Identifying influential nodes in complex networks by adjusted feature contributions and neighborhood impact

Abstract

Determining the spreading ability of nodes is considered a fundamental issue in network science, with numerous applications in controlling system failure, rumors spreading, and product advertising. Many methods have been proposed to identify influential nodes, which, despite their advantages, suffer from high time complexity, low accuracy, and low resolution. This paper presents a feature based on K-Shell and the degree applied to the node and its neighbors. It adjusts the contribution of various features. The number of selected neighbors and the influence of each neighbor are chosen according to the structural features of the graph. The actual spreading ability of the node is measured with the Susceptible-Infected-Recovered (SIR) model, and the evaluations include accuracy, precision, resolution, correlation, Kolmogorov–Smirnov Test, and time complexity. Assessing 14 real-world and 20 artificial networks compared to 12 recent methods, such as the HGSM (Hybrid Global Structure Model), indicates that the proposed method performs best in various aspects.

Appendix

1.1 Preliminaries

The methods presented here will be implemented in different networks and compared with the proposed method from various aspects.

• DC [18]: It is a local and simple feature, counting the number of neighbors for each node without considering the importance of each neighbor.

• KS [19]: It is a global metric with linear time complexity. It has a hierarchical algorithm that operates based on node degree. Initially, it removes nodes with a degree=1 and assigns them a value of K-Shell index equal to one, then continues to remove and assign KS =1 to the remaining nodes until no nodes with a degree of one are left. This process repeats for nodes with a degree of two, assigning them KS-index, and continues iteratively until all nodes are assigned KS.

• GP [23]: It is a global metric based on degree and incorporates two features: SI and GI. SI calculates the spreading capability of each node according to its degree powered by a base e. GI considers the degrees of neighboring nodes in proportion to the inverse of their distance. Finally, the spreading capability of each node is determined by a weighted combination of SI and GI, where the parameter weights SI is $e^{\vartheta }$ and GI is weighed by $\sigma$. It is defined by equation 7,8. Symbols $k_{i}$, $n$, $d_{ij}$ represent the degree of node i, the number of nodes, and the shortest distance between two nodes i and j, respectively.

  $$\begin{aligned} {SI}_{i}= & e^{\frac{1}{n}*k_{i}*\vartheta }, {GI}_{i} = \sum _{i \ne j} \frac{k_{j}*\sigma }{d_{ij}} \end{aligned}$$

  (7)
  $$\begin{aligned} {GP}_{i}= & \left\{ \begin{matrix} {SI}_{i} \times {GI}_{i}\ \ \ \ \ connected\ \ node \\ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ unconnected\ node \\ \end{matrix} \right. \end{aligned}$$

  (8)

• GSM [24]: It is a global metric based on KS, similar to GP, and consists of two features SI (Self-Influence) and GI (Global-Influence). The Self-Influence considers the spreading capability of a node by ‘e’ raised to the power of K-Shell divided by the size of the network. GI is a global metric that considers all nodes and computes the influence of each node based on the K-Shell and the inverse of the distance to the target node. Ultimately, the GSM method’s spreading capability is equal to the product of SI and GI. It is defined by equation 9 where ${ks}_{i}$, $n$, $d_{ij}$ represent the K-Shell of node i, the number of nodes, and the shortest distance between two nodes i and j, respectively.

  $$\begin{aligned} {GSM}_{i} = \ {SI}_{i} \times {GI}_{i} = e^{\frac{{ks}_{i}}{n}} \times \sum _{i \ne j} \frac{{ks}_{j}}{d_{ij}} \end{aligned}$$

  (9)

• HGSM [25]: It is a hybrid method that integrates local and global features. It uses Degree as the local metric and K-Shell as the global metric. It also defines two features of Self-Influence (SI) and Global Influence (GI) similar to GP and GSM. In SI, the spreading capability of each node is calculated by e raised to the power of the multiply of its Degree and K-Shell, divided by the number of nodes. The GI calculates the spreading capability of each node by distance and a metric derived from the Degree and K-Shell. It is defined by equation 10 where ${ks}_{i}$, $n$, $d_{ij}$ represent the K-Shell of node i, the number of nodes, and the shortest distance between nodes i and j, respectively. The $k_{i},$ $ks,\ k$ represents the degree of node i, the K-Shell value in the network, and the average degree in the network, respectively.

