Biased Voter Model Research Articles

Spread of premalignant mutant clones and cancer initiation in multilayered tissue

Over 80% of human cancers originate from the epithelium, which covers the outer and inner surfaces of organs and blood vessels. In stratified epithelium, the bottom layers are occupied by stem and stem-like cells that continually divide and replenish the upper layers. In this work, we study the spread of premalignant mutant clones and cancer initiation in stratified epithelium using the biased voter model on stacked two-dimensional lattices. Our main result is an estimate of the propagation speed of a premalignant mutant clone, which is asymptotically precise in the cancer-relevant weak-selection limit. We use our main result to study cancer initiation under a two-step mutational model of cancer, which includes computing the distributions of the time of cancer initiation and the size of the premalignant clone giving rise to cancer. Our work quantifies the effect of epithelial tissue thickness on the process of carcinogenesis, thereby contributing to an emerging understanding of the spatial evolutionary dynamics of cancer.

Coalescing and branching simple symmetric exclusion process

Motivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph G=(V,E) dual to the biased voter model on G. Our main goal is tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as |V|→∞. Our results allow us to recover very directly and improve to ℓp-mixing, p≥2, and to more general graphs, the mixing time results of Pillai and Smith for the Fredrickson–Andersen one spin facilitated (FA-1f) KCM on the discrete d-dimensional torus. In view of applications to the more complex FA-jf KCM, j>1, we also extend part of the analysis to an analogous process with a more general product state space.

Biased-voter model: How persuasive a small group can be?

We study the voter model dynamics in the presence of confidence and bias. We assume two types of voters. Unbiased voters whose confidence is indifferent to the state of the voter and biased voters whose confidence is biased towards a common fixed preferred state. We study the problem analytically on the complete graph using mean field theory and on an Erdős-Rényi random network topology using the pair approximation, where we assume that the network of interactions topology is independent of the type of voters. We find that for the case of a random initial setup, and for sufficiently large number of voters N, the time to consensus increases proportionally to log(N)/γv, with γ the fraction of biased voters and v the parameter quantifying the bias of the voters (v = 0 no bias). We verify our analytical results through numerical simulations. We study this model on a topology of the network of interactions depending on the bias, and examine two distinct, global average-degree preserving strategies (model I and model II) to obtain such random topologies starting from the random topology independent of bias case as the initial setup. Keeping all other parameters constant, in model I, μBU, the average number of links among biased (B) and unbiased (U) voters is varied at the expense of μUU and μBB, i.e. the average number of links among only unbiased and biased voters respectively. In model II, μBU is kept constant, while μBB is varied at the expense of μUU. We find that if the agents follow the strategy described by model II, they can achieve a significant reduction in the time to reach consensus as well as an increment in the probability to reach consensus to the preferred state. Hence, persuasiveness of the biased group depends on how well its members are connected among each other, compared to how well the members of the unbiased group are connected among each other.

An invariance principle for biased voter model interfaces

We consider one-dimensional biased voter models, where 1’s replace 0’s at a faster rate than the other way round, started in a Heaviside initial state describing the interface between two infinite populations of 0’s and 1’s. In the limit of weak bias, for a diffusively rescaled process, we consider a measure-valued process describing the local fraction of type 1 sites as a function of time. Under a finite second moment condition on the rates, we show that in the diffusive scaling limit there is a drifted Brownian path with the property that all but a vanishingly small fraction of the sites on the left (resp. right) of this path are of type 0 (resp. 1). This extends known results for unbiased voter models. Our proofs depend crucially on recent results about interface tightness for biased voter models.

Equilibrium interfaces of biased voter models

A one-dimensional interacting particle system is said to exhibit interface tightness if starting in an initial condition describing the interface between two constant configurations of different types, the process modulo translations is positive recurrent. In a biological setting, this describes two populations that do not mix, and it is believed to be a common phenomenon in one-dimensional particle systems. Interface tightness has been proved for voter models satisfying a finite second moment condition on the rates. We extend this to biased voter models. Furthermore, we show that the distribution of the equilibrium interface for the biased voter model converges to that of the voter model when the bias parameter tends to zero. A key ingredient is an identity for the expected number of boundaries in the equilibrium voter model interface, which is of independent interest.

