Theory Of Branching Processes Research Articles

Reproductive skew can provide a net advantage in both conditional and unconditional social interactions

We revisit a model for the evolution of costly social behaviour in the presence of reproductive skew. The model population is structured into groups, and reproductive skew is captured by assuming individuals adopt one of two social roles (dominant/subordinate). Unlike previous work, we adopt an ultimate perspective by tracking a mutant allele over the entire course of an invasion. Our main analysis applies the theory of branching processes, but a parallel analysis using the inclusive-fitness approach is also provided. Our first two results are modifications of known inequalities describing selective advantages for behaviours expressed conditional upon social status. We find that altruistic subordinate individuals are favoured more readily than previously thought; spiteful dominant individuals, however, are favoured less readily. Secondly, we identify the condition under which unconditional altruism (performed by both dominant and subordinate) will be adaptive. Our third main result shows that increasing the strength of selection can also change the range of parameters over which costly social behaviours are favoured. We find that stronger selection makes it relatively easier for subordinate altruism to emerge, but more difficult for dominant spite and unconditional altruism to occur. We discuss the possible implications of our results for human social evolution.

Stochastic branching-diffusion models for gene expression.

A challenge to both understanding and modeling biochemical networks is integrating the effects of diffusion and stochasticity. Here, we use the theory of branching processes to give exact analytical expressions for the mean and variance of protein numbers as a function of time and position in a spatial version of an established model of gene expression. We show that both the mean and the magnitude of fluctuations are determined by the protein's Kuramoto length--the typical distance a protein diffuses over its lifetime--and find that the covariance between local concentrations of proteins often increases if there are substantial bursts of synthesis during translation. Using high-throughput data, we estimate that the Kuramoto length of cytoplasmic proteins in budding yeast to be an order of magnitude larger than the cell diameter, implying that many such proteins should have an approximately uniform concentration. For constitutively expressed proteins that live substantially longer than their mRNA, we give an exact expression for the deviation of their local fluctuations from Poisson fluctuations. If the Kuramoto length of mRNA is sufficiently small, we predict that such local fluctuations become approximately Poisson in bacteria in much of the cell, unless translational bursting is exceptionally strong. Our results therefore demonstrate that diffusion can act to both increase and decrease the complexity of fluctuations in biochemical networks.

Stochastic models for competing species with a shared pathogen

The presence of a pathogen among multiple competing species has important ecological implications. For example, a pathogen may change the competitive outcome, resulting in replacement of a native species by a non-native species. Alternately, if a pathogen becomes established, there may be a drastic reduction in species numbers. Stochastic variability in the birth, death and pathogen transmission processes plays an important role in determining the success of species or pathogen invasion. We investigate these phenomena while studying the dynamics of deterministic and stochastic models for n competing species with a shared pathogen. The deterministic model is a system of ordinary differential equations for n competing species in which a single shared pathogen is transmitted among the n species. There is no immunity from infection, individuals either die or recover and become immediately susceptible, an SIS disease model. Analytical results about pathogen persistence or extinction are summarized for the deterministic model for two and three species and new results about stability of the infection-free state and invasion by one species of a system of n-1 species are obtained. New stochastic models are derived in the form of continuous-time Markov chains and stochastic differential equations. Branching process theory is applied to the continuous-time Markov chain model to estimate probabilities for pathogen extinction or species invasion. Finally, numerical simulations are conducted to explore the effect of disease on two-species competition, to illustrate some of the analytical results and to highlight some of the differences in the stochastic and deterministic models.

Analysis of the impact degree distribution in metabolic networks using branching process approximation

Theoretical frameworks to estimate the tolerance of metabolic networks to various failures are important to evaluate the robustness of biological complex systems in systems biology. In this paper, we focus on a measure for robustness in metabolic networks, namely, the impact degree, and propose an approximation method to predict the probability distribution of impact degrees from metabolic network structures using the theory of branching process. We demonstrate the relevance of this method by testing it on real-world metabolic networks. Although the approximation method possesses a few limitations, it may be a powerful tool for evaluating metabolic robustness.

