# Identifying and Ranking Influential Nodes in Complex Networks Based on Dynamic Node Strength
## 2 Methods

Table 1: The ranking lists of the sample network in Figure 1 measured by different methods. \(\alpha\) is set to 0.5 and \(\beta\) is set to 0 simply. The ranks greater than 12 are not shown. Figure 1: A sample network with 26 nodes. The results of different methods in the sample network are shown in Table 1.  
importance of this edge is obviously vital. In this moment, it is unreasonable that this edge and other edges are treated equally. Actually, edges are spreading paths, which are different in the spreading process. Information could spread more widely through some specific paths. Thus, the importance of different edges should be distinguished, which depends on the nodes connected by the edges. To reflect this potential difference, we define the edge weight as  
\[w_{ij}=k_{i}+k_{j}. \tag{1}\]  
According to Equation (1), if an edge connects two nodes with large degree, it is assigned a large weight. In the process of decomposing network, the local characteristics of the network change at each step of decomposition as the nodes are removed [29; 30]. This means that the order of node removal affects the final result. In DNSD method, this effect is taken into account. We define the edges linked to the nodes that have been removed as vanishing edges and the edges linked to the existing nodes as existing edges. The dynamic weight of node \(i\) in the decomposition process is composed of the existing weight, \(w^{e}_{ij}\), and the vanishing weight, \(w^{v}_{ij}\), as follows:  
\[w^{d}_{ij}=w^{e}_{ij}+\alpha\cdot w^{v}_{ij}, \tag{2}\]  
where \(\alpha\) is the variable parameter in the range of 0-1. If \(\alpha\) is set to 1, the dynamic weight is equal to the normal weight. When \(\alpha\) is set to 0, it means that the vanishing weight is not considered. Consequently, we encounter the decomposition criteria by taking the edges weights and two-step weights into account. Each node is assigned to a node strength by the following measure:  
\[S_{i}=\sum_{j\in\gamma(i)}\ \sum_{l\in\gamma(j)}\ w^{d}_{ij}\cdot(w^{d}_{ jl})^{\beta}, \tag{3}\]  
where \(\gamma(i)\) and \(\gamma(j)\) are the set of neighbors of node \(i\) and node \(j\), respectively. \(\beta\) is is the attenuation coefficient in the range between 0 and 1. The farther an edge is from the target node, the smaller the \(\beta\) is. because the effect of information spreading in the network is gradually attenuated. In the measure of node strength, not only the weight of the node's edges but also those of its neighbors are considered. Actually, the wider is the neighborhood considered, the more accurately is the spreading outcome of a node predicted. However, the computations increase as more neighbors are considered [23; 31]. Usually, it is a choice between efficiency and accuracy. In this way, the detailed decomposition according to DNSD can be conducted in the following procedure:  
Step 1. Initially, calculate the \(S_{i}\) of each node in the network. At this moment, \(w^{d}_{ij}\) of each node is equal to \(w^{e}_{ij}\) because there is no node that has been removed. Step 2. Remove nodes with the lowest \(S_{i}\) (denoted as \(S\)), and these nodes are assigned to the _S-shell_. Update \(k^{d}\) of each remaining node by \(k^{d}_{i}=k^{e}_{i}+\alpha\cdot k^{v}_{i}\). Then, calculate the \(S_{i}\) of each node in the network again. Remove all the nodes with \(S_{i}\) lower than or equal to \(S\) and assign them to the _S-shell_ too. Repeat this routine until the \(S_{i}\) of the remaining nodes in the network is greater than \(S\). Step 3. Execute Step 2 recursively until each node has been placed into one _S-shell_. For the nodes removed in the final decomposition, they are placed in the _S-shell_ according to the \(S_{i}\) in the last decomposition. Through the above procedure, DNSD method can rank nodes according to the \(S\) and assign them in the corresponding _S-shell_. The most influential nodes are assigned to the maximal \(S\).