A spatial model for selection and cooperation

Abstract 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 α with respect to the other type (called cooperator). However, a cooperator helps a neighbor (either defector or cooperator) to reproduce at rate γ. We prove that the one-dimensional nearest-neighbor interacting dynamical system exhibits a phase transition at α = γ. A special choice of interaction kernels yield that for α > γ cooperators always die out, but if γ > α, cooperation is the winning strategy.

Genealogies in expanding populations

The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [Theor. Pop. Biol. 73 (2008) 158–170]. The first step, which is easily accomplished using results of Mueller and Tribe [Probab. Theory Related Fields 102 (1995) 519–545], is to show that when space and time are rescaled correctly, our biased voter model converges to a Wright–Fisher SPDE. A simple extension of a result of Durrett and Restrepo [Ann. Appl. Probab. 18 (2008) 334–358] then shows that the dual branching coalescing random walk converges to a branching Brownian motion in which particles coalesce after an exponentially distributed amount of intersection local time. Brunet et al. [Phys. Rev. E (3) 76 (2007) 041104, 20] have conjectured that genealogies in models of this type are described by the Bolthausen–Sznitman coalescent, see [Proc. Natl. Acad. Sci. USA 110 (2013) 437–442]. However, in the model we study there are no simultaneous coalescences. Our third and most significant result concerns “tracer dynamics” in which some of the initial particles in the biased voter model are labeled. We show that the joint distribution of the labeled and unlabeled particles converges to the solution of a system of stochastic partial differential equations. A new duality equation that generalizes the one Shiga [In Stochastic Processes in Physics and Engineering (Bielefeld, 1986) (1988) 345–355 Reidel] developed for the Wright–Fisher SPDE is the key to the proof of that result.

Genealogies in expanding populations

The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [25]. The first step, which is easily accomplished using results of Mueller and Tribe [38], is to show that when space and time are rescaled correctly, our biased voter model converges to a Wright-Fisher SPDE. A simple extension of a result of Durrett and Restrepo [18] then shows that the dual branching coalescing random walk converges to a branching Brownian motion in which particles coalesce after an exponentially distributed amount of intersection local time. Brunet et al. [8] have conjectured that genealogies in models of this type are described by the Bolthausen-Sznitman coalescent, see [39]. However, in the model we study there are no simultaneous coalescences. Our third and most significant result concerns "tracer dynamics" in which some of the initial particles in the biased voter model are labeled. We show that the joint distribution of the labeled and unlabeled particles converges to the solution of a system of stochastic partial differential equations. A new duality equation that generalizes the one Shiga [44] developed for the Wright-Fisher SPDE is the key to the proof of that result.

The Williams–Bjerknes model on regular trees

We consider the Williams–Bjerknes model, also known as the biased voter model on the $d$-regular tree $\mathbb{T} ^{d}$, where $d\geq3$. Starting from an initial configuration of “healthy” and “infected” vertices, infected vertices infect their neighbors at Poisson rate $\lambda\geq1$, while healthy vertices heal their neighbors at Poisson rate $1$. All vertices act independently. It is well known that starting from a configuration with a positive but finite number of infected vertices, infected vertices will continue to exist at all time with positive probability if and only if $\lambda>1$. We show that there exists a threshold $\lambda_{c}\in(1,\infty)$ such that if $\lambda>\lambda_{c}$ then in the above setting with positive probability, all vertices will become eventually infected forever, while if $\lambda<\lambda_{c}$, all vertices will become eventually healthy with probability $1$. In particular, this yields a complete convergence theorem for the model and its dual, a certain branching coalescing random walk on $\mathbb{T} ^{d}$—above $\lambda_{c}$. We also treat the case of initial configurations chosen according to a distribution which is invariant or ergodic with respect to the group of automorphisms of $\mathbb{T} ^{d}$.

