# Identifying and Ranking Influential Nodes in Complex Networks Based on Dynamic Node Strength
## 2 Methods  
Decomposing a network to identify node influence is known as the k-shell method [22], which partitions a network into hierarchical sub-structures related to centrality. After decomposition, each node is assigned an index, \(k_{s}\), which represents the location of the node in the network. Nodes with high/low values of \(k_{s}\) are located to the center/peripher of the network. The innermost nodes constitute the core of the network, while other nodes constitute external layers (k-shells). An example network is given to illustrate how k-shell method works (see Figure 1). First, the nodes with degree \(k=1\) in the network are removed. This process is repeated recursively until the degree of the remaining nodes is greater than 1, and these nodes are assigned to \(k_{s}=1\). Then, nodes with \(k=2\) in the network are removed recursively until the degree of the nodes remaining in the network is greater than 2. The nodes removed in this process are assigned to \(k_{s}=2\). This routine runs until all nodes of the network have been assigned to corresponding \(k_{s}\). Finally, nodes with the same \(k_{s}\) are assigned to the corresponding k-shell. The Mixed Degree Decomposition (MDD) method notes that the dynamic process of decomposing network have influence on the resulting result [25]. Some nodes are removed at each decomposition step and the edges linked to these nodes are removed too, which changes the degree of the remaining nodes. MDD defines an index named as the mixed degree that is the weighted sum of the residual degree (number of edges linked to the remaining nodes) and the exhausted degree (number of edges linked to the removed nodes). The nodes are removed according to the mixed degree in each step of MDD procedure. The effect of MDD method to the sample network in Figure 1 is shown in Table 1. This idea that considers the influence of the removed edge is also adapted in our method. However, the edges are weighted based on the nodes at both ends. The neighborhood coreness centrality method (C\({}_{\text{nc}}\)) considers that the \(k_{s}\) of each node generated by the k-shell method reflects the information of location, and a spreader with more neighbors located in the core of the network is more influential [26]. C\({}_{\text{nc}}\) adopts the k-shell method to obtain the \(k_{s}\) of each node first. Then, the neighborhood coreness centrality of the target node is equal to the sum of its neighbors' \(k_{s}\). The ranking list of node influence is generated based on the C\({}_{\text{nc}}\). There is an improved C\({}_{\text{nc}}\) method named as the extended neighborhood coreness C\({}_{\text{nc+}}\), which denotes the \(k_{s}\) of neighbors and two-step neighbors as the metric of node influence. Inspired by these methods, we propose our method, Dynamic Node Strength Decomposition (DNSD). This method also applies the idea of dynamically decomposing network dynamically, but it is based on an alternative standard for removing the node. In many works, the edges in unweighted networks are treated equally. This means that, for different nodes, there are only differences in the number of edges, but no differences in features of edges. If there is only one edge between two nodes with a large number of edges, the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Rank** & **DC** & **k-shell** & **MDD** & **C\({}_{\mathbf{nc}}\)** & **DNSD** & **DNSD\({}^{+}\)** \\ \hline
1 & 2,7,8 & 1,2, & 1,2,3, & 2 & 1 & 2 \\
2 & 4 & 5,6,7 & 5,6,10, & 3,4 & 4 & 3 \\
3 & 1,3 & others & 19,21 & 1 & 7 & 1 \\
4 & 11,22 & & others & 7 & 2 & 4 \\
5 & 5,6,10, & & & 8 & 3 & 7 \\
6 & others & & & 5,6 & 8 & 8 \\
7 & & & & 22 & 22 & 22 \\
8 & & & & 11,23 & 5,6 & 5,6 \\
9 & & & & 14,15,16, & 10 & 10 \\
10 & & & & 17,10,19,21 & & 14,15, \\  & & & & others & 11 & 16,17 \\
11 & & & & & 9,14,15, & \\  & & & & 16,17,24, & 23 \\  & & & & & 25,26 & \\
12 & & & & & 19,21 & 25,26 \\ \(>\)12 & & & & others & others \\ \hline \hline \end{tabular}
\end{table}