Department of Mathematics, Duke University, Box 90320, Durham, NC 27708, USA.
Proc Natl Acad Sci U S A. 2012 Mar 6;109(10):3682-7. doi: 10.1073/pnas.1200709109. Epub 2012 Feb 21.
We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 - α, one imitates the opinion of the other; otherwise (i.e., with probability α), the link between them is broken and one of them makes a new connection to an individual chosen at random (i) from those with the same opinion or (ii) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting ρ be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how ρ depends on α and the initial fraction u of voters with opinion 1. In case (i), there is a critical value α(c) which does not depend on u, with ρ ≈ u for α > α(c) and ρ ≈ 0 for α < α(c). In case (ii), the transition point α(c)(u) depends on the initial density u. For α > α(c)(u), ρ ≈ u, but for α < α(c)(u), we have ρ(α,u) = ρ(α,1/2). Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.
我们考虑一个简化的社交网络模型，其中个体具有两种观点之一（称为 0 和 1），他们的观点和网络连接共同演变。边是随机选择的。如果两个连接的个体持有不同的观点，那么，以概率 1-α，其中一个会模仿另一个的观点；否则（即概率为α），它们之间的连接会被打破，其中一个会随机选择一个具有相同观点的个体（i）或整个网络中的个体（ii）建立新的连接。当不再有连接具有不同观点的个体的不和谐边时，系统的演化就停止了。令 ρ 为演化停止后持有少数观点的投票者的分数，我们感兴趣的是 ρ 如何依赖于 α 和初始投票者中持有观点 1 的分数 u。在情况 (i) 中，存在一个不依赖于 u 的临界值 α(c)，对于 α>α(c)，有 ρ≈u，对于 α<α(c)，有 ρ≈0。在情况 (ii) 中，转折点 α(c)(u)取决于初始密度 u。对于 α>α(c)(u)，有 ρ≈u，但对于 α<α(c)(u)，我们有 ρ(α,u)=ρ(α,1/2)。通过模拟和近似计算，我们解释了为什么这两个几乎相同的模型会有如此截然不同的相变。
