Hadamard空间和随机图的扩张
Expanders with respect to Hadamard spaces and random graphs
摘要：
证明了存在一个3-正则图序列$\{G_n\}_{n=1}^\infty$和一个Hadamard空间$X$，使得$\{G_n\}_{n=1}^\infty$形成关于$X$的扩展子序列，而随机正则图不是关于$X$的扩展子.这回答了\cite{NS11}的一个问题。$\{G_n\}_{n=1}^\infty$也被证明是相对于随机正则图的扩展器，产生一个确定性的次线性时间常数因子近似算法，用于计算随机图的子集中的平均平方距离。证明使用随机图上的欧几里得锥，这是一个辅助连续几何对象，允许执行鞅方法。
英文摘要：
It is shown that there exists a sequence of 3-regular graphs $\{G_n\}_{n=1}^\infty$ and a Hadamard space $X$ such that $\{G_n\}_{n=1}^\infty$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with respect to $X$. This answers a question of \cite{NS11}. $\{G_n\}_{n=1}^\infty$ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.
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。
