Academic Articles & Supplements

Stationary Measures on Infinite Graphs

Alexandre Baraviera, Pedro Duarte, Maria Joana Torres

Author

Alexandre Baraviera, Pedro Duarte, Maria Joana Torres

Title

Stationary Measures on Infinite Graphs

Description

Supplemental notebook to "Stationary measures on infinite graphs"

Mathematica Code
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Stationary Measures on Infinite Graphs

Alexandre Baraviera, Pedro Duarte, Maria Joana Torres

- Instituto de Matemática e Estatística, UFRGS, Av. Bento Gonçalves 9500, 91500, Porto Alegre, RS, Brazil.
Email: baravi@mat.ufrgs.br
- CMAF-CIO, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal.
Email: pedromiguel.duarte@gmail.com
- CMAT and Departamento de Matemática, Universidade do Minho, Campus de Gualtar, 4700-057 Braga, Portugal
Email: jtorres@math.uminho.pt

Recently, we extended the theory of isospectral reductions of L. Bunimovich and B. Webb to infinite graphs, and described an application of this extension to the problems of existence and approximation of stationary measures on a class of tridiagonal stochastic infinite graphs. In this note, we describe an isospectral reduction-reconstruction algorithm and provide numerical examples to compare its execution times with those of a standard algorithm. This is based on Section 8 in our article “Stationary Measures on Infinite Graphs”, Communications in Contemporary Mathematics, https://doi.org/10.1142/S0219199721500255. For the sake of completeness, we also provide the Mathematica code and numerical examples to perform isospectral reductions of arbitrary complex-valued matrices (and, in particular, isospectral reduction of stochastic matrices) and reconstruction of eigenvectors (in particular, reconstruction of stationary probability measures).

1. Introduction
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2. Isospectral Reduction Theory on Finite Graphs
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3. Isospectral Reduction Examples
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4. Extension of Isospectral Reduction Theory to Infinite Dimension
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5. Tridiagonal Stochastic Infinite Graphs
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6. Computational Algorithm
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7. Iterated isospectral reduction & reconstruction examples
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8. References
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9. Acknowledgments
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Cite this as: Alexandre Baraviera, Pedro Duarte, Maria Joana Torres, "Stationary Measures on Infinite Graphs" from the Notebook Archive (2022), https://notebookarchive.org/2022-07-9oyvhxu

Mathematica Code

Stationary Measures on Infinite Graphs

1. Introduction

2. Isospectral Reduction Theory on Finite Graphs

3. Isospectral Reduction Examples

4. Extension of Isospectral Reduction Theory to Infinite Dimension 

5. Tridiagonal Stochastic Infinite Graphs

6. Computational Algorithm

7. Iterated isospectral reduction & reconstruction examples 

8. References

9. Acknowledgments

Mathematica Code


1. Introduction


2. Isospectral Reduction Theory on Finite Graphs


3. Isospectral Reduction Examples


4. Extension of Isospectral Reduction Theory to Infinite Dimension


5. Tridiagonal Stochastic Infinite Graphs


6. Computational Algorithm


7. Iterated isospectral reduction & reconstruction examples


8. References


9. Acknowledgments
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