J. O. Matthews

Distinguishing population processes by external monitoring

We investigate the statistical and correlation properties of two stochastic population models that give rise to identical first-order probability densities. We assume that the processes are monitored indirectly through measurement of the rate at which individuals emigrate from the population. Formulae characterizing the integrated statistics of these counting processes are derived, and it is shown how they may be used to distinguish the population models.

Generation and monitoring of discrete stable random processes using multiple immigration population models

Some properties of classical population processes that comprise births, deaths and multiple immigrations are investigated. The rates at which the immigrants arrive can be tailored to produce a population whose steady state fluctuations are described by a pre-selected distribution. Attention is focused on the class of distributions with a discrete stable law, which have power-law tails and whose moments and autocorrelation function do not exist. The separate problem of monitoring and characterizing the fluctuations is studied, analysing the statistics of individuals that leave the population. The fluctuations in the size of the population are transferred to the times between emigrants that form an intermittent time series of events. The emigrants are counted with a detector of finite dynamic range and response time. This is modelled through clipping the time series or saturating it at an arbitrary but finite level, whereupon its moments and correlation properties become finite. Distributions for the time to the first counted event and for the time between events exhibit power-law regimes that are characteristic of the fluctuations in population size. The processes provide analytical models with which properties of complex discrete random phenomena can be explored, and in addition provide generic means by which random time series encompassing a wide range of intermittent and other discrete random behaviour may be generated.

Discrete scale-free distributions and associated limit theorems

Consideration is given to the convergence properties of sums of identical, independently distributed random variables drawn from a class of discrete distributions with power-law tails, which are relevant to scale-free networks. Different limiting distributions, and rates of convergence to these limits, are identified and depend on the index of the tail. For indices ≥2, the topology evolves to a random Poisson network, but the rate of convergence can be extraordinarily slow and unlikely to be yet evident for the current size of the WWW for example. It is shown that treating discrete scale-free behaviour with continuum or mean-field approximations can lead to incorrect results.

Fluctuations in a coupled population model

We investigate a discrete Markov process in which the immigration of individuals into one population is controlled by the fluctuations in another. We examine the effect of coupling back the second population to the first through a similar mechanism and derive exact solutions for the generating functions of the population statistics. We show that a stationary state exists over a certain parameter range and obtain expressions for moments and correlation functions in this regime. When more than two populations are coupled, cyclically transient oscillations and periodic behaviour of correlation functions are predicted. We demonstrate that if the initial distribution of either population is stable, or more generally has a power-law tail that falls off like N−(1+α) (0 < α < 1), then for certain parameter values there exists a stationary state that is also power law but not stable. This stationary state cannot be accessed from a single multiple immigrant population model, but arises solely from the nonlinear interaction of the coupled system.

Generation and monitoring of a discrete stable random process

A discrete stochastic process with stationary power law distribution is obtained from a death-multiple immigration population model. Emigrations from the population form a random series of events which are monitored by a counting process with finite-dynamic range and response time. It is shown that the power law behaviour of the population is manifested in the intermittent behaviour of the series of events.

Generating Discrete Power‐Law Distributions from a Death‐ Multiple Immigration Population Process

We consider the evolution of a simple population process governed by deaths and multiple immigrations that arrive with rates particular to their order. For a particular choice of rates, the equilibrium solution has a discrete power‐law form. The model is a generalization of a process investigated previously where immigrants arrived in pairs [1]. The general properties of this model are discussed in a companion paper. The population is initiated with precisely M individuals present and evolves to an equilibrium distribution with a power‐law tail. However the power‐law tails of the equilibrium distribution are established immediately, so that moments and correlation properties of the population are undefined for any non‐zero time. The technique we develop to characterize this process utilizes external monitoring that counts the emigrants leaving the population in specified time intervals. This counting distribution also possesses a power‐law tail for all sampling times and the resulting time series exhibits...

Distinguishing two Population Processes with Identical Equilibrium Densities

We analyze the relationship between the evolution of simple population processes and the rate of emigration of individuals. An external monitoring scheme is defined by counting the number leaving the population in fixed time intervals. This is the analogue of photon counting in quantum optics. It is a reasonable measurement in many situations of interest and also has the merit of being analytically tractable. The formalism we develop is used to investigate the statistical and correlation properties of two stochastic population models that give rise to identical first order probability densities. The first is the birth‐death‐ immigration process for which many well‐known results can be found in the literature. The second is based on a population sustained by multiple immigration. This model is a generalization of the pair process investigated previously [1]. It can be used to generate populations with a range of equilibrium densities including those with power law tails to be described in a companion paper...