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.

翻译：本文研究用于电源法图形的种子图形匹配。 假设两个与边缘有关的图形是独立地从带有电源法度分布的普通父形图中打印出的边缘。 随机选择一组匹配的顶点纸质。 我们的目标是使用种子来恢复两个图形之间剩余的潜伏的顶点对应。 从当前侧重于在$- hop 附近使用高度种子的方法出发, 我们开发一个高效的算法, 将低度种子从一个有适当定义的 $D$- hop 社区独立地复制出来。 具体地说, 我们第一次匹配一组有适当度的顶点纸质面纸质( 我们称之为第一个切片 ) 。 我们的目标是使用种子来恢复两个图区之间其余的隐性顶点的顶点对应。 退出现有的方法, 以 $美元 美元 美元 、 美元 美元 美元 美元 美元 的顶点随机数模式 。 在 Chung- Lu 随机图形模型中, 以美元 美元 、 美元 美元 美元= 种子 的直径 种子 提供我们已知的直径 的直径 的直径 的直径 的直径 。