Before revision:  in the 2-hop neighborhood, instead of only 1-hop neighborhood as in \mbox{%DIFAUXCMD \cite{lubars2018correcting}\hspace{0pt}%DIFAUXCMD . We show that when np^2\leq\frac{1}{135\log n}, our new algorithm can achieve perfect recovery with high probability if \beta\geq\max} 900\sqrt{\frac{np^3(1-s)\log n{s}}, 600\sqrt{\frac{\log n}{ns^4}}, \frac{1200\log n}{n^2p^2s^4}} and nps^2\geq 128\log n }\hspace{0pt}%DIFAUXCMD when p}\ge .
After revision:  in the 2-hop neighborhood, instead of only 1-hop neighborhood as in \mbox{%DIFAUXCMD \cite{lubars2018correcting}\hspace{0pt}%DIFAUXCMD . We show that when np^2\leq\frac{1}{135\log n}, our new algorithm can achieve perfect recovery with high probability if \beta\geq\max} 900\sqrt{\frac{np^3(1-s)\log n{s}}, 600\sqrt{\frac{\log n}{ns^4}}, \frac{1200\log n}{n^2p^2s^4}} and nps^2\geq 128\log n }\hspace{0pt}%DIFAUXCMD when p}\ge n^{-5/6.