# Differentially Private Clustered Federated Load Prediction Based on the Louvain Algorithm
## 2 Related Technology
### Differential Privacy  
Differential privacy protects data by adding noise that follows a certain distribution, thus providing privacy guarantees. It rigorously defines the strength of privacy protection. If two datasets \(D\) and \(D^{\prime}\) differ by only one record, they are referred to as neighboring datasets. Differential privacy ensures that the results of the same query operation on any two neighboring datasets are nearly indistinguishable. Client-level differential privacy FL integrates privacy protection mechanisms into FL, ensuring that the learning model does not reveal whether a specific client participated in the training. This means that the entire dataset of the client is protected. The formal definition of differential privacy is shown below.  
**Definition 1** (Differential Privacy [15]).: _A randomized mechanism \(M\), where \(Dom(M)\) denotes the domain and \(Range(M)\) denotes the range. If for any two neighboring datasets \(D,D^{\prime}\subseteq Dom(M)\) and any subset \(O\subseteq Range(M)\), the following holds:_  
\[\Pr[M(D)\in O]\leq e^{\epsilon}\Pr[M(D^{\prime})\in O]+\delta, \tag{5}\]  
_then \(M\) is said to satisfy \((\epsilon,\delta)\)-\(DP\), where the parameter \(\epsilon\) represents the privacy budget._  
**Definition 2** (\(l_{2}\) Sensitivity [15]).: _For a query function \(f\colon D\to R^{d}\) on a given dataset, where \(D\) and \(D^{\prime}\) are two neighboring datasets, the \(l_{2}\) sensitivity of \(f\) is defined as_  
\[\Delta(f)=\max_{D,D^{\prime}}\|f(D)-f(D^{\prime})\|_{2}. \tag{6}\]  
_Zero-Concentrated Differential Privacy (zCDP) is a new relaxed form of differential privacy. Compared to the standard \((\epsilon,\delta)\)-\(DP\), it offers a clearer and more precise analysis of the privacy loss over multiple iterative computations. The definition of zCDP is shown below._  
**Definition 3** (zCDP [27]).: _For any two neighboring datasets \(D\) and \(D^{\prime}\), and for any \(\alpha>1\), a randomized mechanism \(M\) satisfies \(\rho\)-zCDP if and only if_  
\[D_{\alpha}\Big{(}M(D)\Big{\|}M(D^{\prime})\Big{)}=\frac{1}{\alpha-1}\log\Big{(} E\Big{[}e^{(\alpha-1)L^{(O)}}\Big{]}\Big{)}\leq\rho \tag{7}\]  
_where \(D_{\alpha}\Big{(}M(D)\Big{\|}M(D^{\prime})\Big{)}\) denotes the \(\alpha\)-Renyi divergence between \(M(D)\) and \(M(D^{\prime})\). \(L^{(O)}\) represents the privacy loss incurred by \(M\) between neighboring datasets \(D\) and \(D^{\prime}\) when the outcome is \(O\), that is,_  
\[L^{(O)}_{\Big{(}M(D)\Big{\|}M(D^{\prime})\Big{)}}=\log\frac{\Pr(M(D)=O)}{\Pr(M (D^{\prime})=O)} \tag{8}\]  
The following propositions hold for zCDP:  
**Lemma 1** ([27]).: _The Gaussian mechanism, which returns \(f(D)+N(0,\Delta^{2}(f)\sigma^{2}I)\), satisfies \((1/2\sigma^{2})\)-\(z\)CDP._  
**Lemma 2** ([27]).: _Suppose there are \(k\) mechanisms \(M_{1},\ldots,M_{k}\), and each \(M_{i}\) (for \(i=1,\ldots,k\)) satisfies \(\rho_{i}\)-\(z\)CDP. Then, the composition of these mechanisms satisfies \(\left(\sum_{i=1}^{k}\rho_{i}\right)\)-\(z\)CDP._  
**Lemma 3** ([28]).: _Let \(M\) be composed of adaptive randomized mechanisms \(M_{1},\ldots,M_{E}\), and each \(M_{i}\) (for \(i=1,\ldots,E\)) satisfies \(\rho_{i}\)-\(z\)CDP. When the dataset \(D\) is randomly partitioned into \(D_{1},\ldots,D_{E}\), then \(M(D)=(M_{1}(D_{1}),M_{2}(D_{2}),\ldots,M_{E}(D_{E}))\) satisfies \(\max_{i}\rho_{i}\)-\(z\)CDP._  
**Lemma 4** ([27]).: _If \(M\) satisfies \(\rho\)-\(z\)CDP, then for any \(\delta>0\), \(M\) satisfies \(\left(\rho+2\sqrt{\rho\log(1/\delta)},\delta\right)-\)DP._