11/9/2023 0 Comments Stochastic probability![]() ![]() When the number of agents/particles in the system is very large the dynamics of the full particle system (PS) can be rather complex and expensive to simulate moreover, one is quite often more interested in the collective behaviour of the system rather than in its detailed description. individuals, animals, cells, robots) that interact with each other. Many interesting systems in physics and applied sciences consist of a large number of particles, or agents, (e.g. Applications include the modelling of the spread of computer viruses through computer networks, spread of information through a social network or spread of infection through population. We also study stochastic processes evolving on random graphs. These help us understand lengths of typical or extremal paths through the graph, or the likelihood of observing particular sub-graphs. We have worked on understanding the structure and behaviour of different random graph models, including, for example, various measures of connectivity. Our work in this area includes the study of both exact and asymptotic properties of random graphs, including limit theorems and approximations. Some of these models are static, others can be thought of as evolving in time. TheseÄifferent applications give rise to a variety of random graph models with different mechanisms for deciding which nodes will be connected, and varying properties and characteristics. These are an essential tool in modelling communications, power and computer systems, social media contacts, the internet, and in many other areas. Random graphsĪ random graph consists of a number of nodes, of which some randomly chosen pairs are connected by an edge. via Markov Chain Monte Carlo methods) and optimization (e.g. ![]() Besides their theoretical significance, such questions have multiple applications in fields such as computational statistics and machine learning, in problems of sampling (e.g. We are interested in establishing conditions for existence of stationary distributions of such processes, as well as quantifying the convergence rate of the process to its equilibrium state. Examples of processes under consideration include solutions to stochastic differential equations, Markov chains and birth-death processes. We are interested in research problems related to the question of ergodicity of stochastic processes, with a focus on convergence of Markov processes to equilibrium. We have also worked on the theoretical underpinnings of several useful and widely applied approximation techniques. Applications on which we have recently worked include communications and queueing systems, random networks, insurance, and Covid testing, among others. We work on limit theorems and approximations for a wide range of models and processes with a variety of dependence structures. Central limit theorems, giving a Gaussian limit, are known to hold in a variety of settings, and a Poisson limit often occurs in the approximation of rare events. In a large number of settings we observe well-behaved estimates and approximations from one of a relatively small number of stochastic process, of which the Poisson and Gaussian processes are the best known examples. One solution to this problem is to apply limit theorems, which give asymptotic estimates, or other approximations of quantities of interest. ![]() In many models for real-world stochastic systems, probabilities and other characteristics of practical interest cannot be calculated exactly, perhaps because of a prohibitively complex model or large number of model components. We work in a number of areas of applied and theoretical probability which are listed, in no particular order, below. Members of the group have long-term collaborations with leading research centres in the UK and abroad, as well as with colleagues at the University of Edinburgh and in other schools at Heriot-Watt University. We have interests in both the development of new theoretical ideas and techniques within the realm of probability and stochastic models and the applications of probabilistic ideas to tackle novel problems within the real world. Marketing, Recruitment and Communications Energy, Geoscience, Infrastructure and Society ![]()
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