An improved voterank algorithm to identifying a set of influential spreaders in complex networks

Identifying a set of critical nodes with high propagation in complex networks to achieve maximum influence is an important task in the field of complex network research, especially in the background of the current rapid global spread of COVID-19. In view of this, some scholars believe that nodes with high importance in the network have stronger propagation, and many classical methods are proposed to evaluate node importance. However, this approach makes it difficult to ensure that the selected spreaders are dispersed in the network, which greatly affects the propagation ability. The VoteRank algorithm uses a voting-based method to identify nodes with strong propagation in the network, but there are some deficiencies. Here, we solve this problem by proposing the DILVoteRank algorithm. The VoteRank algorithm cannot properly reflect the importance of nodes in the network topology. Based on this, we redefine the initial voting ability of nodes in the VoteRank algorithm and introduce the degree and importance of the line (DIL) ranking method to calculate the voting score so that the algorithm can better reflect the importance of nodes in the network structure. In addition, the weakening mechanism of the VoteRank algorithm only weakens the information of neighboring nodes of the selected nodes, which does not guarantee that the identified initial spreaders are sufficiently dispersed in the network. On this basis, we consider all the neighbors nodes of the node’s nearest and next nearest neighbors, so that the crucial spreaders identified by our algorithm are more widely distributed in the network with the same initial node ratio. In order to test the algorithm performance, we simulate the DILVoteRank algorithm with six other benchmark algorithms in 12 real-world network datasets based on two propagation dynamics model. The experimental results show that our algorithm identifies spreaders that achieve stronger propagation ability and propagation scale and with more stability compared to other benchmark algorithms.

Introduction

Along with the rapid development of information technology, high-speed networking is increasingly important to human society, and systems in many fields are abstracted as complex networks for research [1, 2]. Many aspects of our lives are covered by a large number of complex networks, such as power networks [3], transportation networks [4], and trade networks [5]. On the one hand, the research applications of complex networks in different fields have greatly facilitated our lives. Biomolecular networks [6] analyze the structure of intermolecular networks and help us understand the relationship between network structures and functions. Social networks [7] are analyzed in an effort to explain social phenomena in psychology and economics and to reveal influential individuals and their effect on others. Online trading networks [8] are used to target advertisements and products to interested groups through complex network community segmentation. On the other hand, the study of critical nodes in complex networks can help people to predict and prevent risk. The normal operation of many systems is greatly affected by a small number of critical nodes [9–12]. In the last decade or so, large-scale grid blackouts have occurred in nine countries around the world, and the maintenance of some critical nodes in power networks will effectively improve their resistance to destruction and the robustness of the network [13, 14]. On the basis of propagation dynamics [15], after the identification of key nodes in social networks, the spread of epidemics [16, 17] and rumors [18] can be effectively predicted. In the context of the global spread of COVID-19, research on controlling the spread of viruses has received extensive attention from various countries around the world.

Identifying the influential spreaders in complex networks is significant for improving the system’s resistance to destruction and preventing the spread of diseases and rumors. This task is known as the influence maximization (IM) problem. The IM problem is defined as sending information to a small group of nodes in the network, which ultimately maximizes the range of information dissemination. Many algorithms have been proposed to solve the IM problem. One method is to evaluate the node importance by metrics and select the top k nodes as the initial spreaders. In the early days, many classic ranking methods were proposed, such as the degree centrality [19], betweenness centrality [20], closeness centrality [21], eigenvector centrality [22], K-shell [23], and h-index [24]. The degree centrality is the most basic local evaluation method, in that it merely takes the number of neighboring nodes into consideration, but fails to reflect the importance of the nodes properly. Many researchers have proposed new methods based on the degree centrality. Chen et al. [25] proposed a semi-local method based on the degree information of nodes and their direct neighbors. Liu et al. [26] proposed an evaluation method using degree information to calculate the importance of nodes and edges, called the DIL method, which can better identify the bridge nodes. Ren et al. [27] combined degree and clustering coefficient information to evaluate node importance. Methods based on the global information of the networks, such as the betweenness centrality and closeness centrality, need to calculate the shortest path based on the global network. This renders the algorithm quite complex and, in large networks, generates an unbearable amount of computation [26]. The K-shell method was proposed by Kitsak et al. [23] to divide nodes into layers according to their locations, but this method is insufficiently hierarchical. Liu et al. [28] argued that important nodes are closer to the center of the network. They distinguished the importance of nodes at the same level by calculating the distance between a node and the nodes with the largest K-core value in the network. Wang et al. [29] proposed an improve K-shell method to distinguish nodes at the same level by information entropy. Yeruva et al. [30] proposed the Pareto-shell decomposition method using a Pareto front function based on the K-shell algorithm. Bae et al. [31] proposed the neighborhood coreness (NC) and extended neighborhood coreness (ENC) based on the K-core value of nodes and their neighbors. The h-index method was first proposed by Hirsch et al. [24] as a method to measure the research influence of scientists. Lü et al. [32] proposed a method to evaluate the importance of nodes based on a combination of the degree, h-index, and coreness methods. Liu et al. [33] proposed the local h-index method, by taking the h-index value of neighbors into consideration. With the in-depth study of complex networks, researchers have proposed more efficient methods to find important nodes in networks, such as PageRank [34, 35], LeaderRank [36], and other algorithms based on random walks. Qian et al. [37] proposed a new measure of node importance by utilizing a redefined entropy centrality model. Sheikhahmadi et al. [38] proposed the MCDE method by mixing the core value, degree centrality, and entropy of nodes together.

