关于六度分离的理论思考
Some Considerations on Six Degrees of Separation from A Theoretical Point of View
摘要：
在这篇文章中，我们讨论了六度分离，这已经提出了米尔格拉姆，从理论的角度来看。简单地说，如果一个人有$k$朋友，那么间接朋友的数量$N$就以$d$分离度的形式增加到$\sim k^d$。所以它很容易达到整个世界的人口。然而，这是不可接受的。主要是由于非零聚类系数$C$，$N$不成为$\sim k^d$。本文首先讨论了Watt和Strogatz \cite{Watt1}，\cite{Watt2}提出的小世界网络中六个分离度与聚类系数之间的关系。特别是从理论和数值两方面探讨了WS模型中出现$(N)>$（美国人口或全世界人口）的条件。其次，我们引入了一个指标，表示传播速度的朋友的数量，并获得了一个解析公式，它的函数$C$，$K$，这是一个平均度在所有节点，和一些参数$P$有关的网络拓扑结构。最后通过数值计算研究了$C$、$K$、$P$和$N$之间的关系。
英文摘要：
In this article we discuss six degrees of separation, which has been proposed by Milgram, from a theoretical point of view. Simply if one has $k$ friends, the number $N$ of indirect friends goes up to $\sim k^d$ in $d$ degrees of separation. So it would easily come up to population of whole world. That, however, is unacceptable. Mainly because of nonzero clustering coefficient $C$, $N$ does not become $\sim k^d$. In this article, we first discuss relations between six degrees of separation and the clustering coefficient in the small world network proposed by Watt and Strogatz\cite{Watt1},\cite{Watt2}. Especially, conditions that $(N)>$ (population of U.S.A or of the whole world) arises in the WS model is explored by theoretical and numerical points of view. Secondly we introduce an index that represents velocity of propagation to the number of friends and obtain an analytical formula for it as a function of $C$, $K$, which is an average degree over all nodes, and some parameter $P$ concerned with network topology. Finally the index is calculated numerically to study the relation between $C$, $K$ and $P$ and $N$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
