Abstract
In this chapter, we first introduce the notations that will be used throughout the thesis. After that, we discuss and state some known results, as background material relevant to our future development.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In this definition, we consider the players as maximizers whose goals are to maximize their individual utility functions. Analogously, one can consider the players as minimizers whose goals are to minimize their individual cost functions \(C_i(\cdot )\). In such a case \(\sigma ^*\) is said to be an NE if \(C_i(\sigma ^*_i,\sigma ^*_{-i})\!\le \! C_i(\sigma _i,\sigma ^*_{-i}), \forall i\!\in \! [n], \sigma _i\!\in \! \Sigma _i\).
References
P. Tetali, “Random walks and the effective resistance of networks,” Journal of Theoretical Probability, vol. 4, pp. 101–109, 1991.
L. Lovász, “Random walks on graphs: A survey,” Combinatorics, Paul Erdos is eighty, vol. 2, no. 1, pp. 1–46, 1993.
M. Catral, S. J. Kirkland, M. Neumann, and N.-S. Sze, “The Kemeny constant for finite homogeneous ergodic Markov chains,” Journal of Scientific Computing, vol. 45, no. 1-3, pp. 151–166, 2010.
D. Aldous and J. Fill, Reversible Markov chains and random walks on graphs. Book in preparation - preprint available at http://www.stat.Berkeley.EDU/users/aldous, 2002.
D. A. Levin, Y. Peres, and E. L. Wilmer, Markov Chains and Mixing Times. American Mathematical Society, 2009.
L. Saloff-Coste, “Lectures on finite Markov chains,” in Lectures on Probability Theory and Statistics. Springer, 1997, pp. 301–413.
C. D. Godsil, G. Royle, and C. Godsil, Algebraic Graph Theory. Springer, New York, 2001.
R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, 2012.
W. N. Anderson Jr and T. D. Morley, “Eigenvalues of the Laplacian of a graph*,” Linear and Multilinear Algebra, vol. 18, no. 2, pp. 141–145, 1985.
J. Shen, “A geometric approach to ergodic non-homogeneous Markov chains,” Wavelet Analysis and Multiresolution Methods, pp. 341–366, 2000.
T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory. SIAM, 1999, vol. 23.
D. Monderer and L. S. Shapley, “Potential games,” Games and Economic Behavior, vol. 14, no. 1, pp. 124–143, 1996.
E. Koutsoupias and C. Papadimitriou, “Worst-case equilibria,” in STACS 99. Springer, 1999, pp. 404–413.
N. Alon and J. H. Spencer, The Probabilistic Method. John Wiley & Sons, 2004.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Etesami, S.R. (2017). Notation and Mathematical Preliminaries. In: Potential-Based Analysis of Social, Communication, and Distributed Networks. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-54289-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-54289-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-54288-1
Online ISBN: 978-3-319-54289-8
eBook Packages: EngineeringEngineering (R0)