# Node Screening Method Based on Federated Learning with IoT in Opportunistic Social Networks
## Appendix A  
According to the definition of the weighted network above, the current degree of community structure is defined as:  
\[\Psi(t)=\frac{\kappa_{a}}{K}-\frac{\varphi_{s}}{\left(2K\right)^{2}} \tag{1}\]  
In an opportunistic social network, increasing the weight can increase the degree of association with the community. At time \(t\), the degree of modularity of the community is \(\Psi(t)\), and the degree of modularity after some time can be expressed as  
\[\Psi(t+1)=\frac{\kappa_{a}+\Delta\kappa}{K+\Delta\kappa}-\frac{\left(\varphi_ {s}+2\Delta\kappa\right)^{2}}{4(K+\Delta\kappa)}\]  
Use \((30)-(1)\) to get  
\[ \begin{array}{l}\Psi(t+1)-\Psi(t)=\frac{\kappa_{a}+\Delta\kappa}{K+\Delta \kappa}-\frac{\left(\varphi_{s}+2\Delta\kappa\right)^{2}}{4(K+\Delta\kappa)}- \left(\frac{\kappa_{a}}{K}-\frac{\varphi_{s}}{\left(2K\right)^{2}}\right)\\ =\frac{4K^{3}\Delta\kappa\left(2K-2K-2K-\varphi_{s}\right)}{4K^{2}(K+\Delta \kappa)^{2}}-\frac{4K\varphi_{s}\Delta\kappa^{2}-\left(\varphi_{s}\Delta \kappa\right)^{2}}{2K^{2}(K+\Delta\kappa)^{2}}\\ \geq\frac{4K^{3}\Delta\kappa-6K^{2}\varphi_{s}\Delta\kappa+2K^{2}\varphi_{s} \Delta\kappa-2K^{2}\varphi_{s}\Delta\kappa+(\varphi_{s}\Delta\kappa)^{2}}{4K^ {2}(K+\Delta\kappa)^{2}}\\ =\Delta\kappa\frac{4K^{3}-6K^{2}\varphi_{s}+2K^{2}\varphi_{s}-2K^{2}\varphi_{ s}+\varphi_{s}^{2}\Delta\kappa}{4K^{2}(K+\Delta\kappa)^{2}}\\ =\Delta\kappa\frac{\left(2K^{2}-2K\varphi_{s}-\varphi_{s}\Delta\kappa \right)\times\left(2K-\varphi_{s}\right)}{4K^{2}(K+\Delta\kappa)^{2}}\\ \end{array}  \tag{2}\]  
Because we assume \(\Delta\kappa>0\), in order to prove that (2)-(1) is greater than zero, we only need to prove  
\[\left(2K^{2}-2K\varphi_{s}-\varphi_{s}\Delta\kappa\right)\times\left(2K- \varphi_{s}\right)>0 \tag{3}\]In other words  
\[\left\{ \begin{array}{l}2K^{2}-2K\varphi_{s}-\varphi_{s}\Delta\kappa>0\\ 2K-\varphi_{s}>0\\ \end{array} \right. \tag{10}\]  
\[\left\{ \begin{array}{l}2K^{2}-2K\varphi_{s}-\varphi_{s}\Delta\kappa>0\\ 2K-\varphi_{s}>0\\ \Delta\kappa>0\end{array} \right.\Rightarrow\ \left\{ \begin{array}{l}0< \Delta\kappa<2K\left(\frac{K}{\varphi_{s}}-1\right)\\ \varphi_{s}<2K\end{array} \right. \tag{11}\]  
\[\left\{ \begin{array}{l}0<\Delta\kappa<2\left(\frac{K}{\varphi_{s}}-1\right)\\ 2\left(\frac{K}{\varphi_{s}}-1\right)>0\\ \varphi_{s}<2K\end{array} \right.\Rightarrow\ \left\{ \begin{array}{l}0< \Delta\kappa<2\left(\frac{K}{\varphi_{s}}-1\right)\\ \varphi_{s}<K\end{array} \right. \tag{12}\]  
Because \(2\Psi\) is the sum of the degrees in the entire network, the sum of the degrees of the communities in the network does not exceed \(2\Psi\), so from the above proof, we can know that in the social opportunistic network, increasing the weight can increase the degree of relevance to the community. \(\square\)  
**Theorem A2. **_If the connection weights of two sub-communities have the following relationship._  
\[\frac{\varphi_{i}\varphi_{j}+\varphi_{i}\Delta\kappa}{2(K+\Delta\kappa)}- \frac{\varphi_{i}\varphi_{j}}{K}=\frac{\varphi_{i}\Delta\kappa(K-\varphi_{i}) }{K(K+\Delta\kappa)}<0 \tag{13}\]  
\[\frac{\varphi_{i}\varphi_{j}}{2K}<\kappa_{ij}<\Delta\kappa+\frac{\varphi_{i} \varphi_{j}+\varphi_{s}\Delta\kappa+\Delta\kappa^{2}}{2(K+\Delta\kappa)} \tag{14}\]  
then these two communities are separate. **Proof of Theorem A2. ** Assuming that the original N of the community is divided into \(\eta_{i}\) and \(\eta_{j}\) due to the decrease in weight, then the relationship is as follows:  
\[\left\{ \begin{array}{l}K_{i}+K_{j}<K\\ \vec{K}-\frac{\varphi_{i}^{2}}{4K^{2}}+\frac{\varphi_{i}}{K}-\frac{\varphi_{i }^{2}}{4K^{2}}<\frac{\varphi_{i}+\varphi_{j}+\kappa_{ij}}{K}-\frac{\left( \varphi_{i}+\varphi_{j}\right)^{2}}{4K^{2}}\\ \kappa_{ij}>\frac{\varphi_{i}\varphi_{j}}{2K}\end{array} \right.