The above ranking algorithms solve the IM problem by identifying key nodes in the network, but they cannot guarantee that the selected nodes are widely distributed in the network. In the process of maximizing the spread of influence, it is often necessary to identify a set of spreaders, and it is desirable that these spreaders are sufficiently dispersed in the network so that the nodes achieve the maximum coverage of the network during the spread. Zeng et al. [39] verified that the distance between the spreaders in a network has a crucial influence on influence maximization. This provides a new way of thinking about the IM problem: selecting influential nodes widely distributed in the network. Zhang et al. [40] proposed a novel method based on a voting mechanism, called VoteRank, to select decentralized critical nodes in the network which the initial spreaders filtered out are more widely distributed, and avoid leading to an unnecessary waste of time and influence. Sun et al. [41] proposed an extension to VoteRank for weighted networks, called WVoteRank. Kumar et al. [42] argued that the node voting ability needs to reflect the position of the node in the network and introduced neighborhood coreness to optimize the VoteRank algorithm; their optimized algorithm is called NCVoteRank. Guo et al. [43] proposed the EnRenew algorithm, which uses the node information entropy as the node voting ability.

The VoteRank algorithm puts forward a new approach to solve the IM problem in the above studies, but the algorithm also has some flaws. First, the VoteRank algorithm treats the initial voting ability of all nodes in the network as the same and does not differentiate between nodes based on their importance. Second, in the voting process, the node scores are obtained only by summing up the voting ability of the neighbors, which fails to reflect the contribution of different neighbors. In Addition, the VoteRank algorithm weakens the voting ability of only the nearest neighbors after selecting the winning node in each round, which does not take into account the effect of non-directly connected nodes and does not guarantee that the identified spreaders are widely distributed in the network.

In response to the above shortcomings, some scholars have made partial improvements. The EnRenew algorithm uses the information entropy as the initial voting ability of a node, but does not propose improvements to calculating the node’s voting score. The NCVoteRank algorithm treats the initial voting ability of nodes as the same and introduces the NC value of nodes in the calculation process of node scores for improvement. However, the NC value does not better reflect the importance of the nodes in the network. To address the above issues, we propose an improved algorithm of VoteRank called DILVoteRank, the main improvements were made as follows:

(i) The initial voting ability of a node is redefined using the degree value, so that the initial voting ability of a node can reflect the importance of the node to some extent.

(ii) The calculation of the node voting score in the VoteRank algorithm is improved by introducing the DIL method, which better reflects the importance of the node in the network local, making the node identified by the improved algorithm have more influence on the network.

(iii) The weakening mechanism of the VoteRank algorithm is optimized to weaken the nearest and next nearest neighbors’ information of the nodes selected in each round of voting, so that the initial spreaders selected by our method are more widely distributed in the network, so that a larger propagation range is achieved in the network.

In the experimental part of this paper, we first compare the DIL method with other different types of traditional importance ranking methods, comparing the computational complexity of the different methods and the correlation with the ranking results obtained by using the node deletion method (NDM) to demonstrate the superiority of the DIL method in reflecting the importance of nodes. Afterwards, we compare our DILVoteRank algorithm with other benchmark algorithms using 12 real datasets under the SIR model and linear threshold model. The experimental results demonstrate that the DILVoteRank algorithm outperforms other benchmark algorithms in terms of both propagation speed and propagation size, as well as stability.

The framework of this paper is as follows. Section 2 reviews some related works. Section 3 describes the steps of the proposed algorithm and the main innovation points. Section 4 presents the performance metrics chosen for the experiment in this paper. Section 5 provides the experimental results and discusses them, and Section 6 concludes the paper.

Related work

There are many methods to evaluate the importance of nodes. A relatively simple method is to use the degree centrality, which considers nodes with more neighbors to be more important than those with fewer neighbors. However, in some cases, nodes located in the center of the network do not necessarily have a high degree value [25, 26]. Therefore, many novel and valid methods have been proposed to evaluate the importance of nodes. In this section, we present some effective methods for evaluating node importance in complex networks, as well as a brief description of the VoteRank algorithm and its improvements.

Assuming that an undirected and unweighted complex network can be characterized as $G\left(V,E\right)$, where $V=\left\{v_{\mathit{1}},v_{\mathit{2}},\cdots ,v_n\right\}$ and $E=\left\{e_{\mathit{1}},e_{\mathit{2}},\cdots ,e_m\right\}$ denote the set of nodes and the set of edges in the network, respectively, and $n$ and $m$ denote the number of nodes and edges, respectively. In Addition, $\Gamma \left(v\right)$ is used to denote the set of neighbors of node $v$ and $k_n$ is used to denote the degree value of node $v_n$.

