arXiv subjects

Rick Durrett

Publications and source records attributed to Rick Durrett.

At least 55 records · Page 3Linked to original sources

arXiv2013

A first order phase transition in the threshold-$θ\ge 2$ contact process on random $r$-regular graphs and $r$-trees

We consider the discrete-time threshold-$θ\ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least θoccupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We show that if $θ\ge 2$ and $r \ge θ+2$, $ε_1$ is small and p is at least $p_1(ε_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2ε_1$ up to time $\exp(γ_1(r)n)$ with probability at least $1 - \exp(-γ_1(r)n)$. In the other direction, we show that for $p_2 < 1$ there is an $ε_2(p_2)>0$ so that if $p \le p_2$ and the number of occupied vertices in the initial configuration is at most $ε_2(p_2)n$, then with high probability all vertices are vacant at time $C_2(p_2) \log(n)$. These two conclusions imply that on the random r-regular graph there cannot be a quasi-stationary distribution with density of occupied vertices between 0 and $ε_2(p_1)$, and allow us to conclude that the process on the r-tree has a first order phase transition.

math.PR↗

arXiv2013

Phase Transitions in the Quadratic Contact Process on Complex Networks

The quadratic contact process (QCP) is a natural extension of the well studied linear contact process where infected (1) individuals infect susceptible (0) neighbors at rate $λ$ and infected individuals recover ($1 \longrightarrow 0$) at rate 1. In the QCP, a combination of two 1's is required to effect a $0 \longrightarrow 1$ change. We extend the study of the QCP, which so far has been limited to lattices, to complex networks. \comment{as a model for the change in a population through sexual reproduction and death.} We define two versions of the QCP -- vertex centered (VQCP) and edge centered (EQCP) with birth events $1-0-1 \longrightarrow 1-1-1$ and $1-1-0 \longrightarrow 1-1-1$ respectively, where `$-$' represents an edge. We investigate the effects of network topology by considering the QCP on random regular, Erdős-Rényi and power law random graphs. We perform mean field calculations as well as simulations to find the steady state fraction of occupied vertices as a function of the birth rate. We find that on the random regular and Erdős-Rényi graphs, there is a discontinuous phase transition with a region of bistability, whereas on the heavy tailed power law graph, the transition is continuous. The critical birth rate is found to be positive in the former but zero in the latter.

physics.soc-ph↗

arXiv2013

A multi-opinion evolving voter model with infinitely many phase transitions

We consider an idealized model in which individuals' changing opinions and their social network coevolve, with disagreements between neighbors in the network resolved either through one imitating the opinion of the other or by reassignment of the discordant edge. Specifically, an interaction between $x$ and one of its neighbors $y$ leads to $x$ imitating $y$ with probability $(1-α)$ and otherwise (i.e., with probability $α$) $x$ cutting its tie to $y$ in order to instead connect to a randomly chosen individual. Building on previous work about the two-opinion case, we study the multiple-opinion situation, finding that the model has infinitely many phase transitions. Moreover, the formulas describing the end states of these processes are remarkably simple when expressed as a function of $β= α/(1-α)$.

physics.soc-ph↗

arXiv2013

Population genetics of neutral mutations in exponentially growing cancer cell populations

In order to analyze data from cancer genome sequencing projects, we need to be able to distinguish causative, or "driver," mutations from "passenger" mutations that have no selective effect. Toward this end, we prove results concerning the frequency of neutural mutations in exponentially growing multitype branching processes that have been widely used in cancer modeling. Our results yield a simple new population genetics result for the site frequency spectrum of a sample from an exponentially growing population.

q-bio.PE↗

arXiv2012

Asymptotic behavior of Aldous' gossip process

Aldous [(2007) Preprint] defined a gossip process in which space is a discrete $N\times N$ torus, and the state of the process at time $t$ is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each and at rate $N^{-α}$ to a site chosen at random from the torus. We will be interested in the case in which $α<3$, where the long range transmission significantly accelerates the time at which everyone knows the information. We prove three results that precisely describe the spread of information in a slightly simplified model on the real torus. The time until everyone knows the information is asymptotically $T=(2-2α/3)N^{α/3}\log N$. If $ρ_s$ is the fraction of the population who know the information at time $s$ and $\varepsilon$ is small then, for large $N$, the time until $ρ_s$ reaches $\varepsilon$ is $T(\varepsilon)\approx T+N^{α/3}\log (3\varepsilon /M)$, where $M$ is a random variable determined by the early spread of the information. The value of $ρ_s$ at time $s=T(1/3)+tN^{α/3}$ is almost a deterministic function $h(t)$ which satisfies an odd looking integro-differential equation. The last result confirms a heuristic calculation of Aldous.