  $$\begin{aligned} {HGSM}_{i} = \ {SI}_{i} \times {GI}_{i} = e^{\frac{{{ks}_{i}*k}_{i}}{n}} \times \sum _{i \ne j} \frac{e^{\frac{{{ks}_{j}*k}_{j}}{n}}}{{d_{ij}}^{\lceil \log _{2}{< e^{\frac{ks*k}{n}}>} \rceil }} \end{aligned}$$

  (10)

• Degree and Neighborhood information Centrality (DNC) [57]: It uses the sum of the local clustering coefficients of first-level neighbors and combines it in a weighted manner with the node degree. Its formula is in equation 11, 12. Symbols $C_{j}$, $k_{i}$, $N_{1}(i)$ denote the local clustering coefficient of node j, the degree of node i and first-level neighbors of node i.

  $$\begin{aligned} {slc}_{i}= & \sum _{j \in N_{1}(i)} C_{j} \end{aligned}$$

  (11)
  $$\begin{aligned} DNC(i)= & k_{i} + \alpha \sum _{j \in N_{1}(i)} C_{j},\ \alpha> 0 \end{aligned}$$

  (12)

• NPIC [26]: It is a combined metric that integrates global and local features. It uses a combination of node degree and KS. The calculated value for each node is divided by the total number of nodes in the network, and for calculating the influence received from other nodes their distance is important. Additionally, two parameters, $\alpha$, and $\beta$, are used to adjust the impact of nodes and their neighbors. It is defined by equation 13 where ${ks}_{i}$, $n$, $d_{ij}$ represent the K-Shell of node i, the number of nodes, and the shortest distance between nodes i and j, respectively. The $k_{i}$ denotes the degree of node i.

  $$\begin{aligned} NPIC(i) = \frac{({ks}_{i}*k_{i}) + \alpha }{n}*\sum _{i \ne j} \frac{({ks}_{j}*k_{j}) + \beta }{d_{ij}} \end{aligned}$$

  (13)

• LGI [28]: In LGI calculation, the number of common neighbors between each node and its neighbors is counted and divided by the node’s degree to create the LC feature. Then, this metric and the degree are normalized based on their sum within their neighbors and combined in a weighted manner to derive the LI feature. LI is a local feature. It has a global feature named GI that is constructed by the KS of nodes and the sum of its neighbor’s KS. Finally, the GI is combined with LI to compute LGI. Its definition is presented in equation 18 where $N_{1}(j),\ k_{i},\ {ks}_{j}$ represent the first-level neighbors of node j, the degree of node i, and k-shell of node j, respectively. The $\alpha$ and $\beta$ are tuning parameters.

  $$\begin{aligned} R(i,j)= & \frac{\left| N_{1}(j) \cap N_{1}(i) \right| }{k_{i}} \end{aligned}$$

  (14)
  $$\begin{aligned} LC(i)= & \sum _{j \in N_{1}(i)} {R(i,j)} \end{aligned}$$

  (15)
  $$\begin{aligned} LI(i)= & \alpha \frac{k_{i}}{\sum _{j \in N_{1}(i)} k_{j}} + \beta \frac{LC(i)}{\sum _{j \in N_{1}(i)} {LC(j)}} \end{aligned}$$

  (16)
  $$\begin{aligned} GI(i)= & {ks}_{i} + \sum _{j \in N_{1}(i)} {ks}_{j} \end{aligned}$$

  (17)
  $$\begin{aligned} LGI(i)= & \sum _{j \in N_{1}(i)} {LI(j)*GI(j)} \end{aligned}$$

  (18)

• GGC [34]: Similar to the Gravity formula, it uses degree centrality and a coefficient composed of the local clustering coefficient as weight. Its definition is in equation 19 where $k_{i},\ C_{j},\ d_{ij}$ denotes the degree of node i, local clustering coefficient of node j, and the distance between two nodes i and j, respectively. The $R$ and $\alpha$ represent the half of the average shortest path and tuning parameter.

  $$\begin{aligned} GGC(i) = \sum _{d_{ij} \le R,\ \ \ j \ne i} \frac{e^{- \alpha \left( C_{i} + C_{j} \right) }k_{i}k_{j}}{d_{ij}^{2}} \end{aligned}$$