On the Fixation Process of a Beneficial Mutation in a Variable Environment

A population that adapts to gradual environmental change will typically experience temporal variation in its population size and the selection pressure. On the basis of the mathematical theory of inhomogeneous branching processes, we present a framework to describe the fixation process of a single beneficial allele under these conditions. The approach allows for arbitrary time-dependence of the selection coefficient s(t) and the population size N(t), as may result from an underlying ecological model. We derive compact analytical approximations for the fixation probability and the distribution of passage times for the beneficial allele to reach a given intermediate frequency. We apply the formalism to several biologically relevant scenarios, such as linear or cyclic changes in the selection coefficient, and logistic population growth. Comparison with computer simulations shows that the analytical results are accurate for a large parameter range, as long as selection is not very weak.

T. E. Harris and branching processes

T. E. Harris was a pioneer par excellence in many fields of probability theory. In this paper, we give a brief survey of the many fundamental contributions of Harris to the theory of branching processes, starting with his doctoral work at Princeton in the late forties and culminating in his fundamental book “The Theory of Branching Processes,” published in 1963.

Self-organized model of cascade spreading

We study simultaneous price drops of real stocks and show that for high drop thresholds they follow a power-law distribution. To reproduce these collective downturns, we propose a minimal self-organized model of cascade spreading based on a probabilistic response of the system elements to stress conditions. This model is solvable using the theory of branching processes and the mean-field approximation. For a wide range of parameters, the system is in a critical state and displays a power-law cascade-size distribution similar to the empirically observed one. We further generalize the model to reproduce volatility clustering and other observed properties of real stocks.

Random Networks with Preferential Growth and Vertex Death

A dynamic model for a random network evolving in continuous time is defined, where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a function b of the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a function d of its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution {pk} is derived and analyzed for a number of specific choices of b and d. When b(i) = i + α and d(i) = β, that is, linear preferential attachment for the newborn and random deaths, then pk ∼ k-(2+α). When b(i) = i + 1 and d(i) = β(i + 1), with β < 1, then pk ∼ (1 + β)−k, that is, if the death rate is also proportional to the fitness, then the power-law distribution is lost. Furthermore, when b(i) = i + 1 and d(i) = β(i + 1)γ, with β, γ < 1, then logpk ∼ -kγ, a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.

Random Networks with Preferential Growth and Vertex Death

A dynamic model for a random network evolving in continuous time is defined, where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a functionbof the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a functiondof its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution {pk} is derived and analyzed for a number of specific choices ofbandd. Whenb(i) =i+ α andd(i) = β, that is, linear preferential attachment for the newborn and random deaths, thenpk∼k-(2+α). Whenb(i) =i+ 1 andd(i) = β(i+ 1), with β < 1, thenpk∼ (1 + β)−k, that is, if the death rate is also proportional to the fitness, then the power-law distribution is lost. Furthermore, whenb(i) =i+ 1 andd(i) = β(i+ 1)γ, with β, γ < 1, then logpk∼ -kγ, a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.

A composite likelihood approach to the analysis of longitudinal clonal data on multitype cellular systems under an age-dependent branching process.

A recurrent statistical problem in cell biology is to draw inference about cell kinetics from observations collected at discrete time points. We investigate this problem when multiple cell clones are observed longitudinally over time. The theory of age-dependent branching processes provides an appealing framework for the quantitative analysis of such data. Likelihood inference being difficult in this context, we propose an alternative composite likelihood approach, where the estimation function is defined from the marginal or conditional distributions of the number of cells of each observable cell type. These distributions have generally no closed-form expressions but they can be approximated using simulations. We construct a bias-corrected version of the estimating function, which also offers computational advantages. Two algorithms are discussed to compute parameter estimates. Large sample properties of the estimator are presented. The performance of the proposed method in finite samples is investigated in simulation studies. An application to the analysis of the generation of oligodendrocytes from oligodendrocyte type-2 astrocyte progenitor cells cultured in vitro reveals the effect of neurothrophin-3 on these cells. Our work demonstrates also that the proposed approach outperforms the existing ones.

SOME RESULTS FOR SKIP-FREE RANDOM WALK

A random walk that is skip-free to the left can only move down one level at a time but can skip up several levels. Such random walk features prominently in many places in applied probability including queuing theory and the theory of branching processes. This article exploits the special structure in this class of random walk to obtain a number of simplified derivations for results that are much more difficult in general cases. Although some of the results in this article have appeared elsewhere, our proof approach is different.

