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.
References
Avin, C., Koucký, M., Lotker, Z.: How to explore a fast-changing world (cover time of a simple random walk on evolving graphs). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5125, pp. 121–132. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70575-8_11
Bruin, H., Deane, J.H.B.: Piecewise contractions are asymptotically periodic. Proc. Am. Math. Soc. 137(4), 1389–1395 (2009)
Chazelle, B.: Toward a theory of Markov influence systems and their renormalization. In: Proceedings of the 9th ITCS, pp. 58:1–58:18 (2018)
Condon, A., Hernek, D.: Random walks on colored graphs. Random Struct. Algorithms 5, 285–303 (1994)
Denysyuk, O., Rodrigues, L.: Random walks on directed dynamic graphs. In: Proceedings of the 2nd International Workshop on Dynamic Networks: Algorithms and Security (DYNAS10), Bordeaux, France, July 2010. arXiv:1101.5944 (2011)
Denysyuk, O., Rodrigues, L.: Random walks on evolving graphs with recurring topologies. In: Kuhn, F. (ed.) DISC 2014. LNCS, vol. 8784, pp. 333–345. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-45174-8_23
Holme, P., Saramäki, J.: Temporal networks. Phys. Rep. 519, 97–125 (2012)
Iacobelli, G., Figueiredo, D.R.: Edge-attractor random walks on dynamic networks. J. Complex Netw. 5, 84–110 (2017)
Nguyen, G.H., Lee, J.B., Rossi, R.A., Ahmed, N., Koh, E., Kim, S.: Dynamic network embeddings: from random walks to temporal random walks. In: 2018 IEEE International Conference on Big Data, pp. 1085–1092 (2018)
Perra, N., Baronchelli, A., Mocanu, D., Gonçalves, B., Pastor-Satorras, R., Vespignani, A.: Random walks and search in time-varying networks. Phys. Rev. Lett. 109, 238701 (2012)
Petit, J., Gueuning, M., Carletti, T., Lauwens, B., Lambiotte, R.: Random walk on temporal networks with lasting edges. Phys. Rev. E 98, 052307 (2018)
Ramiro, V., Lochin, E., Sénac, P., Rakotoarivelo, T.: Temporal random walk as a lightweight communication infrastructure for opportunistic networks. In: 2014 IEEE 15th International Symposium (WoWMoM), June 2014
Seneta, E.: Non-Negative Matrices and Markov Chains, 2nd edn. Springer, New York (2006)
Starnini, M., Baronchelli, A., Barrat, A., Pastor-Satorras, R.: Random walks on temporal networks. Phys. Rev. E 85(5), 056115–26 (2012)
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Chazelle, B. (2019). Some Observations on Dynamic Random Walks and Network Renormalization. In: Gąsieniec, 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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