# On the Secrecy Sum-Rate of Internet of Things Networks: Scheduling and Power Control +
## 4 Secrecy Sum-Rate Scaling Analysis  
Now, we analyze the secrecy performance of proposed scheduling algorithm in terms of secrecy sum-rate scaling. We show that the proposed scheme asymptotically achieves the optimal secrecy sum-rate scaling (It means optimal degree-of-freedom gain \(M\) and multiuser diversity gain \(\log\log N\) can be achieved. ), where the secrecy sum-rate scales as \(M\log(\rho\log N)\) when the number of users (\(N\)) increases with SNR (\(\rho\)). In other words, we analyze how \(N\) scales with \(\rho\) to achieve the optimal secrecy sum-rate scaling. We investigate a lower bound of the achievable secrecy sum-rate and prove the optimal secrecy sum-rate scaling using the lower bound. Note that we have \(R_{\mathrm{Int}}^{\mathrm{sec}}\geq R_{\mathrm{Ext}}^{\mathrm{sec}}\) since we do not limit eavesdroppers' capability in the case of the external eavesdropping scenario. Further, the lower bound of \(R_{\mathrm{Ext}}^{\mathrm{sec}}\) is given by  
\[R_{\mathrm{Ext}}^{\mathrm{sec}}=\sum_{S}^{S}\Bigl{[}r_{\mathrm{ BS}}^{[s]}-r_{\mathrm{Ext}}^{[s]}\Bigr{]}^{+} \geq\left[\sum_{s}^{S}r_{\mathrm{BS}}^{[s]}-\sum_{s}^{S}r_{\mathrm{ Ext}}^{[s]}\right]^{+}\] \[=[R_{\mathrm{BS}}^{\mathrm{sum}}-R_{\mathrm{Ext}}^{\mathrm{sum}}]^ {+}\] \[\geq\left[R_{\mathrm{BS}}^{\mathrm{sum}}-\max_{k\in\mathcal{K}} \left\{\log\det\left(\mathbf{I}_{\mathrm{M}}+\rho\sum_{s\in\mathcal{N}_{S}} \mathbf{g}_{sk}\mathbf{g}_{sk}^{H}\right)\right\}\right]^{+}\] \[=[R_{\mathrm{BS}}^{\mathrm{sum}}-C_{\mathrm{Ext}}^{\mathrm{sum}}]^ {+}\] \[\geq R_{\mathrm{BS}}^{\mathrm{sum}}-C_{\mathrm{Ext}}^{\mathrm{sum}}, \tag{12}\]  
where \(C_{\mathrm{Ext}}^{\mathrm{sum}}\) denotes \(\max_{k\in\mathcal{K}}\{\log\det\left(\mathbf{I}_{\mathrm{M}}+\rho\sum_{s\in \mathcal{N}_{S}}\mathbf{g}_{sk}\mathbf{g}_{sk}^{H}\right)\}\) and \(R_{\mathrm{BS}}^{\mathrm{sum}}\) and \(R_{\mathrm{Ext}}^{\mathrm{sum}}\) denote \(\sum_{S}^{S}r_{\mathrm{BS}}^{[s]}\) and \(\sum_{S}^{S}r_{\mathrm{Ext}}^{[s]}\), respectively. In (12), the first and the third inequalities hold due to characteristics of \([\cdot]^{+}\) function and the second inequality holds since we have \(C_{\mathrm{Ext}}^{\mathrm{sum}}\geq R_{\mathrm{Ext}}^{\mathrm{sum}}\). Thus, we consider \(R_{\mathrm{BS}}^{\mathrm{sum}}-C_{\mathrm{Ext}}^{\mathrm{sum}}\) instead of \(R_{\mathrm{Ext}}^{\mathrm{sec}}\) during the main proof. Next, we consider a slightly modified version of the proposed scheduling algorithm to prove the achievability of the optimal secrecy sum-rate scaling. We do not consider power control mechanism in the modified version. Instead, the modified scheduling scheme only utilizes the information of transmitter index \(n\) and beam index \(l^{*}\) satisfying the following scheduling criteria:  
\[ \begin{split}(\mathbf{C1})&\eta_{Q}^{[n,l^{*}]}\geq \eta_{Q^{*}}^{*},\\ (\mathbf{C2})&\eta_{1}^{[n,l^{*}]}\leq\eta_{1}^{*}, \\ (\mathbf{C3})&\eta_{\mathrm{L}}^{[n]}\leq\eta_{ \mathrm{L}}^{*},\end{split}  \tag{13}\]  
where \(\eta_{Q}^{[n,l^{*}]}\), \(\eta_{1}^{[n,l^{*}]}\), and \(\eta_{\mathrm{L}}^{[n]}\) are defined in (9). Definitely, the modified scheduling scheme shows degraded performance compared with the original proposed scheduling scheme since the modified version does not utilize all transmitters in the system. Therefore, the proof for the achievability of the modified scheduling scheme is enough to show the achievability of the proposed scheme. To prove the achievability, we first show that there exists at least one transmitter satisfying all criteria in (13) with high probability and next verify the optimal secrecy sum-rate scaling. We introduce the following lemma in order to prove our main theorem  
Lemma 1: _Let \(f(x)\) denote a continuous function of \(x\in[0,\infty)\), where \(0<f(x)\leq 1\).