Semi-local method

The semi-local centrality was a method proposed by Chen [25] to evaluate the importance of nodes by considering the number of nearest and next-nearest neighbors, by weighing the low relevance of degree centrality and the large calculating complexity of global methods. The local centrality $C_L\left(v\right)$ is defined as:

$Q\left(u\right)={\displaystyle \sum }_{w\in \Gamma \left(u\right)}N\left(w\right)(1)$

$C_L\left(v\right)={\displaystyle \sum }_{u\in \Gamma \left(v\right)}Q\left(u\right)(2)$

where $N\left(w\right)$ is the nearest and the next-nearest neighbors of node $w$, respectively. The semi-local centrality is more effective than the degree centrality at identifying key nodes because it utilizes more information about the nodes. Also it has much lower calculation complexity than the betweenness and closeness centralities.

DIL method

Liu et al. [26] proposed a new method, called the DIL method, which computes the importance of a node by using the degree centrality and the importance of lines. This method initially considers that the importance of the edges in the network as being proportional to the connectivity of the edges and inversely proportional to the alternative ability of the connected nodes. For example, if edge $e_{mn}$ connects node $v_n$ and node $v_m$, the importance of edge $I_{mn}$ can be defined as:

$I_{mn}=\frac{U}{\lambda }(3)$

where $U=\left(k_m-p-\mathit{1}\right)\left(k_n-p-\mathit{1}\right)$ reflects the connectivity of the edge $e_{mn}$; $\lambda$ is the alternative ability index of edge $e_{mn}$, which is defined as $\lambda =\frac{p}{\mathit{2}}+\mathit{1}$, and $p$ indicates the number of triangles formed by the edge. After calculating the importance of the edge, the weight of the contribution to the node is calculated based on the degree of the node. For instance, the weight of the importance contribution of edge $e_{mn}$ to node $v_m$ can be defined as:

$W_{v_m{v}_n}=I_{mn}\cdot \frac{k_m-1}{k_m+k_n-2}(4)$

Then the importance of node $v_m$ can be defined as:

$L_{v_m}=k_m+{\displaystyle \sum }_{v_n\in \text{Γ}\left(m\right)}{W}_{v_m{v}_n}(5)$

Nodes with larger $L_{v_m}$ values are considered to be more important in the network. This method can better identify the bridge nodes utilizing local characteristics.

NC method

Bae et al. [31] argued that important nodes have more neighbors located at the center of the network, and proposed NC by taking the K-core value of neighbors into consideration:

$NC\left(v\right)={\displaystyle \sum }_{w\in \text{Γ}\left(v\right)}ks\left(w\right)(6)$

where $ks\left(w\right)$ denotes the K-core value of node $w$. Further, the ENC was proposed based on the NC:

$ENC\left(v\right)={\displaystyle \sum }_{w\in \text{Γ}\left(v\right)}NC\left(w\right)(7)$

where $NC\left(w\right)$ is the neighborhood coreness of neighbor $w$.

VoteRank algorithm

Zhang et al. [40] proposed the VoteRank algorithm, based on a voting mechanism, to select the most influential nodes based on the scores of the nodes in each round of voting. Each node in the network contains two attributes, ${\{S}_v,V_{a_v}\}$, where $S_v$ is used to record the voting score of nodes after each iteration, and $V_{a_v}$ indicates the voting ability of the nodes during each iteration. The voting score of a node is equal to the sum of its neighbors’ voting ability, and the VoteRank algorithm goes through the following five steps:

Step 1: Initialize. Initialize the voting score $S_v$ and voting ability $V_{a_v}$ of all nodes in the network to 0 and 1.

Step 2: Vote. In this phase, each node votes on their neighbors, and each receives all votes from its neighbors. The voting score of a node $S_v$ in the Tth round of voting can be expressed as:

$S_v\left(T\right)=S_v\left(T-\mathit{1}\right)+{\displaystyle \sum }_{i\in \Gamma \left(v\right)}{V}_{a_i}\left(T-\mathit{1}\right)(8)$

Step 3: Select. The node with the highest voting score is selected based on the results of the current round of voting. The selected node $V_{Tmax}$ will not participate in the next round of voting, thus changing the voting ability of this node to 0.

Step 4: Update. In order to make the selected nodes as diffuse as possible, the voting ability of the selected node’s neighbors needs to be diminished. The diminished node voting ability can be defined as:

$V_{a_v}=\{\begin{array}{c}{V_a}_v-\delta if{V_a}_v-\delta >0\\ 0otherwise\end{array}(9)$

where $\delta =<k>/\left(<k^{\mathit{2}}>-<k>\right)$ denotes the reduction coefficient of the node voting ability, and $<k>$ denotes the average degree of the network.

Step 5: Repeat. Repeat the process from Steps 2 to 4 until the top k initial spreaders are filtered out.

