# A Multi-Objective Degree-Based Network Anonymization Method
## 6 Security Analysis  
The essential features of our generalized multi-objective approach are the flexibility and practicality while applying the least modifications needed for a graph to be anonymous. Depending on the sensitivity of the data to be anonymized, the value of the parameter \(t\) differs, as it can be set to balance between privacy and utility. If we want to anonymize very sensitive data, then setting \(t\) to zero and applying our anonymization approach gives better results in terms of preserving data utility, compared to using traditional approaches. For each vertex, we are applying the minimum number of edge-editing operations. However, if the data are not critical enough, but the statistical information (for data mining purposes) is essential, we can assign a value for \(t\) that keeps the data utility as we can see in Table 2, where we present experimental results on the \"GrQc\" network when \(t=0\), versus those that are found in [11]. We measured the average error induced by each algorithm on the data utility measures for different \(k\) values. When, \(t=0\), parameterized kDA produced an average error that is 47% less than that produced by NC on the harmonic mean of the shortest path. For transitivity, our average error is 10% less than that of NC. Having this flexibility makes our approach a general umbrella for the existing algorithms, because we can preserve data utility and fine tune by setting the parameter \(t\). Of course, our approach focused on enhancing the preservation of data utility without affecting privacy. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Network** & **Algorithm** & **Precision Error** \\ \hline \multirow{3}{*}{American college football} & NC & 0.053 \\  & Parameterized kDA & 0.003 \\  & NC & 0.187 \\ Erdos & Parameterized kDA & 0.114 \\  & NC & 0.121 \\ Enron & Parameterized kDA & 0.081 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The precision error produced by our approach versus that of NC on different data sets. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Measure** & **Algorithm** & **Error** \\ \cline{2-3}  & NC & 0.134 \\ \cline{2-3}  & Parameterized kDA & 0.127 \\ \hline \multirow{3}{*}{The second smallest eigenvalue of the Laplacian matrix} & NC & 0.242 \\ \cline{2-3}  & Parameterized kDA & 0.223 \\ \cline{2-3}  & NC & 0.097 \\ \cline{2-3}  & Parameterized kDA & 0.09 \\ \hline \multirow{3}{*}{Harmonic mean of the shortest distance} & NC & 0.15 \\ \cline{2-3}  & Parameterized kDA & 0.08 \\ \cline{2-3}  & NC & 0.03 \\ \cline{2-3}  & Parameterized kDA & 0.027 \\ \cline{2-3}  & NC & 0.757 \\ \cline{2-3}  & Parameterized kDA & 0.754 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental results on the GrQc dataset using NC and our algorithm when \(t=0\). To evaluate the robustness of our model in preserving privacy and security, we analyze and contradict the behavior of an intruder. If we consider the example of a medical data set where an intruder may have some information about a targeted individual like age, gender, or address, among other things collected from external resources, then the (potential) intruder tries to match this information with nodes in the released network. Using our constrained anonymization model to mask the data before releasing it has a better chance of hindering the intruder from identifying his/her target, because the given information will be matched with many more individuals. To illustrate, if an individual in a network has 500 links, and \(t=3\), then our approach can yield more vertices whose degrees fall in the interval [497; 503]. In practice, distinguishing between an individual with 500 links and another with 503 links is challenging. Furthermore, an intruder may try to link his/her a priori information with released nodes and estimate the probability of correct matching. In this context, a node is at risk if its identification probability exceeds a certain value.