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We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include contingency tables, logistic regression, and spectral analysis of permutation data. The algorithms involve computations in polynomial rings using Grobner bases.

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Thisis an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous time. Some aspects of the theory simplify considerably with finite state spaces and we are able to give a self-contained development.Examples of applications include the study of a Metropolis chain for the binomial distribution, sharp results for natural chains on the box of side n in d dimensions and improved rates for exclusion processes. We also show that for most r-regular graphs the log-Sobolev constant is of smaller order than the spectral gap. The log-Sobolev constant of the asymmetric twopoint space is computed exactly as well as the log-Sobolev constant of the complete graph on n points.

A bstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average.

4 The social environment is a pervasive influence on the ecological and evolutionary dynamics 5 of animal populations. Recently, social network analysis has provided an increasingly 6 powerful and diverse toolset to enable animal behaviour researchers to quantify the social 7 environment of animals and the impact that it has on ecological and evolutionary processes. 8 However, there is considerable scope for improving these methods further. We outline an 9 approach specifically designed to model the formation of network links, exponential random 10 graph models (ERGMs), which have great potential for modelling animal social structure. 11 ERGMs are generative models that treat network topology as a response variable. This 12 makes them ideal for answering questions related directly to how and why social 13 associations or interactions occur, from the modelling of population-level transmission, 14 through within-group behavioural dynamics to social evolutionary processes. We discuss 15 how ERGMs have been used to study animal behaviour previously, and how recent 16 developments in the ERGM framework can increase the scope of their use further. We also 17 highlight the strengths and weaknesses of this approach relative to more conventional 18 methods, and provide some guidance on the situations and research areas in which they can 19 be used appropriately. ERGMs have the potential to be an important part of an animal 20 behaviour researcher's toolkit and fully integrating them into the field should enhance our 21 ability to understand what shapes animal social interactions, and identify the underlying 22 processes that lead to the social structure of animal populations. 23 2 assortative behaviour, transitivity 25 26 the study of the social and spatial components of dispersal behaviours (Blumstein et al., 47 2009; Fletcher et al., 2011). 48 The statistical analysis of social networks is complicated by the non-independence of 49 individuals within a population that results from linking individuals together within a 50 network (Croft et al., 2011; Farine & Whitehead, 2015). This confounds the use of the 51 conventional statistical approaches used in ecology such as the linear model and linear 52 mixed model, as these methods assume independence of the residuals, which is an invalid 53 assumption for individuals that are linked in a network. In light of this, numerous statistical 54 methodologies have been developed to analyse social network structure. Typically, the 55 analysis of animal social networks has revolved around randomisation-based approaches to 56 significance testing (Croft et al., 2011; Farine & Whitehead, 2015). The data used to 57 construct networks is permuted to generate uncertainty around the null hypothesis (e.g. 58 social interactions are assorted by a phenotype of interest), with permutations typically 59 constrained to produce biologically plausible null models. For example, if researchers are 60 studying how body size relates to social network connections in a population spread over 61 several ...

We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the Markov chain, is determined by the second largest (in magnitude) eigenvalue of the transition matrix. In this paper we address the problem of assigning probabilities to the edges of the graph in such a way as to minimize the second largest magnitude eigenvalue, i.e., the problem of finding the fastest mixing Markov chain on the graph.We show that this problem can be formulated as a convex optimization problem, which can in turn be expressed as a semidefinite program (SDP). This allows us to easily compute the (globally) fastest mixing Markov chain for any graph with a modest number of edges (say, 1000) using standard numerical methods for SDPs. Larger problems can be solved by exploiting various types of symmetry and structure in the problem, and far larger problems (say 100000 edges) can be solved using a subgradient method we describe. We compare the fastest mixing Markov chain to those obtained using two commonly used heuristics: the maximum-degree method, and the Metropolis-Hastings algorithm. For many of the examples considered, the fastest mixing Markov chain is substantially faster than those obtained using these heuristic methods.We derive the Lagrange dual of the fastest mixing Markov chain problem, which gives a sophisticated method for obtaining (arbitrarily good) bounds on the optimal mixing rate, as well the optimality conditions. Finally, we describe various extensions of the method, including a solution of the problem of finding the fastest mixing reversible Markov chain, on a fixed graph, with a given equilibrium distribution.