algorithm - 谢林隔离模型的数学原理

Question

对于那些不知道模型的人。你可以阅读这个pdf .我想知道当算法收敛时(即当所有节点都满意时)2 个节点彼此相邻的概率是多少。

Here's the model in a gist. You have a grid (say 10x10). You have nodes of two kind (red and green) 45 each. So we have 10 empty spaces. We randomly place the nodes on the grid. Now we scan through this grid (Exact order does not matter according to Schelling). Each node wants a specific percentage of people of same kind in its Moore neighborhood (say b = 50% for each red and green). We calculate the happiness of each node (a = Number of neighbors of same kind/Number of neighbors of different kind). If a node is unhappy (a < b) it moves to an empty cell where it knows it will be happy. This movement can change the dynamics of old as well as new neighborhood. Algorithm converges when all nodes are happy.

PS - 我正在寻找谢林模型的任何数学分析的链接。

Answer 1

Easley 和 Kleinberg 在“网络、人群和市场:高度互联世界的推理”中对此模型进行了说明，- 参见 http://www.cs.cornell.edu/home/kleinber/networks-book/ 这是一本很好的书。

但是他们说“作为最后一点，我们注意到虽然该模型在数学上是精确且独立的，但讨论是在模拟和定性观察方面进行的。这是因为对谢林模型进行了严格的数学分析似乎相当困难，并且在很大程度上是一个开放的研究问题。”他们确实引用了 Young、Mobius 和 Rosenblat，以及 Vinkovic 和 Kirman 的一些工作。