Abstract
We recently developed a general bifurcation analysis framework for establishing the periodicity of certain time-varying random walks. In this work, we look at the special case of lazy uniform-inflow random walks and show how a much simpler version of the argument can be used to resolve their analysis. We also revisit a renormalization technique for network sequences that we introduced earlier and we propose a few simplifications. This work can be viewed as a gentle introduction to Markov influence systems.
The Research was sponsored by the Army Research Office and the Defense Advanced Research Projects Agency and was accomplished under Grant Number W911NF-17-1-0078. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office, the Defense Advanced Research Projects Agency, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
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Notes
- 1.
The discontinuities can also be chosen to be real-algebraic varieties.
- 2.
For example, the lazy random walk specified by the matrix \(\bigl ( {\begin{matrix}1 &{} 0\\ 0.5 &{} 0.5\end{matrix}}\bigr )\) is not irreducible. Also, unlike in [3], we do not require the matrices to be rational.
- 3.
The constants in this paper may depend on any of the input parameters, such as the dimension n, the number of hyperplanes, the hyperplane coefficients, and the matrix elements.
- 4.
Indeed, if that were not the case, then some \(\mathbf {x}\) in a cell of Z, thus outside of Z, would be such that \(f(\mathbf {x})\in Z= Z_{\nu -1}\). It would follow that \(f^{k}(\mathbf {x})\in D\), for \(k\le \nu \); hence \(\mathbf {x}\in Z_\nu =Z\), a contradiction.
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Chazelle, B. (2019). Some Observations on Dynamic Random Walks and Network Renormalization. In: GÄ…sieniec, L., Jansson, J., Levcopoulos, C. (eds) Fundamentals of Computation Theory. FCT 2019. Lecture Notes in Computer Science(), vol 11651. Springer, Cham. https://doi.org/10.1007/978-3-030-25027-0_2
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