NCVoteRank algorithm

Kumar et al. [42] argued that the voting ability of nodes should be distinguished based on the topological location of nodes in the network. Therefore, they proposed the NCVoteRank algorithm to improve the voting ability of nodes by considering the neighborhood coreness value of nodes, which is obtained by calculating the node coreness value by Eq. 6. The NCVoteRank algorithm also first initializes the voting ability $V_{a_v}$ and the voting scores $S_v$ of all nodes in the network to 1 and 0. The following formula is used to calculate the node scores in the voting phase:

$S_v={\displaystyle \sum }_{i\in \Gamma \left(v\right)}\left(V_{a_i}\cdot NC\left(i\right)\cdot \left(1-\theta \right)+V_{a_i}\cdot \theta \right)(10)$

where $\theta$ is an adjustable parameter in the range of [0, 1], which is used to adjust the weight of the node neighborhood coreness value $NC\left(i\right)$. After that, the nodes with the highest voting scores in this round are selected, the nodes and their neighbor node information are updated, and the above steps are repeated until the top k initial nodes are selected.

EnRenew algorithm

Inspired by the VoteRank algorithm, Guo et al. [43] argued that node information entropy better reflects the position of nodes in the network. Their EnRenew algorithm thus uses the node information entropy as the initial propagation ability of a node. The node information entropy can be calculated by:

$E_v={\displaystyle \sum }_{u\in \text{Γ}\left(v\right)}{H}_{uv}={\displaystyle \sum }_{u\in \text{Γ}\left(v\right)}-P_{uv}\log P_{uv}(11)$

where $P_{uv}=\frac{d_u}{{\displaystyle \sum }_{l\in \Gamma \left(v\right)}{d}_l}$, and $H_{uv}$ denotes the propagation ability that node $v$ receives from node $u$. The EnRenew algorithm selects the node in the network with the largest propagation ability as the selected node, and weakens the propagation ability of the node’s l-length reachable nodes. The weakened propagation ability can be calculated by:

$H_{u^{l-1}{u}^l}=H_{u^{l-1}{u}^l}-\frac{1}{2^{l-1}}\frac{H_{u^{l-1}{u}^l}}{E_{<k>}}(12)$

where $E_{<k>}=-<k>\cdot \frac{<k>}{n}\mathit{log}\frac{<k>}{n}$ is the information entropy of any node in the <k>-regular graph network, $u^l$ indicates that the distance between node $u^l$ and the selected node is $l$.

Proposed work

The VoteRank algorithm selects influential nodes in complex networks by using a voting mechanism. In the traditional VoteRank algorithm, however, the initial voting ability of nodes is set to be the same, and the score contributions of nodes to their neighbors are their own voting ability. To a certain extent, then, the final score of the nodes has a greater correlation with the degree index of the nodes. We believe that a more suitable node-importance method should be used to improve the VoteRank algorithm—one that better reflects the position of the nodes in the network topology during the voting process. Therefore, we propose an algorithm called DILVoteRank to identify the key nodes in the network. We improve the VoteRank algorithm in different aspects, and the following are the specific details for the proposed DILVoteRank.

Algorithm steps

The DILVoteRank algorithm proceeds as follows:

(i) Calculating the local importance of nodes. The local importance of nodes $L_{v_i}$ is calculated based on the DIL algorithm, according to Eq. 5, and normalized as follows:

$L_{v_i}=\frac{L_{v_i}-\min \left\{L_{v_i}\right\}}{\max \left\{L_{v_i}\right\}-\min \left\{L_{v_i}\right\}}(13)$

where $\max \left\{L_{v_i}\right\}$ and $\min \left\{L_{v_i}\right\}$ denote the maximum and minimum values in the list, respectively.

(ii) Initialize node score and voting ability. We consider that the voting ability is related to the degree of the node. In this phase, the node voting score is initialized to 0, and the voting ability $V{a}_i$ is calculated according to the following formula:

$V_{a_v}=\log \left(e+\frac{k_v}{k_{\mathit{max}}}\right)(14)$

where $k_{\mathit{max}}$ denotes the maximum value of the node degree in the network.

(iii) Voting phase. In the voting phase, each node receives votes from its neighbors, and sends votes to the neighbors that voted for them. The score of the current round for each node can be calculated as:

$S_v={\displaystyle \sum }_{i\in \text{Γ}\left(v\right)}\left(V_{a_i}\cdot \frac{L_{v_i}}{\sqrt{{\sum }_{j=1}^N{L}_{v_i}^2}}\right)(15)$

To better reflect the local importance indicators of nodes, we use the homotopy function $u\left(x\right)=x/\sqrt{\sum x^2}$ to process the indicators.

(iv) Update node attribute values. Firstly, the node with the top voting score in this round is selected, and its voting ability is set to 0. We assume that node $v_T$ is the selected node for the Tth round of voting. Then, we update the voting ability values of the nearest and the next-nearest neighbors of node $v_T$ as follows:

$V_{a_v}=\{\begin{array}{c}{V_a}_v-\delta if{V_a}_v-\delta >0\\ 0otherwise\end{array}(16)$

where $\delta =\frac{1}{<k>\cdot d\left(v_k,v_T\right)}$ denotes the reduction coefficient of the voting ability, and $d\left(v,v_T\right)$ denotes the distance between $v$ and $v_T$.

(v) Iteration phase. Repeat Steps (iii) to (iv) until the top k initial spreaders are selected.

Algorithm description and complexity analysis

The detailed procedure for DILVoteRank is shown in Algorithm 1.

