對演化投票模型的探討

Abstract

在這篇論文中，我們深入研究了一種廣義的投票模型，其中個體有能力與持不同意見的人斷絕聯繫，同時與志同道合的人建立聯繫。我們的分析集中在隨機正則圖上考慮 Durrett 等人於 [6] 研究過的兩個模型上。在每次迭代中，以一定機率進行傳統投票模型步驟或者個體終止與持有不同意見的鄰居的關係，並以下列兩種方式建立新連結： 1. 完全隨機模型：個體於所有人中隨機選擇一人建立聯繫。 2. 同溫層模型：個體於和自身持有相同意見的群體中隨機選擇一人建立聯繫。 在兩種模型中，我們證明了模型參數的相變。對於較低的參數值，兩種意見人數幾乎與原始分布一致，且人際網絡往往分化為具有不同意見的兩個互斥派別。相反，對於較高的參數值，少數意見顯著的減少。另外，在完全隨機模型中，我們證明了少數意見始終存在。

In this thesis, we study a variant of the voter model on a network where each individual may break his/her relationship with those holding different opinions, and befriend those holding the same opinion. We analyze two models considered in Durrett et al [6] on random regular graphs. In each step, we randomly choose to do a voter model step, or an individual breaks his/her relation with one of his/her neighbors holding a different opinion, and establishes a new one with one of the others in two ways: 1. the rewire-to-random model : the individual builds rapport with a randomly chosen person, 2. the rewire-to-same model : the individual builds rapport with a randomly chosen person holding the same opinion. In both models, we exhibit a phase transition in the model parameter, where the duration for this process to cease alters. For sufficiently small parameter, the network is likely to divide into two groups with disparate opinions, whereas for sufficiently large parameter, the process endures for a prolonged period, and the proportion of the minority opinion diminishes substantially before halting. In the rewire-to-random model, we demonstrate that the minority opinion persists with overwhelming probability.