We present the SILVERRUSH program strategy and clustering properties investigated with $sim 2,000$ Ly$alpha$ emitters at $z=5.7$ and $6.6$ found in the early data of the Hyper Suprime-Cam (HSC) Subaru Strategic Program survey exploiting the carefully designed narrowband filters. We derive angular correlation functions with the unprecedentedly large samples of LAEs at $z=6-7$ over the large total area of $14-21$ deg$^2$ corresponding to $0.3-0.5$ comoving Gpc$^2$. We obtain the average large-scale bias values of $b_{rm avg}=4.1pm 0.2$ ($4.5pm 0.6$) at $z=5.7$ ($z=6.6$) for $gtrsim L^*$ LAEs, indicating the weak evolution of LAE clustering from $z=5.7$ to $6.6$. We compare the LAE clustering results with two independent theoretical models that suggest an increase of an LAE clustering signal by the patchy ionized bubbles at the epoch of reionization (EoR), and estimate the neutral hydrogen fraction to be $x_{rm HI}=0.15^{+0.15}_{-0.15}$ at $z=6.6$. Based on the halo occupation distribution models, we find that the $gtrsim L^*$ LAEs are hosted by the dark-matter halos with the average mass of $log (left < M_{rm h} right >/M_odot) =11.1^{+0.2}_{-0.4}$ ($10.8^{+0.3}_{-0.5}$) at $z=5.7$ ($6.6$) with a Ly$alpha$ duty cycle of 1 % or less, where the results of $z=6.6$ LAEs may be slightly biased, due to the increase of the clustering signal at the EoR. Our clustering analysis reveals the low-mass nature of $gtrsim L^*$ LAEs at $z=6-7$, and that these LAEs probably evolve into massive super-$L^*$ galaxies in the present-day universe.
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