A survey of random processes with reinforcement

The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Modeling the ballistic-to-diffusive transition in nematode motility reveals variation in exploratory behavior across species

A quantitative understanding of organism-level behavior requires predictive models that can capture the richness of behavioral phenotypes, yet are simple enough to connect with underlying mechanistic processes. Here we investigate the motile behavior of nematodes at the level of their translational motion on surfaces driven by undulatory propulsion. We broadly sample the nematode behavioral repertoire by measuring motile trajectories of the canonical lab strain $C. elegans$ N2 as well as wild strains and distant species. We focus on trajectory dynamics over timescales spanning the transition from ballistic (straight) to diffusive (random) movement and find that salient features of the motility statistics are captured by a random walk model with independent dynamics in the speed, bearing and reversal events. We show that the model parameters vary among species in a correlated, low-dimensional manner suggestive of a common mode of behavioral control and a trade-off between exploration and exploitation. The distribution of phenotypes along this primary mode of variation reveals that not only the mean but also the variance varies considerably across strains, suggesting that these nematode lineages employ contrasting ``bet-hedging'' strategies for foraging.Comment: 46 pages, 18 figures, 6 table

Universality classes of interaction structures for NK fitness landscapes

Kauffman's NK-model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems. Genotypes are represented as sequences of $L$ binary loci. The fitness assigned to a genotype is a sum of contributions, each of which is a random function defined on a subset of $k \le L$ loci. These subsets or neighborhoods determine the genetic interactions of the model. Whereas earlier work on the NK model suggested that most of its properties are robust with regard to the choice of neighborhoods, recent work has revealed an important and sometimes counter-intuitive influence of the interaction structure on the properties of NK fitness landscapes. Here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible (that is, fitness-monotonic) mutational pathways. In particular, we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of $L$, and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large $k$. Moreover, we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in $L$ for a large class of interaction structures that we characterize as locally bounded. Finally, we discuss the impact of the NK interaction structures on the dynamics of evolution using adaptive walk models.Comment: 61 pages, 9 figure

Who Replaces Whom? Local versus Non-local Replacement in Social and Evolutionary Dynamics

In this paper, we inspect well-known population genetics and social dynamics models. In these models, interacting individuals, while participating in a self-organizing process, give rise to the emergence of complex behaviors and patterns. While one main focus in population genetics is on the adaptive behavior of a population, social dynamics is more often concerned with the splitting of a connected array of individuals into a state of global polarization, that is, the emergence of speciation. Applying computational and mathematical tools we show that the way the mechanisms of selection, interaction and replacement are constrained and combined in the modeling have an important bearing on both adaptation and the emergence of speciation. Differently (un)constraining the mechanism of individual replacement provides the conditions required for either speciation or adaptation, since these features appear as two opposing phenomena, not achieved by one and the same model. Even though natural selection, operating as an external, environmental mechanism, is neither necessary nor sufficient for the creation of speciation, our modeling exercises highlight the important role played by natural selection in the interplay of the evolutionary and the self-organization modeling methodologies.Comment: 14 pages, 11 figure