Erdos Renyi图的局部结构定理及其数学应用
Local Structure Theorems for Erdos Renyi Graphs and their Algorithmic Application
摘要：
我们分析了稀疏Erdos-Renyi图的一些局部性质，其中$d(n)/n$是边概率。特别是我们研究的行为非常短的路径。对于$d(n)=n^{o(1)}$，我们证明了$G(n,d(n)/n)$具有渐近几乎必然（a.a.s.~）有界局部树宽，因此是a.a.s.~无处密集。我们还发现了$G(n,d/n)$具有a.a.s.~的一个新的、更简单的证明。常数~ $d$的有界展开。稀疏Erdos-Renyi Gaphs的局部结构非常特殊：顶点的$r$ -邻域是一棵带有一些附加边的树，其中存在$m$附加边的概率随着~ $m$而减小。这意味着有效的算法的子图同构，特别是寻找小直径的子图。最后，我们注意到，实验表明，偏好连接图可能有类似的性质后，删除少量的顶点。
英文摘要：
We analyze some local properties of sparse Erdos-Renyi graphs, where $d(n)/n$ is the edge probability. In particular we study the behavior of very short paths. For $d(n)=n^{o(1)}$ we show that $G(n,d(n)/n)$ has asymptotically almost surely (a.a.s.~) bounded local treewidth and therefore is a.a.s.~nowhere dense. We also discover a new and simpler proof that $G(n,d/n)$ has a.a.s.~bounded expansion for constant~$d$. The local structure of sparse Erdos-Renyi Gaphs is very special: The $r$-neighborhood of a vertex is a tree with some additional edges, where the probability that there are $m$ additional edges decreases with~$m$. This implies efficient algorithms for subgraph isomorphism, in particular for finding subgraphs with small diameter. Finally we note that experiments suggest that preferential attachment graphs might have similar properties after deleting a small number of vertices.
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