Maximum-entropy moment-closure for stochastic systems on networks

Rogers, Tim

doi:10.1088/1742-5468/2011/05/p05007

Cited by 33 publications

(40 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the mathematical exploration of a network model, the central challenge is often to find an approach that maps the network problem onto a tractable set of equations. One prominent class of methods for approximating network dynamics are moment expansions [9,19,[23][24][25][26]. The central idea is to write evolution equations for the abundance of certain motifs in the network.…”

Section: Introductionmentioning

confidence: 99%

Exploring the adaptive voter model dynamics with a mathematical triple jump

et al. 2014

View full text Add to dashboard Cite

Progress in theoretical physics is often made by the investigation of toy models, the model organisms of physics, which provide benchmarks for new methodologies. For complex systems, one such model is the adaptive voter model. Despite its simplicity, the model is hard to analyze. Only inaccurate results are obtained from well-established approximation schemes that work well on closely-related models. We use the adaptive voter model to illustrate a new approach that combines (a) the use of a heterogeneous moment expansion to approximate the network model by an infinite system of ordinary differential equations (ODEs), (b) generating functions to map the ODE system to a twodimensional partial differential equation (PDE), and (c) solution of this partial differential equation by the tools of PDE-theory. Beyond the adaptive voter models, the proposed approach establishes a connection between network science and the theory of PDEs and is widely applicable to the dynamics of networks with discrete node-states.

show abstract

Section: Introductionmentioning

confidence: 99%

Exploring the adaptive voter model dynamics with a mathematical triple jump

et al. 2014

View full text Add to dashboard Cite

show abstract

“…, which has been observed numerically [31]. In spite of this it has been shown to yield accurate approximations in these probabilistic systems [31,33,36].…”

Section: A Contact Conditioning Approximation Modelmentioning

confidence: 96%

Methods for approximating stochastic evolutionary dynamics on graphs

Overton

Broom

Hadjichrysanthou

et al. 2019

Journal of Theoretical Biology

View full text Add to dashboard Cite

Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the evolutionary process. However, for more complicated heterogeneous structures, computationally intensive methods are required such as individual-based stochastic simulations. By adapting methods from statistical physics, including moment closure techniques, we first show how to derive existing homogenised pair approximation models and the exact neutral drift model. We then develop node-level approximations to stochastic evolutionary processes on arbitrarily complex structured populations represented by finite graphs, which can capture the different dynamics for individual nodes in the population. Using these approximations, we evaluate the fixation probability of invading mutants for given initial conditions, where the dynamics follow standard evolutionary processes such as the invasion process. Comparisons with the output of stochastic simulations reveal the effectiveness of our approximations in describing the stochastic processes and in predicting the probability of fixation of mutants on a wide range of graphs. Construction of these models facilitates a systematic analysis and is valuable for a greater understanding of the influence of population structure on evolutionary processes.

show abstract

“…, n. For such models to be tractable, these hierarchies of equations need to be truncated, or closed, before an approximate solution describing the dynamics of the system of interest can be obtained (Singer 2004). As a result, the development of new closure approximations or the development of new understanding of existing closure approximations is an active area of research, and results are often published in the physics and mathematics literature (Murrell et al 2004;Karrer and Newman 2010;Raghib et al 2011;Rogers 2011;Wilkinson and Sharkey 2014). Another front of research concerns the application of moment dynamics descriptions to dynamic processes taking place on networks.…”

Section: Figmentioning

confidence: 99%

Special Issue on Spatial Moment Techniques for Modelling Biological Processes

Simpson

Baker

2015

Bull Math Biol

View full text Add to dashboard Cite

Over the last two decades, there has been an increasing awareness of, and interest in, the use of spatial moment techniques to provide insight into a range of biological and ecological processes. Models that incorporate spatial moments can be viewed as extensions of mean-field models. These mean-field models often consist of systems of classical ordinary differential equations and partial differential equations, whose derivation, at some point, hinges on the simplifying assumption that individuals in the underlying stochastic process encounter each other at a rate that is proportional to the average abundance of individuals. This assumption has several implications, the most striking of which is that mean-field models essentially neglect any impact of the spatial structure of individuals in the system. Moment dynamics models extend traditional mean-field descriptions by accounting for the dynamics of pairs, triples and higher n-tuples of individuals. This means that moment dynamics models can, to some extent, account for how the spatial structure affects the dynamics of the system in question.Previous applications of moment dynamics models have focused on several application areas, such as: ecological dynamics

show abstract

Maximum-entropy moment-closure for stochastic systems on networks

Cited by 33 publications

References 32 publications

Exploring the adaptive voter model dynamics with a mathematical triple jump

Exploring the adaptive voter model dynamics with a mathematical triple jump

Methods for approximating stochastic evolutionary dynamics on graphs

Special Issue on Spatial Moment Techniques for Modelling Biological Processes

Contact Info

Product

Resources

About