随机图的刚性转移
The rigidity transition in random graphs
摘要：
当我们在平面上的点之间添加刚性杆时，在哪一点上有一个巨大的（线性大小的）刚性组件，它可以旋转和平移，但没有内部柔性？如果这些点是通用的，则这仅取决于由条形形成的图的组合。我们证明，如果这个图是一个Erdos-Renyi随机图G（n，c/n），那么存在一个尖锐的阈值出现一个巨大的刚性组件。当c < c_2，w. h. p.时，所有刚性分支都跨越一个、两个或三个顶点，当c > c_2，w. h. p.时，存在一个巨刚性分支。常数c_2 \approx 3.588是2-可定向性的阈值，由Fernholz和Ramachandran以及Cain、Sanders和Wormald在SODA'07中独立发现。我们还给出了当巨刚性分支出现时它的大小的定量界，证明了它跨越（3+2）-核中顶点的（1-o（1））-分数。非正式地，（3+2）-核心是通过从3-核心开始，然后归纳地添加到目前为止获得的图中的具有2个邻居的顶点而获得的最大诱导子图。
英文摘要：
As we add rigid bars between points in the plane, at what point is there a giant (linear-sized) rigid component, which can be rotated and translated, but which has no internal flexibility? If the points are generic, this depends only on the combinatorics of the graph formed by the bars. We show that if this graph is an Erdos-Renyi random graph G(n,c/n), then there exists a sharp threshold for a giant rigid component to emerge. For c < c_2, w.h.p. all rigid components span one, two, or three vertices, and when c > c_2, w.h.p. there is a giant rigid component. The constant c_2 \approx 3.588 is the threshold for 2-orientability, discovered independently by Fernholz and Ramachandran and Cain, Sanders, and Wormald in SODA'07. We also give quantitative bounds on the size of the giant rigid component when it emerges, proving that it spans a (1-o(1))-fraction of the vertices in the (3+2)-core. Informally, the (3+2)-core is maximal induced subgraph obtained by starting from the 3-core and then inductively adding vertices with 2 neighbors in the graph obtained so far.
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