On least fixed points of systems of positive polynomials

). Obviously, 1 = (1,...,1) is a solution for everySPrP. By Kleene’s theorem, every SPrP has a least nonnegative solution (called just least solution in what follows).SPrPsareimportantindifferentareasofthetheoryofstochasticprocessesandcomputationalmodels. Afundamentalresultof the theory of branching processes, with numerous applications in physics and biology (see e.g. [6, 1]), states that extinctionprobabilities of species are equal to the least solution of a SPrP. The same result has been recently shown for the probability ofterminationofcertainprobabilisticrecursiveprograms([5,4]). Theconsistencyofstochasticcontext-freegrammars,aproblemof interest in statistical natural language processing, also reduces to checking whether the least solution of a SPrP equals 1 (seee.g. [8]).We fix an SPrP with function f and denote its least solution by µ

Connectivity, Dynamics, and Memory in Reservoir Computing with Binary and Analog Neurons

Reservoir computing (RC) systems are powerful models for online computations on input sequences. They consist of a memoryless readout neuron that is trained on top of a randomly connected recurrent neural network. RC systems are commonly used in two flavors: with analog or binary (spiking) neurons in the recurrent circuits. Previous work indicated a fundamental difference in the behavior of these two implementations of the RC idea. The performance of an RC system built from binary neurons seems to depend strongly on the network connectivity structure. In networks of analog neurons, such clear dependency has not been observed. In this letter, we address this apparent dichotomy by investigating the influence of the network connectivity (parameterized by the neuron in-degree) on a family of network models that interpolates between analog and binary networks. Our analyses are based on a novel estimation of the Lyapunov exponent of the network dynamics with the help of branching process theory, rank measures that estimate the kernel quality and generalization capabilities of recurrent networks, and a novel mean field predictor for computational performance. These analyses reveal that the phase transition between ordered and chaotic network behavior of binary circuits qualitatively differs from the one in analog circuits, leading to differences in the integration of information over short and long timescales. This explains the decreased computational performance observed in binary circuits that are densely connected. The mean field predictor is also used to bound the memory function of recurrent circuits of binary neurons.

Exact and asymptotic n-tuple laws at first and last passage

Understanding the space–time features of how a Lévy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory of branching processes, to name but a few. In Doney and Kyprianou [Ann. Appl. Probab. 16 (2006) 91–106] a new quintuple law was established for a general Lévy process at first passage below a fixed level. In this article we use the quintuple law to establish a family of related joint laws, which we call n-tuple laws, for Lévy processes, Lévy processes conditioned to stay positive and positive self-similar Markov processes at both first and last passage over a fixed level. Here the integer n typically ranges from three to seven. Moreover, we look at asymptotic overshoot and undershoot distributions and relate them to overshoot and undershoot distributions of positive self-similar Markov processes issued from the origin. Although the relation between the n-tuple laws for Lévy processes and positive self-similar Markov processes are straightforward thanks to the Lamperti transformation, by interplaying the role of a (conditioned) stable processes as both a (conditioned) Lévy processes and a positive self-similar Markov processes, we obtain a suite of completely explicit first and last passage identities for so-called Lamperti-stable Lévy processes. This leads further to the introduction of a more general family of Lévy processes which we call hypergeometric Lévy processes, for which similar explicit identities may be considered.

A Viral Branching Model for Predicting the Spread of Electronic Word of Mouth

In a viral marketing campaign, an organization develops a marketing message and encourages customers to forward this message to their contacts. Despite its increasing popularity, there are no models yet that help marketers to predict how many customers a viral marketing campaign will reach and how marketers can influence this process through marketing activities. This paper develops such a model using the theory of branching processes. The proposed viral branching model allows customers to participate in a viral marketing campaign by (1) opening a seeding e-mail from the organization, (2) opening a viral e-mail from a friend, and (3) responding to other marketing activities such as banners and offline advertising. The model parameters are estimated using individual-level data that become available in large quantities in the early stages of viral marketing campaigns. The viral branching model is applied to an actual viral marketing campaign in which over 200,000 customers participated during a six-week period. The results show that the model quickly predicts the actual reach of the campaign. In addition, the model proves to be a valuable tool to evaluate alternative what-if scenarios.

