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 2-hop algorithms, showing that our new 2-hop algorithm requires substantially fewer correct seeds than the 1-hop algorithm when graphs are sparse. Moreover, by combining our new performance guarantees for the 1-hop and 2-hop algorithms, we attain the best-known results (in terms of the required fraction of correct seeds) across the entire range of graph sparsity and significantly improve the previous results in \mbox{%DIFAUXCMD \cite{10.14778/2794367.2794371,lubars2018correcting }\hspace{0pt}%DIFAUXCMD when p}\ge .