# Identifying and Ranking Influential Nodes in Complex Networks Based on Dynamic Node Strength
## 3 Experimental Results
### Applications to the Real Networks  
The monotonicity M of different methods is summarized in Table 2, where the basic characteristics of the real networks containing the mount of nodes and edges are shown in the first two columns. The value of \(\alpha\) is set to \(0.5\), because the performances of our methods are best when the value of \(\alpha\) is between \(0.45\) and \(0.6\), and the better choice only accounts for \(1\%\). Meanwhile, \(\beta\) is set to \(0\), which is further considered in subsequent experiments, although it is enough to show the superiority of our methods in monotonicity. Note that the monotonic indexes of DNSD and DNSD\({}^{+}\) are more than \(0.9\) in four networks, which outperform the other four methods. The C\({}_{\rm nc}\) is also competitive, but there is still a gap compared with our methods. The performance of the DC is not much different from MDD, because the two are equivalent when the coefficient before the exhausted degree in MDD equals to \(1\). The performance of k-shell is the worst in any case, especially in Power Grid and Karate Club. This result shows that the ranking lists generated by our methods have strong monotonicity. To more clearly show the advantage of our methods in resolution ratio, we plot complementary cumulative distribution functions (CCDF) of the rankings, as shown in Figure 3. Note that the curves of DNSD and DNSD\({}^{+}\) always descend slowly, which means that our methods can distribute nodes in more ranks compared with other methods. Specifically, DNSD generates \(27\) ranks in Karate Club, \(136\) ranks in Email, \(141\) ranks in Co-authors and \(102\) ranks in Power Grid, while DNSD\({}^{+}\) generates \(27\) ranks in Karate Club, \(153\) ranks in Email, \(362\) ranks in Co-authors and \(391\) ranks in Power Grid. Obviously, DNSD\({}^{+}\) is much better than DNSD, because the former considers the location of neighbor nodes. This reflects that the influential nodes always have neighbors close to the core of network. The curve of k-shell descends quickly because the mount of ranks obtained by the k-shell method is limited while the number of nodes in each rank is quite large, which indicatesthat the node influence is not well identified by the k-shell method. The performances of MDD in Email, Co-authors and Power Grid are much better than those of the other methods, except ours. Next, we investigate the effectiveness of different ranking methods by comparing the ranking results with the spreading capacity \(\mu(i)\) obtained from the SIR simulation, which are shown as Figure 4. The \(\theta\) is set to \(1/<\)k\(>\), where \(<\)k\(>\) denotes average degree of the network. Two comparison methods and the proposed methods for each network are shown. The number of nodes in the Karate Club is limited, and the difference among methods is not obvious. In Email, the \(\mu(i)\) fluctuates greatly at each rank by \(\mathrm{C_{nc}}\) and k-shell. The correlation between \(\mu(i)\) and ranking measured by DNSD is much higher, and DNSD\({}^{+}\) is even better. In Co-authors, the distributions of points by methods concentrate at low ranks. Some nodes with low ranks have high \(\mu(i)\) and few nodes with high ranks only have below average \(\mu(i)\) by DC and k-shell. Note that the distribution of the points is very diffuse for DC and MDD in Power Grid. On the contrary, the points gather on the trend that slants upward to the right for DNSD and DNSD\({}^{+}\). There is a long tail phenomenon in the (p), which shows that the effect of DNSD\({}^{+}\) is not very good at high ranks in Power Grid. The reason may be that the structure of Power Grid is simple. In other words, it is obvious that our methods outperform the other methods by showing that rankings by DNSD and DNSD\({}^{+}\) are highly correlated with the spreading capacity \(\mu(i)\) by SIR simulation in most cases.