Algorithm 1. DILVoteRank

In lines 2–4, we calculate the connectivity of each edge in the network. We use n and m to denote the number of nodes and edges in the network, respectively. The time complexity of this process is O(m), according to Eq. 3. In lines 5–9, we calculate the local importance of nodes and initialize voting ability. In line 6, we consider the neighbors of all nodes during the calculation, so the time complexity is O(n<k>), where <k> represents the average degree of the network, <k ≥ 2m/n. In lines 11–22, we first calculate the voting scores of all nodes and choose the nodes with the highest scores as the selected nodes. Then, we update the attributes values of the nearest and the next-nearest neighbors. This process needs to be repeated s times, where s indicates the number of initial spreaders. So the time complexity of this process is O(sn<k > ²). In summary, the computational complexity of the algorithm is O(m + n <k> + sn<k > ²), which can be expressed as O(n <k> + sn<k > ²). Since the value of s is much smaller than n, the complexity of the algorithm can be eventually approximated as O(n<k > ²).

Example explanation

In this section, we use a small example network, shown in Figure 1, as an illustration to demonstrate the DILVoteRank algorithm in detail, which has also been adopted by other scholars [29, 44]. In Figure 1, the node colors are labeled according to the degree of the nodes. After the first round of voting, the results of node local importance $L_{v_i}$, voting ability $V_{a_i}$, and voting score $S_i$ of the nodes are shown in Table 1. According to the voting score results of the nodes, node 4 is the selected node generated by the first round of voting. To prevent node 4 from participating in the next round of voting, the voting ability of node 4 is set to 0, and the attribute values of the nearest neighbors $\left\{\mathit{1,2,3,23}\right\}$ and the next-nearest neighbors $\left\{\mathit{5,6,7,8,11,12,16,17,18,19,20,21,22}\right\}$ of node 4 are weakened, according to Eq. 16. A second round of voting is then conducted, and node 17 is chosen as the selected node.

FIGURE 1

FIGURE 1. An example network to explain DILVoteRank algorithm.

TABLE 1

TABLE 1. The importance of the nodes calculated by DIL $L_v$, initial voting ability $V_a$, and the voting score of nodes $S$ after the first round of voting in example network.

Performance metrics

Evaluating different ranking methods

There are two approaches to evaluating the accuracy of nodes selected by different algorithms [45]. One is based on the propagation dynamics model, with the selected node as the propagation source. The propagation process is simulated in the model and analyzed by the initial node influence range. The other approach is based on the NDM, in which the importance of a node is judged by the effectiveness of the network after its deletion. The method based on propagation dynamics can more intuitively simulate the transmission process of information in the network, and has less computational complexity than the NDM. It has become the main method to evaluate the accuracy of the identification results of different algorithms. Therefore, this paper adopts two widely used propagation dynamics evaluation models, and uses the NDM to evaluate the performance of different algorithms. The details of the above methods are described as follows.

Susceptible, infected, recovered model

The susceptible, infected, recovered (SIR) model [46] offers accurate evaluations of the propagation ability of initial spreaders selected by different algorithms by simulating the propagation process of viruses, information, etc. in a network. This model is an effective tool for evaluating complex-network ranking methods by virtue of its operability and applicability to large networks. In the SIR model, nodes are divided into three categories: susceptible nodes (S), infected nodes (I), and recovered nodes (R). In the starting phase, a small portion of nodes in the network are selected as infected nodes, which are in state I, and other nodes are set as susceptible nodes, in state S. During each subsequent iteration step, the infected nodes have the ability to transform the susceptible nodes in their neighboring nodes into infected nodes, with probability $\beta$. The infected nodes in the network nodes in the network may also recover to become recovered nodes, with probability $\lambda$. The infected probability has a threshold ${\beta }_{th}$. When the infected probability is set to less than ${\beta }_{th}$, information cannot spread in the network. Therefore, in order to make the information spread more rapidly in the network, we set the infected probability $\beta =\mathit{1.5}{\beta }_{th}$, where ${\beta }_{th}=\frac{<k>}{<k^{\mathit{2}}>-<k>}$. The infection rate is defined as the ratio of the node infection probability to the recovery probability, $\zeta =\frac{\beta }{\lambda }$, which has a high impact on the dissemination of network information.

Under the SIR model, in each iteration step, infected nodes infect neighboring susceptible nodes to achieve propagation in the network. At the same time, infected nodes have a certain probability of becoming recovered nodes in the propagation process, so the number of infected nodes in the network will gradually increase with time and then decrease. When the number of infected nodes decreases to 0, only susceptible nodes and recovered nodes are left in the network and the network stops spreading. Based on this, we can use $F\left(t\right)$ to denote the ratio of infected nodes and recovered nodes to the total number of nodes, which is a curve that changes with time during network propagation. It can be used as an indicator to evaluate the propagation ability of the initial spreaders. $F\left(t\right)$ can be expressed as:

$F\left(t\right)=\frac{n_I\left(t\right)+n_R\left(t\right)}{n}(17)$

where $n_I\left(t\right)$ and $n_R\left(t\right)$ denote the number of infected nodes and recovered nodes in the network, respectively, at time t. When the number of infected nodes drops to 0, $F\left(t\right)$ reaches its maximum value $F\left(t_c\right)$, which can be expressed as:

$F\left(t_c\right)=\frac{n_R\left(t_c\right)}{n}(18)$

where $t_c$ indicates that the number of infected nodes drops to 0 at the moment $t_c$. This can be used as a metric to evaluate the propagation scale of the initial spreaders.