Phase and multifractality analyses of random price time series by finite-range interacting biased voter system

A random financial price process which is developed by mechanisms of finite-range interacting biased voter model is considered in the present paper. Voter model is one of statistical physics systems as well as a continuous time Markov process, which originally represents a voter's attitude on a particular topic, namely, voters reconsider their opinions at times distributed according to independent exponential random variables. The empirical mode decomposition method is employed to analyze the behaviors of logarithmic returns for the simulation data of the model and the two real market indexes, Shanghai Composite Index and Deutscher Aktien Index. The multifractal characteristics of the original returns and first 3 intrinsic mode functions (IMFs) after empirical mode decomposition are explored by the multifractal detrended function analysis. The instantaneous phase, amplitude probability distribution of first 4 IMFs, and the multifractal properties of instantaneous amplitude are investigated.

Ancestral Graph with Bias in Gene Conversion

Gene conversion is a genetic mechanism by which one gene is ‘copied and pasted’ onto another gene, where the direction can be biased between the different types. In this paper, a stochastic model for biased gene conversion within ad-unlinked multigene family and its diffusion approximation are developed for a finite Moran population. A connection with ad-island model is made. A formula for the fixation probability in the absence of mutation is given. A two-timescale argument is applied in the case of the strong conversion limit. The dual process is generally shown to be a biased voter model, which generates an ancestral bias graph for a given sample. An importance sampling algorithm for computing the likelihood of the sample is deduced.

Ancestral Graph with Bias in Gene Conversion

Gene conversion is a genetic mechanism by which one gene is ‘copied and pasted’ onto another gene, where the direction can be biased between the different types. In this paper, a stochastic model for biased gene conversion within ad-unlinked multigene family and its diffusion approximation are developed for a finite Moran population. A connection with ad-island model is made. A formula for the fixation probability in the absence of mutation is given. A two-timescale argument is applied in the case of the strong conversion limit. The dual process is generally shown to be a biased voter model, which generates an ancestral bias graph for a given sample. An importance sampling algorithm for computing the likelihood of the sample is deduced.

Voter Model and Biased Voter Model in Heterogeneous Environments

With the rapid adoption of transgenic crops, gene flow from transgenic crops to wild relatives through pollen dispersal is of significant concern and warrants both empirical and theoretical studies to assess the risk of introduction of transgenes into wild populations. We propose to use the (biased) voter model in a heterogeneous environment to investigate the effects of recurrent gene flow from transgenic crop to wild relatives. The model is defined on the d-dimensional integer lattice that is divided into two parts, Δ and Z d \ Δ. Individuals carrying the transgene and individuals carrying the wild type gene compete according to the evolution rules of a (biased) voter model on Z d \ Δ, while the process is conditioned to have only individuals carrying the transgene on Δ. Our main findings suggest that unless transgenes confer increased fitness in wild relatives, introgression of transgenes into populations of wild plants is slow and may even be reversible without intervention. Our study also addresses the effects of different spatial planting patterns of transgenic crops on the rate of introgression.

Voter Model and Biased Voter Model in Heterogeneous Environments

With the rapid adoption of transgenic crops, gene flow from transgenic crops to wild relatives through pollen dispersal is of significant concern and warrants both empirical and theoretical studies to assess the risk of introduction of transgenes into wild populations. We propose to use the (biased) voter model in a heterogeneous environment to investigate the effects of recurrent gene flow from transgenic crop to wild relatives. The model is defined on the d-dimensional integer lattice that is divided into two parts, Δ and Zd \ Δ. Individuals carrying the transgene and individuals carrying the wild type gene compete according to the evolution rules of a (biased) voter model on Zd \ Δ, while the process is conditioned to have only individuals carrying the transgene on Δ. Our main findings suggest that unless transgenes confer increased fitness in wild relatives, introgression of transgenes into populations of wild plants is slow and may even be reversible without intervention. Our study also addresses the effects of different spatial planting patterns of transgenic crops on the rate of introgression.

Designing a contact process: the piecewise-homogeneous process on a finite set with applications

We consider how to choose the reproduction rates in a one-dimensional contact process on a finite set to maximize the growth rate of the extinction time with the population size. The constraints are an upper bound on the average reproduction rate, and that the rate profile must be piecewise constant. We show that the optimum growth rate is achieved by a rate profile with at most two rates, and we characterize the solution in terms of a “spatial correlation length” of the supercritical process. We examine the analogous problem for the simpler biased voter model, for which we completely characterize the optimum profile. The contact process proofs make use of a planar-graph duality in the graphical representation, due to Durrett and Schonmann.