math.PR↗

arXiv2011

Traveling waves of selective sweeps

The goal of cancer genome sequencing projects is to determine the genetic alterations that cause common cancers. Many malignancies arise during the clonal expansion of a benign tumor which motivates the study of recurrent selective sweeps in an exponentially growing population. To better understand this process, Beerenwinkel et al. [PLoS Comput. Biol. 3 (2007) 2239--2246] consider a Wright--Fisher model in which cells from an exponentially growing population accumulate advantageous mutations. Simulations show a traveling wave in which the time of the first $k$-fold mutant, $T_k$, is approximately linear in $k$ and heuristics are used to obtain formulas for $ET_k$. Here, we consider the analogous problem for the Moran model and prove that as the mutation rate $μ\to0$, $T_k\sim c_k\log(1/μ)$, where the $c_k$ can be computed explicitly. In addition, we derive a limiting result on a log scale for the size of $X_k(t)={}$the number of cells with $k$ mutations at time $t$.

math.PR↗

arXiv2010

Evolution in predator-prey systems

We study the adaptive dynamics of predator-prey systems modeled by a dynamical system in which the traits of predators and prey are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long term prey coexistence and the trait of the resident prey type converges to the solution of an ODE. When only the predators are allowed to evolve, coexistence of predators occurs. In this case, depending on the parameters being varied, we see (i) the number of coexisting predators remains tight and the differences in traits from a reference species converge in distribution to a limit, or (ii) the number of coexisting predators tends to infinity, and we calculate the asymptotic rate at which the traits of the least and most "fit" predators in the population increase. This last result is obtained by comparison with a branching random walk killed to the left of a linear boundary and a finite branching-selection particle system.

math.PR↗

arXiv2010

Evolutionary dynamics of tumor progression with random fitness values

Most human tumors result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Mutations that confer a fitness advantage to the cell are known as driver mutations and are causally related to tumorigenesis. Other mutations, however, do not change the phenotype of the cell or even decrease cellular fitness. While much experimental effort is being devoted to the identification of the different functional effects of individual mutations, mathematical modeling of tumor progression generally considers constant fitness increments as mutations are accumulated. In this paper we study a mathematical model of tumor progression with random fitness increments. We analyze a multi-type branching process in which cells accumulate mutations whose fitness effects are chosen from a distribution. We determine the effect of the fitness distribution on the growth kinetics of the tumor. This work contributes to a quantitative understanding of the accumulation of mutations leading to cancer phenotypes.

math.PR↗

arXiv2009

Contact processes on random graphs with power law degree distributions have critical value 0

If we consider the contact process with infection rate $λ$ on a random graph on $n$ vertices with power law degree distributions, mean field calculations suggest that the critical value $λ_c$ of the infection rate is positive if the power $α>3$. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399--1404]. Here, we show that the critical value $λ_c$ is zero for any value of $α>3$, and the contact process starting from all vertices infected, with a probability tending to 1 as $n\to\infty$, maintains a positive density of infected sites for time at least $\exp(n^{1-δ})$ for any $δ>0$. Using the last result, together with the contact process duality, we can establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability $ρ(λ)$. It is expected that $ρ(λ)\sim Cλ^β$ as $λ\to0$. Here we show that $α-1\leβ\le2α-3$, and so $β>2$ for $α>3$. Thus even though the graph is locally tree-like, $β$ does not take the mean field critical value $β=1$.

math.PR↗

arXiv2009

Persistence of Activity in Threshold Contact Processes, an "Annealed Approximation" of Random Boolean Networks

We consider a model for gene regulatory networks that is a modification of Kauffmann's (1969) random Boolean networks. There are three parameters: $n =$ the number of nodes, $r =$ the number of inputs to each node, and $p =$ the expected fraction of 1's in the Boolean functions at each node. Following a standard practice in the physics literature, we use a threshold contact process on a random graph on $n$ nodes, in which each node has in degree $r$, to approximate its dynamics. We show that if $r\ge 3$ and $r \cdot 2p(1-p)>1$, then the threshold contact process persists for a long time, which correspond to chaotic behavior of the Boolean network. Unfortunately, we are only able to prove the persistence time is $\ge \exp(cn^{b(p)})$ with $b(p)>0$ when $r\cdot 2p(1-p)> 1$, and $b(p)=1$ when $(r-1)\cdot 2p(1-p)>1$.

math.PR↗

arXiv2009

Chaos in a spatial epidemic model

We investigate an interacting particle system inspired by the gypsy moth, whose populations grow until they become sufficiently dense so that an epidemic reduces them to a low level. We consider this process on a random 3-regular graph and on the $d$-dimensional lattice and torus, with $d\geq2$. On the finite graphs with global dispersal or with a dispersal radius that grows with the number of sites, we prove convergence to a dynamical system that is chaotic for some parameter values. We conjecture that on the infinite lattice with a fixed finite dispersal distance, distant parts of the lattice oscillate out of phase so there is a unique nontrivial stationary distribution.

