匹配幂律图中$D$ -跳的幂
The Power of $D$-hops in Matching Power-Law Graphs
摘要：
研究幂律图的种子图匹配问题。假设两个边相关的图是从一个具有幂律度分布的公共父图中独立边采样的。随机选择一组正确匹配的顶点对，并显示为初始种子。我们的目标是使用种子来恢复两个图之间剩余的潜在顶点对应关系。从现有的方法，专注于使用高度的种子在$1$跳邻域出发，我们开发了一个有效的算法，利用适当定义的$D$跳邻域的低度种子。具体来说，我们首先匹配一组顶点对与适当的程度（我们称之为第一片）的基础上的数量低度种子在其$D$跳的邻居。这大大减少了触发级联过程以匹配图的其余部分所需的初始种子数量。在具有$n$顶点、最大度$Θ(\sqrt{n})$和幂律指数$2<β<3$的Chung-Lu随机图模型下，我们证明了只要$D> \frac{4-β}{3-β}$，通过最优地选择第一片，我们的算法就可以以很高的概率正确地匹配一个常数部分的真对而没有任何错误，只提供$Ω((\log n)^{4-β})$初始种子。我们的结果实现了种子大小要求的指数级降低，因为先前已知的最佳结果需要$n^{1/2+ε}$种子（对于任何小常数$ε>0$）。合成和真实数据的性能评估进一步证实了我们的算法的性能改善。
英文摘要：
This paper studies seeded graph matching for power-law graphs. Assume that two edge-correlated graphs are independently edge-sampled from a common parent graph with a power-law degree distribution. A set of correctly matched vertex-pairs is chosen at random and revealed as initial seeds. Our goal is to use the seeds to recover the remaining latent vertex correspondence between the two graphs. Departing from the existing approaches that focus on the use of high-degree seeds in $1$-hop neighborhoods, we develop an efficient algorithm that exploits the low-degree seeds in suitably-defined $D$-hop neighborhoods. Specifically, we first match a set of vertex-pairs with appropriate degrees (which we refer to as the first slice) based on the number of low-degree seeds in their $D$-hop neighborhoods. This significantly reduces the number of initial seeds needed to trigger a cascading process to match the rest of the graphs. Under the Chung-Lu random graph model with $n$ vertices, max degree $Θ(\sqrt{n})$, and the power-law exponent $2<β<3$, we show that as soon as $D> \frac{4-β}{3-β}$, by optimally choosing the first slice, with high probability our algorithm can correctly match a constant fraction of the true pairs without any error, provided with only $Ω((\log n)^{4-β})$ initial seeds. Our result achieves an exponential reduction in the seed size requirement, as the best previously known result requires $n^{1/2+ε}$ seeds (for any small constant $ε>0$). Performance evaluation with synthetic and real data further corroborates the improved performance of our algorithm.
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