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State Aggregations in Markov Chains and Block Models of Networks.
Mauro Faccin1, Michael T Schaub2,3, Jean-Charles Delvenne1,4
1ICTEAM, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium.
Physical Review Letters
|August 30, 2021
Summary
We developed a novel information-theoretic approach to aggregate Markov chain states, maximizing mutual information over time. This method reveals long-range dynamical modules in complex systems like ocean currents.
Area of Science:
- Information Theory
- Dynamical Systems
- Network Science
- Markov Chains
Background:
- State aggregation in Markov chains is crucial for simplifying complex systems.
- Understanding long-range dependencies in dynamical systems is challenging.
- Existing network models often lack a dynamical perspective.
Purpose of the Study:
- To develop an information-theoretic framework for state aggregation in Markov chains.
- To maximize mutual information between aggregated states separated by T time steps.
- To uncover coherent, long-range dynamical modules in complex systems.
Main Methods:
- Formulated state aggregation as an optimization problem maximizing mutual information.
- Analyzed the case T=1, connecting it to the degree-corrected stochastic block model.
- Extended the framework to T≫1 to identify long-timescale dynamics.
Main Results:
- For T=1, the method recovers the maximum-likelihood estimator of the degree-corrected stochastic block model.
- Identified long-range dynamical modules by considering timescales T≫1.
- Successfully applied to synthetic data and real-world ocean current data, capturing key features.
Conclusions:
- The information-theoretic approach provides a powerful lens for analyzing Markov chain dynamics.
- This framework unifies network modeling and dynamical systems analysis.
- The method effectively reveals hidden, long-timescale structures in complex systems.
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