类Erdös-Rényi随机图过程的爆炸渗流
Explosive Percolation in Erdös-Rényi-Like Random Graph Processes
摘要：
从1960年的Erdös-Rényi过程开始，最大分量的演化在各种随机图过程中得到了深入研究。众所周知，这个过程在n/2个边缘发生相变，渐近地几乎肯定会出现线性大小的分量。此外，该相变是连续的，即，在极限情况下，表示在CN边缘插入之后的过程中的最大分量中的顶点的分数的函数F（c）是连续的。埃尔德什-雷尼过程的一个变体是所谓的Achlioptas过程，在每一步中都会随机绘制一对边，并使用固定的边选择规则选择其中一个包含在图中，而另一个则放回去。最近，Achlioptas，D 'Souza和Spencer（2009）给出了强有力的数值证据，表明各种边缘选择规则表现出不连续的相变。然而，Riordan和Warnke（2011）最近表明，所有Achlioptas过程都有连续的相变。在这项工作中，我们证明了一类Erdös-Rényi-like过程的不连续相变，在每一步中，我们连接两个顶点，一个随机选择的所有顶点，和一个随机选择的一个有限的顶点集。
英文摘要：
The evolution of the largest component has been studied intensely in a variety of random graph processes, starting in 1960 with the Erdös-Rényi process. It is well known that this process undergoes a phase transition at n/2 edges when, asymptotically almost surely, a linear-sized component appears. Moreover, this phase transition is continuous, i.e., in the limit the function f(c) denoting the fraction of vertices in the largest component in the process after cn edge insertions is continuous. A variation of the Erdös-Rényi process are the so-called Achlioptas processes in which in every step a random pair of edges is drawn, and a fixed edge-selection rule selects one of them to be included in the graph while the other is put back. Recently, Achlioptas, D'Souza and Spencer (2009) gave strong numerical evidence that a variety of edge-selection rules exhibit a discontinuous phase transition. However, Riordan and Warnke (2011) very recently showed that all Achlioptas processes have a continuous phase transition. In this work we prove discontinuous phase transitions for a class of Erdös-Rényi-like processes in which in every step we connect two vertices, one chosen randomly from all vertices, and one chosen randomly from a restricted set of vertices.
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