho) is presented. In this approach, the law of the conformal map associated to the hull is invariant under a SLE induced flow. The full trace of a chordal SLE(kappa) can be studied using this approach. Some example calculations are presented.Comment: 14 pages with 1 figur

Convergence to extremal processes in random environments and extremal ageing in SK models

This paper extends recent results on aging in mean field spin glasses on short time scales, obtained by Ben Arous and Gun [2] in law with respect to the environment, to results that hold almost surely, respectively in probability, with respect to the environment. It is based on the methods put forward in Gayrard [8,9] and naturally complements Bovier and Gayrard [6].Comment: Revised version contains minor change

Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games

Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Such non-transitive relations lead to an evolution with central features represented by the `rock-paper-scissors' game, where rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of temporal evolution. We establish that this phenomenon is robust, it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and evolution of ecological systems and are relevant for the formation and propagation of patterns in excitable media, such as chemical kinetics or epidemic outbreaks.Comment: Final submitted version; the printed version can be found at http://dx.doi.org/10.1038/nature06095 Supplementary movies are available at http://www.theorie.physik.uni-muenchen.de/lsfrey/images_content/movie1.AVI and http://www.theorie.physik.uni-muenchen.de/lsfrey/images_content/movie2.AV

Random copying in space

Random copying is a simple model for population dynamics in the absence of selection, and has been applied to both biological and cultural evolution. In this work, we investigate the effect that spatial structure has on the dynamics. We focus in particular on how a measure of the diversity in the population changes over time. We show that even when the vast majority of a population's history may be well-described by a spatially-unstructured model, spatial structure may nevertheless affect the expected level of diversity seen at a local scale. We demonstrate this phenomenon explicitly by examining the random copying process on small-world networks, and use our results to comment on the use of simple random-copying models in an empirical context.Comment: 26 pages, 11 figures. Based on invited talk at AHRC CECD Conference on "Cultural Evolution in Spatially Structured Populations" at UCL, September 2010. To appear in ACS - Advances in Complex System