考慮由數個節點以及節點間的連線所構成的圖形，並討論在這些圖形上 的馬可夫鏈：若任兩個節點之間有連線，則兩節點間會有一 個相對應的機率相互轉移；反之則不會轉移。對於這樣一 種形態的馬可夫鏈，它每個時間點的機率分佈近似到其穩定分佈所需的時間我們稱之為混合時間。在文獻上，對於馬可夫鏈的收斂及其收斂速度，已有許多討論，其中之 一 為：馬可鏈收斂的速度與其機率轉移矩陣的第二大絕對特徵值 (SLEM) 有關。

在本文中，在某些特定圖形上 的混合時間將會是我們討論的主題。這些特定圖形上 的馬可夫鏈，我們考慮它的收斂速度，也 就是轉移矩陣的特徵值，來估計它們的混合時間。我們 也 將藉由這些特徵值，去討論混合時間與節點個數間的關係。

Consider a Markov chain on a given connected graph, where each edge is labeled with a given transition probability between two adjacent vertices. For the above Markov chain, the time up to its equilibrium distribution is usually called the mixing time. The rates of convergence of Markov chains have been studied in literature. In fact, the rate of convergence of a given Markov chain can be bounded by the second largest eigenvalue in modulus of the corresponding transition probability matrix.

In this thesis, our goal is to discuss the mixing times on some particular graphs. For
each above Markov chain, we will discuss the relation between eigenvalues of its transition probability matrix and the number of the corresponding vertices.

論文審定書 i
誌謝 ii
摘要 iii
Abstract iv
1 Introduction 1
2 Preliminary 2
3 Markov Chains on Specific Graphs and their Mixing Times 6
3.1 Mixing times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Complete Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Path Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References 17

[1] S. Boyd, P. Diaconis and L. Xiao, Fastest Mixing Markov Chain on a Graph, SIAM,
Vol. 46, 667-689, 2004.
[2] S. Boyd, A. Ghosh, B. Prabhakar and D. Shah, Mixing Times for Random Walks on
Geometric Random Graphs, The Proceedings of SIAM ANALCO, 240-249, 2005
[3] P. Bremaud, Markov Chains, Gibbs Fields, Monte Carlo Simulation and Queues,
Springer-Verlag, Berlin, 1999.
[4] S. H. Friedberg, A. J. Insel and L. E. Spence, Linear Algebra, 4th ed, Pearson, 2002.
[5] J. J. Hunter, Coupling and Mixing Times in a Markov Chain, Linear Algebra and its
Applications, Vol. 430, 2607-2621, 2009.
[6] C. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed, Springer, 2004.