# Identifying and Ranking Influential Nodes in Complex Networks Based on Dynamic Node Strength
## 3 Experimental Results
### Applications to the Real Networks
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Networks** & **N** & **E** & **M(DC)** & **M(k-Shell)** & **M(MDD)** & **M(\(\mathrm{C_{nc}}\))** & **M(DNSD)** & **M(DNSD\({}^{+}\))** \\ \hline Karate Club & 34 & 78 & 0.708 & 0.496 & 0.708 & 0.839 & 0.954 & 0.954 \\ Email & 162 & 1772 & 0.922 & 0.776 & 0.916 & 0.987 & 0.995 & 0.998 \\ Co-authors & 1589 & 2742 & 0.707 & 0.663 & 0.709 & 0.844 & 0.904 & 0.915 \\ Power Grid & 3706 & 6594 & 0.593 & 0.246 & 0.587 & 0.740 & 0.920 & 0.984 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The Monotonicity of ranking lists generated by different methods. N is the number of nodes and E is the number of edges in the network. M(DC) represents the monotonicity of ranking obtained by Degree Centrality method and others have the similar meaning. Figure 3: The complementary cumulative distribution functions (CCDF) of the rankings obtained by different methods in Karate Club, Email, Co-authors and Power Grid \(\alpha\) is set to 0.5 and \(\beta\) is set to 0. Based on above observation, we measure the Kendall's \(\tau\) of these methods, as shown in Figure 5, where the \(\theta\) is between 0.06 and 0.2 with an interval of 0.1. If the infection possibility is large enough, the range of infection will cover almost the whole network; thus, the spreading origin is no longer important. In Figure 5a-d, the \(\alpha\) in Equation (2) is set to 0.5. The effect of DNSD and DNSD\({}^{+}\) are shown in Figure 5e-l when \(\alpha\) is set between 0.3 and 0.7 with an interval of 0.1. Note that the \(\tau\) is very volatile in the Karate Club, because homogeneous nodes could have very different ranks when many measurement factors are considered. When \(\alpha\) is set to 0.5 or 0.6, the effect of DNSD and DNSD\({}^{+}\) is better than those with other \(\alpha\) in the corresponding network. We compare DNSD and DNSD\({}^{+}\) with other methods in each network when \(\alpha\) is 0.5. One can observe that the superiority of our methods is not obvious when the infection possibility is lower than 0.1 because it is hard to spread infection far away when the infection possibility is low. However, the \(\tau\) of our methods exceeds 0.8 in most cases with the increase of infection probability. Although the k-shell method is considered to be able to identify influential nodes, our experimental results show that there is weak correlation between the \(k_{s}\) measured by k-shell and the influence of spreaders. These results show that our methods are more effective for identifying node influence. In the above experiments, \(\beta\) is set to 0 simply, which denotes the influence of two-step neighbors is not considered temporarily. Here, we set \(\beta\) to 0-1 with the interval 0.1 and measure the Kendall's \(\tau\) between the rankings obtained by our methods and node influence simulated by the SIR model again. The results of the \(\tau\) under different infection possibility and \(\beta\) are shown in Figure 6 in the form of heat map, where Figure 6a-d shows the ones measured by DNSD and Figure 6e-h those measured by DNSD\({}^{+}\). Note that the influence of \(\beta\) in Karate Club is not obvious (there is no obvious concentration trend of red spots) because of small network size. In Email, When \(\beta\) is set close to 0.6 the performance of DNSD is better (the areas are almost red in the middle right part of the heat map), while the performance of DNSD\({}^{+}\) is better with \(\beta\) between 0.3 and 0.6. In Co-authors, \(\tau\) is higher than others when \(\beta\) is set between 0.4 and 0.6 for our methods. In Power Grid, DNSD is not sensitive to \(\beta\), and it is improved a little when \(\beta\) is close to 0.3. For DNSD\({}^{+}\), \(\beta\) close to 0.7 outperforms others. One can observe \(\tau\) become higher (the color of the areas changes  
Figure 4: The spreading capacity measured by the SIR simulation in the different networks plotted with the different ranking methods: Karate Club (**a–d**); Email (**e–h**); Co-authors (**i–l**); and Power Grid (**m–p**).