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Aggregation of Markov flows I: theory.

R S MacKay1, J D Robinson2

  • 1Mathematics Institute and Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, UK r.s.mackay@warwick.ac.uk.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|March 21, 2018
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This study introduces a new node aggregation method for Markov flows, a type of stationary measure used in semi-Markov processes. This technique enables efficient computation and provides macroscopic behavior insights.

Keywords:
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Area of Science:

  • Mathematics
  • Probability Theory
  • Stochastic Processes

Background:

  • Markov flows are stationary measures for continuous-time regular jump homogeneous semi-Markov processes.
  • Existing methods for simplifying Markov flows include node elimination, with improvements suggested for Wales' methods.
  • Simplifying these processes is crucial for computational efficiency and understanding system dynamics.

Purpose of the Study:

  • To present an alternative method for simplifying Markov flows by aggregating nodes.
  • To enable iterative application of this aggregation for hierarchical schemes.
  • To demonstrate the potential for efficient computation and macroscopic behavior analysis.

Main Methods:

  • A novel node aggregation method is proposed to create factor Markov flows.
  • The aggregation method can be iterated to build hierarchical structures.
  • Improvements to existing node elimination techniques are also discussed.

Main Results:

  • The proposed aggregation method provides a way to produce smaller, factor Markov flows.
  • Iterated aggregation allows for hierarchical schemes, enhancing structural analysis.
  • The method facilitates efficient recomputation and analysis of local changes.

Conclusions:

  • Node aggregation offers an effective alternative to node elimination for simplifying Markov flows.
  • The developed method supports efficient computation and provides insights into macroscopic behavior.
  • This work contributes to the understanding and application of semi-Markov processes, particularly within the context of Hilbert's sixth problem.