Linear threshold model

The linear threshold (LT) model [44] is also an evaluation model based on propagation dynamics, which is different from the SIR model in terms of propagation mechanism. In the LT model, each node in the network has two states of active and inactive, and an activation probability $\left(t_i,t_i\in \left[\mathit{0,1}\right]\right)$ is set to indicate the activation difficulty of a node. In this paper, we set the activation probability of each node to be a random number evenly distributed between 0 and 1. For an undirected network, it is defined that when there is an edge between node $v_i$ and node $v_j$ , the influence of $v_i$ on $v_j$ is the inverse of the number of neighbors of $v_j$, that is, $I_{ij}=\raisebox{1ex}{$\mathit{1}$}\!\left/ \!\raisebox{-1ex}{$k_j$}\right.$. When the degree value of a node is less, its single neighbor has more influence on it. In the initial stage, the model also sets some nodes as the first activated nodes. In the process of simulation propagation, if the sum of the influences on a node is greater than its activation probability, the node is activated. The number of active nodes grows over time and eventually stabilizes. Therefore, the number of active nodes in the network in the final stable stage can be used as an evaluation index for the propagation ability of nodes.

Network efficiency

Through the NDM, we can calculate the proportion of the decrease in network efficiency after the deletion of nodes and effectively evaluate the importance of nodes. Network efficiency reflects the connectivity of the network: the higher the network efficiency, the closer the network is connected and the more quickly information can spread in the network. The network efficiency $\eta$ is calculated by the shortest distance of any node pair in the network, which can be expressed as follows:

$\eta =\frac{\mathit{1}}{n\left(n-\mathit{1}\right)}{\displaystyle \sum }_{v_i\ne v_j}{\eta }_{ij}(19)$

where ${\eta }_{ij}$ denotes the connectivity efficiency between nodes $v_i$ and $v_j$ ${\eta }_{ij}=\mathit{1}/d_{ij}$, and $d_{ij}$ denotes the shortest distance between nodes $v_i$ and $v_j$. The deletion of nodes in the network causes a decrease in network efficiency, so the rate of decrease in network efficiency $\mu$ can be used as an index to evaluate the importance of nodes:

${\mu }_i=\mathit{1}-\frac{{\eta }_i}{{\eta }_{\mathit{0}}}(20)$

where ${\eta }_{\mathit{0}}$ denotes the network efficiency before removing nodes, and ${\eta }_i$ denotes the network efficiency after removing node $v_i$.

Network efficiency can be used to evaluate not only the importance of individual nodes but also the importance of a certain set of nodes. However, the network efficiency requires the shortest path algorithm in the calculation process, which has high computational complexity and is therefore unsuitable for node evaluations of large networks.

Correlation coefficient

Based on the network efficiency, the NDM can produce a convincing ranking of node importance. However, the shortest path algorithm is required to calculate the network efficiency, and the computational complexity is nonlinearly related to the number of nodes, which is not applicable to large network structures. Therefore, this method is not generally used to solve IM problems in real networks, although the calculation results of this method can provide a better reference for evaluating other algorithms. Calculating the correlation between the ranking derived from different algorithms and the ranking based on the NDM can be used to evaluate the performance of the algorithms. There are many coefficients used in statistics to measure the correlation of variables, such as the Pearson, Spearman, and Kendall correlation coefficients, and the selection of appropriate evaluation coefficients according to the type and distribution of variables can make the results more reliable. The Spearman correlation and Kendall correlation are more suitable for the correlation evaluation of ranking algorithms.

Spearman correlation coefficient for ranking data

The Spearman rank correlation coefficient is widely used to evaluate the correlation between two different indicator rankings. It is calculated based on the difference between the different indicator ranking levels, which can be expressed as:

$\rho =\mathit{1}-\frac{\mathit{6}\sum d_i^{\mathit{2}}}{n\left(n^{\mathit{2}}-\mathit{1}\right)}(21)$

where $d_i$ denotes the level deviation of different indicators on the ith sample, and n denotes the number of samples, which can be considered as the number of nodes when applied to complex-network ranking methods. The value of the Spearman correlation coefficient is in the range of [−1,1]. A negative result indicates a negative correlation, a positive result shows a positive correlation, and the larger the absolute value of the result, the stronger the correlation. The Spearman correlation coefficient is widely applicable and does not require much data. As long as the measured values between different indicators appear in pairs, the Spearman correlation coefficient can be used for research.

Kendall correlation coefficient

The Kendall correlation is significantly different from the Spearman correlation. The Kendall correlation classifies all node pairs in the network into concordant pairs and discordant pairs, and evaluates the relevance of different ranking methods by the number of different types of node pairs. For any node pair $\left\{v_i,v_j\right\}$ in the network, $\left\{v_i,v_j\right\}$ is said to be a concordant pair if both methods A and B consider the former to be more (or less) important than the latter, and vice versa for discordant pairs. The Kendall correlation coefficient can be calculated as:

$\tau =\frac{\mathit{2}\left(N_c-N_d\right)}{n\left(n-\mathit{1}\right)}(22)$

where $N_c$ denotes the number of concordant pairs, and $N_d$ denotes the number of discordant pairs. The values of the Kendall correlation are also in [−1,1].

