Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping
Abstract We propose a unified framework to represent a wide range of continuous-time discrete-state Markov processes on networks, and show how many network dynamics models in the literature can be represented in this unified framework. We show how a particular sub-set of these models, referred to he...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-11-01
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| Series: | Applied Network Science |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1007/s41109-019-0206-4 |
| _version_ | 1818161093488410624 |
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| author | Jonathan A. Ward Martín López-García |
| author_facet | Jonathan A. Ward Martín López-García |
| author_sort | Jonathan A. Ward |
| collection | DOAJ |
| description | Abstract We propose a unified framework to represent a wide range of continuous-time discrete-state Markov processes on networks, and show how many network dynamics models in the literature can be represented in this unified framework. We show how a particular sub-set of these models, referred to here as single-vertex-transition (SVT) processes, lead to the analysis of quasi-birth-and-death (QBD) processes in the theory of continuous-time Markov chains. We illustrate how to analyse a number of summary statistics for these processes, such as absorption probabilities and first-passage times. We extend the graph-automorphism lumping approach [Kiss, Miller, Simon, Mathematics of Epidemics on Networks, 2017; Simon, Taylor, Kiss, J. Math. Bio. 62(4), 2011], by providing a matrix-oriented representation of this technique, and show how it can be applied to a very wide range of dynamical processes on networks. This approach can be used not only to solve the master equation of the system, but also to analyse the summary statistics of interest. We also show the interplay between the graph-automorphism lumping approach and the QBD structures when dealing with SVT processes. Finally, we illustrate our theoretical results with examples from the areas of opinion dynamics and mathematical epidemiology. |
| first_indexed | 2024-12-11T16:12:17Z |
| format | Article |
| id | doaj.art-44204df4b34a4784a93539bafe410650 |
| institution | Directory Open Access Journal |
| issn | 2364-8228 |
| language | English |
| last_indexed | 2024-12-11T16:12:17Z |
| publishDate | 2019-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Applied Network Science |
| spelling | doaj.art-44204df4b34a4784a93539bafe4106502022-12-22T00:59:02ZengSpringerOpenApplied Network Science2364-82282019-11-014112810.1007/s41109-019-0206-4Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumpingJonathan A. Ward0Martín López-García1Department of Applied Mathematics, School of Mathematics, University of LeedsDepartment of Applied Mathematics, School of Mathematics, University of LeedsAbstract We propose a unified framework to represent a wide range of continuous-time discrete-state Markov processes on networks, and show how many network dynamics models in the literature can be represented in this unified framework. We show how a particular sub-set of these models, referred to here as single-vertex-transition (SVT) processes, lead to the analysis of quasi-birth-and-death (QBD) processes in the theory of continuous-time Markov chains. We illustrate how to analyse a number of summary statistics for these processes, such as absorption probabilities and first-passage times. We extend the graph-automorphism lumping approach [Kiss, Miller, Simon, Mathematics of Epidemics on Networks, 2017; Simon, Taylor, Kiss, J. Math. Bio. 62(4), 2011], by providing a matrix-oriented representation of this technique, and show how it can be applied to a very wide range of dynamical processes on networks. This approach can be used not only to solve the master equation of the system, but also to analyse the summary statistics of interest. We also show the interplay between the graph-automorphism lumping approach and the QBD structures when dealing with SVT processes. Finally, we illustrate our theoretical results with examples from the areas of opinion dynamics and mathematical epidemiology.http://link.springer.com/article/10.1007/s41109-019-0206-4Continuous-time Markov chainStochastic processNetworkGraph-automorphismLumpingSummary statistics |
| spellingShingle | Jonathan A. Ward Martín López-García Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping Applied Network Science Continuous-time Markov chain Stochastic process Network Graph-automorphism Lumping Summary statistics |
| title | Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping |
| title_full | Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping |
| title_fullStr | Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping |
| title_full_unstemmed | Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping |
| title_short | Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping |
| title_sort | exact analysis of summary statistics for continuous time discrete state markov processes on networks using graph automorphism lumping |
| topic | Continuous-time Markov chain Stochastic process Network Graph-automorphism Lumping Summary statistics |
| url | http://link.springer.com/article/10.1007/s41109-019-0206-4 |
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