math.PR↗

arXiv2009

A waiting time problem arising from the study of multi-stage carcinogenesis

We consider the population genetics problem: how long does it take before some member of the population has $m$ specified mutations? The case $m=2$ is relevant to onset of cancer due to the inactivation of both copies of a tumor suppressor gene. Models for larger $m$ are needed for colon cancer and other diseases where a sequence of mutations leads to cells with uncontrolled growth.

math.PR↗

arXiv2009

Coexistence in stochastic spatial models

In this paper I will review twenty years of work on the question: When is there coexistence in stochastic spatial models? The answer, announced in Durrett and Levin [Theor. Pop. Biol. 46 (1994) 363--394], and that we explain in this paper is that this can be determined by examining the mean-field ODE. There are a number of rigorous results in support of this picture, but we will state nine challenging and important open problems, most of which date from the 1990's.

math.PR↗

arXiv2009

Degenerate diffusions arising from gene duplication models

We consider two processes that have been used to study gene duplication, Watterson's [Genetics 105 (1983) 745--766] double recessive null model and Lynch and Force's [Genetics 154 (2000) 459--473] subfunctionalization model. Though the state spaces of these diffusions are two and six-dimensional, respectively, we show in each case that the diffusion stays close to a curve. Using ideas of Katzenberger [Ann. Probab. 19 (1991) 1587--1628] we show that one-dimensional projections converge to diffusion processes, and we obtain asymptotics for the time to loss of one gene copy. As a corollary we find that the probability of subfunctionalization decreases exponentially fast as the population size increases. This rigorously confirms a result Ward and Durrett [Theor. Pop. Biol. 66 (2004) 93--100] found by simulation that the likelihood of subfunctionalization for gene duplicates decays exponentially fast as the population size increases.

math.PR↗

arXiv2006

Limiting behavior of the distance of a random walk

This investigation is motivated by a result we proved recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time $n/2$. Here, we study three new examples. It is trivial that the distance for random walk on the hypercube is smooth and is given by one simple formula. In the case of random adjacent transpositions, we find that there is no phase transition even though the distance has different scalings in three different regimes. In the case of a random 3-regular graph, there is a phase transition from linear growth to a constant equal to the diameter of the graph, at time $3\log_2 n$.

math.PR↗

arXiv2006

Power laws for family sizes in a duplication model

Qian, Luscombe and Gerstein [J. Molecular Biol. 313 (2001) 673--681] introduced a model of the diversification of protein folds in a genome that we may formulate as follows. Consider a multitype Yule process starting with one individual in which there are no deaths and each individual gives birth to a new individual at rate 1. When a new individual is born, it has the same type as its parent with probability $1-r$ and is a new type, different from all previously observed types, with probability $r$. We refer to individuals with the same type as families and provide an approximation to the joint distribution of family sizes when the population size reaches $N$. We also show that if $1\ll S\ll N^{1-r}$, then the number of families of size at least $S$ is approximately $CNS^{-1/(1-r)}$, while if $N^{1-r}\ll S$ the distribution decays more rapidly than any power.

math.PR↗

arXiv2005

Two Phase Transitions for the Contact Process on Small Worlds

In our version of Watts and Strogatz's small world model, space is a d-dimensional torus in which each individual has in addition exactly one long-range neighbor chosen at random from the grid. This modification is natural if one thinks of a town where an individual's interactions at school, at work, or in social situations introduces long-range connections. However, this change dramatically alters the behavior of the contact process, producing two phase transitions. We establish this by relating the small world to an infinite "big world" graph where the contact process behavior is similar to the contact process on a tree.

math.PR↗

arXiv2005

Random partitions approximating the coalescence of lineages during a selective sweep

When a beneficial mutation occurs in a population, the new, favored allele may spread to the entire population. This process is known as a selective sweep. Suppose we sample $n$ individuals at the end of a selective sweep. If we focus on a site on the chromosome that is close to the location of the beneficial mutation, then many of the lineages will likely be descended from the individual that had the beneficial mutation, while others will be descended from a different individual because of recombination between the two sites. We introduce two approximations for the effect of a selective sweep. The first one is simple but not very accurate: flip $n$ independent coins with probability $p$ of heads and say that the lineages whose coins come up heads are those that are descended from the individual with the beneficial mutation. A second approximation, which is related to Kingman's paintbox construction, replaces the coin flips by integer-valued random variables and leads to very accurate results.

math.PR↗