  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.