Spatial perturbations of one-dimensional spin systems

We consider nearest-neighbor, additive, spin systems on Z + , and show that changing the flip rates at a finite number of sites does not affect survival of the process. We also extend this result to the case of the biased voter model on Z .

Branching random walks on trees

Let p( x, y) be the transition probability of an isotropic random walk on a tree, where each site has d ⩾3 neighbors. We define a branching random walk by letting a particle at site x give birth to a new particle at site y at rate λ dp( x, y), jump to y at rate vdp( x, y), and die at rate δ. Let λ 2 (respectively, μ 2) be the infimum of λ such that the process starting with one particle has positive probability of surviving forever (respectively, of having a fixed site occupied at arbitrarily large times). We compute λ 2 and μ 2 exactly, proving that λ 2<μ 2: i.e., the process has two phase transitions. We characterize λ 2 (respectively, μ 2) in terms of the expected number of particles on the tree (respectively, at a fixed site). We also prove similar results for the biased voter model. Finally, for the contact process, branching random walk and biased voter model on the tree, we prove that the second phase transition has a discontinuity which is absent in Euclidian space

The probability of survival for the biased voter model in a random environment

In this paper we consider a version of the biased voter model in S, the set of all subsets of Z , in which the recovery rates, δ x , x∈ Z , are i.i.d. random variables and λ>0 is fixed. We prove a result about the convergence of the probability of survival of the process when λ tends to the critical value λ c. As a corollary we find that the critical exponent, β, associated with survival probability is ∞ in contrast to the nonrandom case in which β = 1.

The Contact Process on a Finite Set. II

In this paper we show that the phase transition in the contact process manifests itself in the behavior of large finite systems. To be precise, if we let $\sigma_N$ denote the time the process on $\{1, \cdots, N\}$ first hits $\varnothing$ starting from all sites occupied, then there is a critical value $\lambda_c$ so that (i) for $\lambda \lambda_c$ there are constants $\alpha (\lambda), \beta(\lambda) \in (0, \infty)$ so that as $N \rightarrow \infty$, $P(\alpha(\lambda)/2 - \varepsilon \leq (\log \sigma_N)/N \leq \beta (\lambda) + \varepsilon) \rightarrow 1,$ for all $\varepsilon > 0$. Our results improve upon an earlier work of Griffeath but as the reader can see the second one still needs improvement. To help decide what should be true for the contact process we also consider the analogous problem for the biased voter model. For this process we can show $(\log \sigma_N)/N \rightarrow \alpha(\lambda) = \beta(\lambda)$ in probability, and it seems likely that the same result is true for the contact process.

The Contact Process on a Finite Set

In this paper we show that the phase transition in the contact process manifests itself in the behavior of large finite systems. To be precise, if we let $\sigma_N$ denote the time the process on $\{1, \cdots, N\}$ first hits $\varnothing$ starting from all sites occupied, then there is a critical value $\lambda_c$ so that (i) for $\lambda < \lambda_c$ there is a constant $\gamma(\lambda) \in (0, \infty)$ so that as $N \rightarrow \infty, \sigma_n /\log N \rightarrow 1/\gamma(\lambda)$ in probability and (ii) for $\lambda > \lambda_c$ there are constants $\alpha (\lambda), \beta(\lambda) \in (0, \infty)$ so that as $N \rightarrow \infty$, $P(\alpha(\lambda)/2 - \varepsilon \leq (\log \sigma_N)/N \leq \beta (\lambda) + \varepsilon) \rightarrow 1,$ for all $\varepsilon > 0$. Our results improve upon an earlier work of Griffeath but as the reader can see the second one still needs improvement. To help decide what should be true for the contact process we also consider the analogous problem for the biased voter model. For this process we can show $(\log \sigma_N)/N \rightarrow \alpha(\lambda) = \beta(\lambda)$ in probability, and it seems likely that the same result is true for the contact process.