  (19)

• DC+, DCGM+, DCGM++: In [22], three metrics are introduced. Initially, the DC+ feature is defined as the sum of the degrees of first-level neighbors divided by the node’s degree. Then, it is utilized as a weight in the gravity formula, forming DCGM+. Finally, DC+ is combined with DCGM+ to introduce DCGM++, which integrates both metrics. They are defined as equations (20-23) where $k_{j},\ d_{ij},\ N_{1}(i)$ represents the degree of node j, distance between two nodes i and j, and first-level neighbors of node i, respectively.

  $$\begin{aligned} {DC}^{+}(i)= & \frac{\sum _{j \in N_{1}(i)} k_{j}}{k_{i}} \end{aligned}$$

  (20)
  $$\begin{aligned} {DCGM}^{+}(i)= & \sum _{d_{ij} \le R,\ \ \ j \ne i} \frac{{DC}^{+}(i){DC}^{+}(j)}{{d_{ij}}^{2}} \end{aligned}$$

  (21)
  $$\begin{aligned} {DCGM}^{+ +}(i)= & {DC}^{+}(i)DCGM(i) \end{aligned}$$

  (22)
  $$\begin{aligned} {DCGM}(i)= & \sum _{d_{ij} \le R,\ \ \ j \ne i} \frac{{DC}^{*}(i){DC}^{*}(j)}{{d_{ij}}^{2}}, \ \ \ {DC}^{*}(i)= \frac{k_{i}}{n-1} \end{aligned}$$

  (23)

Despite the benefits of the mentioned methods, they also have some limitations. Some of them have high time complexity; others have poor accuracy or low resolution. For instance, the computational complexity of Global Perspective, GSM and HGSM, is quadratic. The GGC and DNC use a combination of degree and local clustering coefficients but do not consider the position of nodes in the network adequately. We proposed a method that performs well in various evaluation aspects while maintaining high accuracy, precision, and resolution, with low time complexity.

1.2 Other evaluation metrics

• SIR Model: The SIR model [17] is employed to determine the actual dissemination capability of each node [14,15,16,17,18,19]. Nodes are classified into three conditions: removed, infected, or susceptible. Initially, all nodes are in the susceptible state except one infected node. In each step, the infected node infects its susceptible neighbors with a beta probability and goes to the removed state with a probability of $\alpha$. Once a node enters the R state, it becomes immune forever. This process continues until there are no infected nodes in the network. The average percentage of enhanced nodes after 100 iterations is considered the real influence ability of the node (the same node that was initially infected).

• Kolmogorov–Smirnov (K-S) Test [58]: A nonparametric statistical test is used to compare two distributions. In this paper, the two-sample test version is employed, where the accuracy curve of one of the recent methods in the given network is selected and compared with the proposed method. This test aims to determine the maximum difference between the empirical distribution functions of the accuracy curves of the two methods and whether this difference is statistically significant. Additionally, it identifies which method is superior in this test.

• Omega [27]: Real-world networks represent activities in the real world which can be modeled as complex networks. Complex networks have several categories, like small-world, and random networks, scale-free. Scale-free networks follow a power law degree distribution, featuring a few high-degree hub nodes connected to many low-degree nodes. Connections in random networks are created randomly with a constant probability. The small-world networks have a high clustering coefficient and a short average shortest path, adhering to the concept of six degrees of separation. Small-world networks are similar to regular networks in their high clustering but resemble random graphs in their short path lengths between nodes. The omega metric quantifies these features and indicates the similarity of a network to a regular network and a random network. In other words, omega shows the small-world coefficient, as shown in equation 24.

  $$\begin{aligned} \omega = \frac{L_{rand}}{L} - \frac{C}{C_{latt}} \end{aligned}$$

  (24)

  In equation 24, L represents the average shortest path, and C indicates the average clustering coefficient. $L_{rand}$ denotes the value of L in a random network with the same number of nodes, and $C_latt$ denotes C in a lattice network with the same number of nodes in the network under investigation. Figure 15 illustrates these points. The $\omega$ is in (-1, 1). If $\omega$ is close to 0, it indicates that the network is small-world. Positive values show randomness, while negative values indicate a regular network, such as a lattice-like network.

Fig. 15

The relationship between values of Omega and its parts (L, C) to different types of networks