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 $\Theta(\sqrt{n})$, and the power-law exponent $2<\beta<3$, we show that as soon as $D> \frac{4-\beta}{3-\beta}$, 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 $\Omega((\log n)^{4-\beta})$ initial seeds. Our result achieves an exponential reduction in the seed size requirement, as the best previously known result requires $n^{1/2+\epsilon}$ seeds (for any small constant $\epsilon>0$). Performance evaluation with synthetic and real data further corroborates the improved performance of our algorithm.

中文翻译：

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匹配幂律图中$ D $ -hops的力量

本文研究幂律图的种子图匹配。假设从具有幂律度分布的公共父图中独立地对两个边缘相关图进行边缘采样。随机选择一组正确匹配的顶点对，并将其显示为初始种子。我们的目标是使用种子恢复两个图之间剩余的潜在顶点对应。与专注于在$$$-hop邻域中使用高度种子的现有方法不同，我们开发了一种有效算法，可在适当定义的$ D $ -hop邻域中利用低度种子。具体来说，我们首先根据$ D $ -hop邻域中的低度种子数，将一组具有适当度数的顶点对（我们称为第一个切片）进行匹配。这显着减少了触发级联过程以匹配其余图所需的初始种子数。在具有$ n $个顶点，最大度数$ \ Theta（\ sqrt {n}）$和幂律指数$ 2 <\ beta <3 $的Chung-Lu随机图模型下，我们显示出只要$ D > \ frac {4- \ beta} {3- \ beta} $，通过最佳选择第一个切片，我们的算法极有可能正确匹配常数对的真对，而没有任何错误，仅提供$ \ Omega（ （\ log n）^ {4- \ beta}）$初始种子。我们的结果实现了种子大小要求的指数减小，因为以前最好的结果是需要$ n ^ {1/2 + \ epsilon} $个种子（对于任何小的常数$ \ epsilon> 0 $）。综合和真实数据的性能评估进一步证实了我们算法的改进性能。