Average distance between spreaders ($\mathit{L}\mathit{s}$)

The average distance is an important index to evaluate the dispersion of the initial spreaders, which has an important impact on maximizing influence. With the limited number of initially selected nodes, we want the selected nodes to be as dispersed as possible in the network to improve the coverage area during propagation. In most real networks, the node distribution shows the phenomenon of community aggregation [47], and if the selected spreaders are too concentrated, it is difficult to spread the information to other communities effectively. The average shortest path can be found by the distance between any two nodes in the node set, which is calculated as follows：

$Ls=\frac{\mathit{2}{\displaystyle \sum }_{v_i\ne v_j\in S}{D}_{ij}}{s\left(s-\mathit{1}\right)}(23)$

where $S$ denotes the initial spreaders selected by different algorithms, $s$ denotes the number of nodes in $S$, and $D_{ij}$ denotes the shortest distance between nodes $v_i$ and $v_j$. Larger values of $Ls$ indicate that the spreaders are more widely distributed and have better coverage in the network.

Datasets and result analysis

In this section, to verify the performance of DILVoteRank compared to other benchmark algorithms, we first compare the DIL with other node-importance evaluation methods to illustrate the superiority of the DIL method at reflecting the importance of nodes. Second, we compare DILVoteRank with other algorithms in a real-world network based on the SIR model in a dataset and analyze the experimental results.

DIL compared with other centralities

In this section, we use six node-importance ranking methods—the betweenness centrality (BC), closeness centrality (CC), semi-local centrality (SL), DIL, NC, and ENC—and the NDM to rank the nodes in the network of Figure 1. The ranking results are shown in Table 2. The nodes in this table are sorted according to the results of the NDM. Table 3 shows the categories, the computational complexity, the Spearman correlation $\rho$, and the Kendall correlation $\tau$ of the six different ranking methods.

TABLE 2

TABLE 2. The ranking of the nodes in the example network is calculated by the BC, CC, SL, DIL, NC, and ENC methods, as well as by the ranking of the nodes and the rate of decrease in the efficiency of the network after node deletion using the NDM.

TABLE 3

TABLE 3. Categories and computational complexity of the six methods and their correlation coefficients of the ranking results using the NDM method, where $\rho$ and $\tau$ denote the Spearman correlation coefficient and the Kendall correlation coefficient, respectively, and n denotes the number of nodes in the network.

From the results of the correlation coefficients in Table 3, the correlation between DIL and the NDM is better than the other methods (except the BC method) in both correlation indexes. In terms of computational complexity, the BC and CC methods calculate the importance of nodes through the global information of the network, and the computational complexity is much higher than the other four algorithms, such that the computational burden is unfeasible for large networks. The DIL method is based exclusively on the local information of the nodes in the network, and the computational complexity is linearly correlated with the number of nodes in the network. Compared to other methods of the same complexity, the DIL method has the greatest correlation with the sorting results of the NDM, and even outperforms the CC method, which is much more complex than the DIL method. From the above results, it can be seen that the DIL algorithm can evaluate the local importance of nodes in the network more accurately by virtue of less computational complexity.

Data description

To test the performance of the algorithm, we performed operations using 12 real network datasets, selected with different data sizes and data sources. These datasets are frequently used in research on complex networks. The following is a description of the datasets used for the tests. 1) Karate: a small social network dataset containing interpersonal relationships and interconnections among 34 members of the Karate Club of America [48]. 2) Dolphin: an undirected social network that portrays the interactions and community distribution of 62 dolphins [49]. 3) Jazz Music: this dataset contains the interactions of a network of jazz musicians [50]. 4) CEnew: a biological metabolic network [51]. 5) Email: a network of email exchanges among members of Rovira i Virgili University [52]. 6) Netscience: a coauthorship network of scientists working on network theory and experiments [53]. 7) USAir: a network of the US air transportation system in 2010 [54]. 8) Hamster: a friendship network between users of the website hamsterster.com [55]. 9) Facebook Social: a crowd-sourced dataset containing information about the social circles of Facebook users [56]. 10) Power: a power grid network in the USA [57]. 11) Astro-ph: a collaboration network of scientists posting preprints on the astrophysics archive at www.arxiv.org [58]. 12) Cond-Mat: a coauthorship network between researchers on the topic of condensed matter [54]. Some of their basic network properties are listed in Table 4.

TABLE 4

TABLE 4. Basic characteristics of the 12 complex network datasets, where <k> denotes the average degree of the network, and ${\beta }_{th}$ denotes the threshold of infected probability in the SIR model.

