# A Privacy Measurement Framework for Multiple Online Social Networks against Social Identity Linkage
## 4 The Measurement Method
### Visibility  
Through the abovementioned work, we can quantify extraction difficulty, accessibility, reliability, and privacy awareness. To calculate visibility, we carefully choose to use the half-suppressed fuzzy C-means clustering algorithm [39], which is a clustering algorithm based on FCM (Fuzzy C-means Algorithm).  
However, the original algorithm has great time complexity and space complexity, so we simplified it to improve these problems. First, we remove the step for training an SVM (Support Vector Machine) because our input is only four dimensions and removing the SVM can enormously increase efficiency; meanwhile, the clustering result is sufficient. Second, we change the iteration of the algorithm: If the last step determines that the iteration does not end, we restore the new cluster centre to our pre-set cluster centre.  
We chose the half-suppressed fuzzy C-means clustering method because it can achieve sufficient results using very few training samples, especially when considering the great need for an open large-scale sample set and data with unique needs such as ours. In addition, this method divides the sample space into several categories according to the value of the sample cluster membership to convert the scoring problem into a classification problem that uses an A-F rating instead of the hundred-mark system, as in educational systems, which can yield better generalization.  
Moreover, the algorithm can manually specify the clustering centre, which is extremely easy to perform in our research. According to common sense, we can formulate a perfect clustering centre that is similar to the difference between (1A, 1B, 1C, 1D, 1E) and (4A, 4A, 4A, 4A). When we specify the cluster centre, we can determine the visibility of a sample using the cluster centre to which the sample is aggregated.  
In a sample space of algorithm, \(X=\left\{x_{j}|j=1,2,\cdots,n;x_{j}\in R^{p}\right\}\) is a vector of \(n\) dimensions. \(v_{i}(i=1,2,\cdots,c)\) represents the center of each cluster. \(u_{ij}\) is the membership of sample \(x_{j}\) belonging to the class which satisfies:  
\[\begin{array}{l}\sum\limits_{i=1}^{c}u_{ij}=1;j=1,2,\cdots,n,\\ u_{ij}\geq 0;i=1,2,\cdots,c;j=1,2,\cdots,n,\\ 0<\sum\limits_{j=1}^{n}u_{ij}<n;i=1,2,\cdots,c.\end{array}\]  
The specific process of simplified algorithm is as follows:  
1. Initialize cluster centers \(\mathrm{v}_{\mathrm{i}}^{(0)}\), the inhibiting factor is \(\alpha\), inhibition threshold is \(\beta\), prime index factor is \(m\), error threshold is \(\varepsilon>0\) and the maximum number of iterations is K, set the number of iterations \(k=0\).
2. Use \(u_{\mathrm{ij}}=\frac{(\frac{1}{|x_{j}-x_{\mathrm{i}}|^{2}})^{\frac{1}{n-1}}}{ \sum\limits_{k=1}^{n}(\frac{1}{|x_{j}-x_{\mathrm{i}}|^{2}})^{\frac{1}{n-1}}}\) to calculate \(U^{(\mathrm{k})}=[u_{ij}^{(k)}]\).
3. According to the above correction equation to get \(U^{(\mathrm{k})^{\prime}}\) by fuzzy classification matrix \(u\).
4. Calculate new cluster center from \(\mathrm{v}_{i}=\frac{\sum\limits_{j=1}^{n}u_{ij}^{n}x_{j}}{\sum\limits_{j=1}^ {n}u_{ij}^{n}}\) and \(U^{(\mathrm{k})^{\prime}}\).
5. If \(\left(\sum\limits_{i=1}^{n}\|v_{i}^{(k+1)}\|^{2}\right)^{\frac{1}{2}}<\varepsilon\) or \(k<K\), the iteration is over; Otherwise \(k=k+1\), \(\mathrm{v}_{\mathrm{i}}\)\(=\mathrm{v}_{\mathrm{i}}^{(0)}\), return to the step 2.  
In this paper, we use accessibility, reliability, extraction difficulty, and privacy awareness as the four-dimensional sample input; then, we can obtain the visibility using the class into which the data are clustered. After we conduct a number of experiments, the better model parameters obtained are as follows: The inhibiting factor was 0.5, the inhibiting threshold was 0.5, the index prime factor\(m=2\), the error threshold D = 0.01, and the maximum number of iterations K = 30. We used the above parameters in the subsequent comparative analysis and set the outcome to six categories. The initial clustering centres have preset values.