\begin{equation}
    \begin{aligned}
        &\begin{cases}
	if\,\,k_{1}^{*}=k_{2}^{*}, S_M\cup k_{2}^{*};\\
	if\,\,k_{1}^{*}\ne k_{2}^{*}, \left\{ \begin{array}{c}
	S_I\cup k_{2}^{*},if\,\,\rho _{I}^{k}>\chi\\
	S_M\cup k_{2}^{*},if\,\,\rho _{I}^{k}\leqslant \chi\\
\end{array} \right.\\
\end{cases}
\\
&k_{2}^{*}\in \underset{k\in \mathrm{rand}\left( V\backslash S_M\backslash S_I,\mathrm{s} \right)}{\mathrm{arg}\max}\left[ \begin{array}{c}
	\bar{G}\left( S_M\cup S_I \right) -\\
	\bar{G}\left( \left\{ k \right\} \cup S_M\cup S_I \right)\\
\end{array} \right] 
    \end{aligned}
    \label{eq:select uni-client}
\end{equation}