The effect of correlations on the population dynamics of lymphocytes

Recent studies of the population dynamics of a system of lymphocytes in an in vitro immune response have reported strong correlations in cell division times, both between parents and their progeny, and between those of sibling cells. The data also show a high level of correlation in the ultimate number of divisions achieved by cells within the same clone. Such correlations are often ignored in mathematical models of cell dynamics as they violate a standard assumption in the theory of branching processes, that of the statistical independence of cells. In this article we present a model in which these correlations can be incorporated, and have used this model to study the effect of these correlations on the population dynamics of a system of cells. We found that correlation in the division times between parents and their progeny can alter the mean population size of clones within the system, while all of the correlations can affect the variance in the sizes of different clones. The model was then applied to experimental data obtained from time-lapse video microscopy of a system of CpG stimulated B lymphocytes and it was found that inclusion of the correct correlation structure is necessary to accurately reproduce the observed population dynamics. We conclude that correlations in the dynamics of cells within an ensemble will affect the population dynamics of the system, and the effects will become more pronounced as the number of divisions increases.

Uncertain survival and time discounting: intertemporal consumption plans for family trusts

We derive expressions for optimal consumption for family trusts with random wealth and uncertain survival. Using UK birth statistics and the theory of branching processes, we compute size and survival probabilities for single- and multiple-branch families. Survival for a single-branch family is approximated by a Pareto distribution and consumption policies exhibit decreasing discount rates, but multiple-branch families use non-monotonic discount rates. When trust distributions depend on the number of beneficiaries rather than the survival of the whole family unit, spending paths depend on expected membership and the elasticity of intertemporal substitution. We report examples of consumption paths for a range of family trusts with constant relative risk aversion preferences.

Coiled-Coil Hydrogels. Effect of Grafted Copolymer Composition and Cyclization on Gelation.

A mean-field theoretical approach was developed to model gelation of solutions of hydrophilic polymers with grafted peptide motifs capable of forming associates of coiled-coil type. The model addresses the competition between associates engaged in branching and cyclization. It results in relative concentrations of intra- and intermolecular associates in dependence on associate strength and motif concentration. The cyclization probability is derived from the model of equivalent Gaussian chain and takes into account all possible paths connecting the interacting motifs. Examination of the association-dissociation equilibria, controlled by the equilibrium constant for association taken as input information, determines the fractions of inter- and intramolecularly associated motifs. The gelation model is based on the statistical theory of branching processes and in combination with the cyclization model predicts the critical concentration delimiting the regions of gelled and liquid states of the system. A comparison between predictions of the model and experimental data available for aqueous solutions of poly[N-(2-hydroxypropyl)methacrylamide] grafted with oppositely charged pentaheptad peptides, CCE and CCK, indicates that the association constant of grafted motifs by four orders of magnitude lower than that of free motifs. It is predicted that at the critical concentration of each motif of about 6×10(-7) mol/cm(3), about half of motifs in associated state is engaged in intramolecular bonds.

POLLING SYSTEMS WITH TWO-PHASE GATED SERVICE

We study an asymmetric cyclic polling system with Poisson arrivals, general service-time and switch-over time distributions, and so-called two-phase gated service at each queue, an interleaving scheme that aims to enforce some level of “fairness” among the different customer classes. For this model, we use the classical theory of multitype branching processes to derive closed-form expressions for the Laplace–Stieltjes transform of the waiting-time distributions when the load tends to 1, in a general parameter setting and under proper heavy-traffic scalings. This result is strikingly simple and provides new insights in the behavior of two-phase polling systems. In particular, the result provides insight in the waiting-time performance and the trade-off between efficiency and fairness of two-phase gated polling compared to the classical one-phase gated service policy.

The effect of superspreading on epidemic outbreak size distributions

Recently, evidence has been presented to suggest that there are significant heterogeneities in the transmission of communicable diseases. Here, a stochastic simulation model of an epidemic process that allows for these heterogeneities is used to demonstrate the potentially considerable effect that heterogeneity of transmission will have on epidemic outbreak size distributions. Our simulation results agree well with approximations gained from the theory of branching processes. Outbreak size distributions have previously been used to infer basic epidemiological parameters. We show that if superspreading does occur then such distributions must be interpreted with care. The simulation results are discussed in relation to measles epidemics in isolated populations and in predominantly urban scenarios. The effect of three different disease control policies on outbreak size distributions are shown for varying levels of heterogeneity and disease control effort.