## 5 Simulation and Experiment
### Experiments in Real Networks
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Index** & **Equation** & **Description** \\ \hline \multirow{2}{*}{CN} & \multirow{2}{*}{\(s_{\text{CN}}(i,j)=|\Gamma(i)\cap\Gamma(j)|\)} & \(\Gamma(i)\) is the set of neighbors of node \(i\). \(|\cdot|\) represents cardinality of a set. CN index denotes the common neighbors between nodes \(i\) and \(j\). \\ \hline \multirow{2}{*}{AA} & \multirow{2}{*}{\(s_{\text{AA}}(i,j)=\sum\limits_{z\in\Gamma(i)\cap\Gamma(j)}\frac{1}{\log k_{z}}\)} & AA index weights the common neighbors by the reciprocal of the logarithm of each node’s degree. \\ \hline \multirow{2}{*}{RA} & \multirow{2}{*}{\(s_{\text{RA}}(i,j)=\sum\limits_{z\in(i)\cap\Gamma(j)}\frac{1}{k_{z}}\)} & RA index weights the common neighbors by the reciprocal of each node’s degree. \\ \hline \multirow{2}{*}{PA} & \multirow{2}{*}{\(s_{\text{PA}}(i,j)=k_{z}k_{j}\)} & PA index expresses preferential attachment by node’s degree. \\ \hline \multirow{2}{*}{Katz} & \multirow{2}{*}{\(s_{\text{Katz}}(i,j)=\left[\lim\limits_{n\to\infty}^{n}(\alpha\mathbf{A})^{m}\right]\)} & \(\mathbf{A}\) is adjacent matrix of network. Katz index considers all path between two nodes and gives more weights, \(\alpha\), to the shorter paths. \\ \hline \multirow{2}{*}{ACT} & \multirow{2}{*}{\(s_{\text{ACT}}(i,j)=\frac{1}{l_{x}^{2}+l_{y}^{2}-2l_{y}^{2}}\)} & \(l_{xy}^{+}\) is the corresponding element in \(\mathbf{L}^{+}\), and \(\mathbf{L}^{+}\) denotes the pseudo-inverse of laplacian matrix. \\ \hline \multirow{2}{*}{Cos+} & \multirow{2}{*}{\(s_{\text{Cos}}(i,j)=\frac{l_{y}^{2}l_{y}^{2}}{\left[\mathbf{p}_{i}\right]\cdot \left[\mathbf{p}_{j}\right]}=\frac{l_{y}^{2}}{\sqrt{l_{x}^{2}+l_{y}^{2}}}\)} & \multirow{2}{*}{According to \(\mathbf{L}^{+}\), Cos+ calculates cosine similarity of two vectors in matrix \(\mathbf{L}^{+}\).} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Definitions and descriptions of similarity indices. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Data** & \(1\,V\)1 & \(1\,E\)1 & \(<\)\(k\)\(>\) & \(r\) & \(C\) & \(H\) \\ \hline FWEW & 69 & 880 & 25.51 & \(-\)0.298 & 0.560 & 1.275 \\ FWFB & 128 & 2075 & 32.42 & \(-\)0.112 & 0.335 & 1.24 \\ PPI\_Cell & 127 & 237 & 3.732 & 0.035 & 0.455 & 1.649 \\ CKM-3 & 246 & 423 & 3.439 & 0.102 & 0.356 & 1.335 \\ Yeast & 2375 & 11,693 & 9.85 & 0.469 & 0.378 & 3.48 \\ PB & 1222 & 16,717 & 27.36 & \(-\)0.221 & 0.361 & 2.97 \\ NS & 1589 & 2742 & 3.451 & 0.462 & 0.889 & 2.011 \\ Email & 1133 & 5451 & 9.622 & 0.078 & 0.297 & 1.942 \\ CG & 5242 & 14,496 & 1.11 & 0.659 & 0.720 & 3.05 \\ CD & 425,957 & 1,049,866 & 4.93 & 0.267 & 0.267 & 4.412 \\ EE & 36,692 & 183,831 & 10.02 & \(-\)0.111 & 0.746 & 13.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Basic topological features of 6 example networks. \(|\,V|\) and \(|\,E|\) are the total numbers of nodes and links, respectively. \(<\)\(k\)\(>\) represents the average degree of nodes in a network. \(C\) and \(r\) are the clustering coefficient and assortative coefficient respectively. \(H\) is the degree heterogeneity, defined as \(H=\frac{<\mathbf{L}^{2}>}{<k>^{2}}\). The results show us that TLF method performs better than other fusion methods such as naive Bayes and logistic regression, no matter what evaluation metric use. Almost all combination methods are better than 4 basic indices. From the limit theorem, combination methods are dependent with each dimension. The promotion of fusion index is restrict to each similarity index. Experiment results also exposed this problem: if the single similarity indices perform not well, the fusion index cannot significantly improve the accuracy. For example, in the CKM-3 network, though we use TLF method to fuse 4 basic similarity indices can improve the AUC obviously, it cannot be better than Katz index (0.928).