Comparison with benchmark algorithms

In this part, we compare the DILVoteRank algorithm with six benchmark algorithms—DC, DIL, K-shell, VoteRank, EnRenew, and NCVoteRank—in real networks to test the performance of different algorithms under different experimental conditions and evaluation metrics. Figure 2 shows the network infection scale $F\left(t\right)$ change curve over time under the SIR model with different algorithms. The experimental results are taken as the average of 1,000 calculations, setting $\beta =\mathit{1.5}{\beta }_{th}$, infected rate $\zeta =\mathit{1.25}$, and initial node selection ratio $\rho =\mathit{0.02}$. The purpose of this experiment is to test the propagation speed of initial spreaders selected by different algorithms under the same conditions and the final infection scale. The results show that the DILVoteRank algorithm has a stronger propagation ability compared to other algorithms under the same number of initially infected nodes. Specifically, in the CEnew, Email, USAir, Hamster, Facebook, and Power networks, the DILVoteRank algorithm has a larger slope compared with other algorithms, which means that the initial nodes selected by this algorithm have faster propagation speed in the network and can reach stability more quickly. On the other hand, except for Netscience, the DILVoteRank algorithm achieves the maximum propagation size in all other networks, which means that the spreaders selected by this algorithm have stronger propagation ability. From the results, we can also conclude that the DILVoteRank algorithm outperforms other algorithms in terms of stability. In Dolphins and Jazz, there is little difference between the spreaders selected by different algorithms in terms of propagation speed and propagation ability. This is probably because, with a small number of network nodes, the initial propagation nodes selected by different algorithms have higher repetition. The NCVoteRank algorithm performs better than other algorithms in Karate, CEnew, and Hamster (except our algorithm), but it ranks fourth in Netscience, USAir, Facebook, Astro-ph, and Cond-Mat. Similarly, the EnRenew algorithm has good performance in Email, Netscience, Power, and Cond-Mat, but even lower performance in Facebook than the degree centrality. Thus, our algorithm not only outperforms other algorithms in terms of propagation speed and propagation ability but also has more stability.

FIGURE 2

FIGURE 2. The infection scale $F\left(t\right)$ change curve over time under the SIR model with different algorithms. The experimental results are taken as the average of 1,000 calculations, setting $\beta =1.5{\beta }_{th}$, infected rate $\zeta =1.25$, and initial spreaders selection ratio $\rho =0.02$. Subgraphs (A–L) respectively represent the experimental results of 12 datasets in Table 4.

We selected three node-importance-based ranking algorithms, DC, DIL, and K-shell, as references in our experiments. From the experimental results, compared to other algorithms based on voting mechanisms, the performance of these three algorithms is poor. This may be due to the fact that these three algorithms rely excessively on the nodes’ own information, and the selected nodes do not surely have high propagation ability in the global network. Further, these algorithms are based on the local information of the nodes and ignore the weakening of the importance of the surrounding nodes of the selected node in the process of node selection, thus causing the initial set of nodes to have a high clustering coefficient. In particular, the K-shell algorithm is more susceptible to the rich club phenomenon, which is detrimental to the propagation of the initial nodes in the network [38]. Algorithms based on voting mechanisms weaken the voting ability of the neighbors of selected nodes to reduce the aggregation of the initial spreaders, and therefore have better performance in the experiments.

Figure 3 shows the variation curve of the final infection scale with the initial infected node ratio. The results of this experiment take the average of 1,000 calculations, with the infected rate $\zeta =1.25$, because the Karate, Dolphins, and Jazz networks are too small. For the initial spreaders, we set the ratio in [0.02, 0.16], and the other network ratio is set in [0.005, 0.04]. The purpose of the experiment is to evaluate the propagation ability of different algorithms with different initial spreader ratios by the final infection scale. Figure 3 shows that the final infection scale of different algorithms all increase with the increase of the initial node ratio, and the discrepancy between different algorithms is gradually obvious in the process of increasing the initial node ratio. From the results of this experiment, it can also be seen that the DILVoteRank algorithm has better performance compared to other algorithms with the same initial spreader ratio. In CEnew, Email, Hamster, and Facebook, our algorithm significantly outperforms other algorithms at different initial node ratios. The DILVoteRank algorithm also has better stability in other networks.

FIGURE 3

FIGURE 3. The variation curve of the final infection scale with the initial infected node ratio, the results of this experiment take the average of 1,000 calculations, set the infected rate $\zeta =1.25$. Subgraphs (A–L) respectively represent the experimental results of 12 datasets in Table 4.

In the SIR model, the node infection rate $\zeta =\frac{\beta }{\lambda }$, which is also a crucial parameter affecting the propagation of information in the network. In our experiments, we set the infection probability $\beta =\mathit{1.5}{\beta }_{th}$ which is adjusted by changing the recover probability $\lambda$. When this value is smaller, it is more difficult for infected nodes to recover, the rate of decline of infected nodes slows, and the network propagates more widely. Form Figure 4, we can see that the final infection scale of the network keeps increasing as the infection rate increases, but the ranking of different algorithms is not much affected by the change of the infection rate. By analyzing of the experimental results of different networks, the DILVoteRank algorithm can also maintain better performance under different infection rates. And it achieves better stability. For example, the EnRenew algorithm has the best performance in Netscience, but there is no surprising result in the Facebook network.

FIGURE 4

FIGURE 4. The distribution of the final infection size of the network with different infection rates. The average of the results of 1,000 runs was calculated by setting the proportion of initial nodes $\rho =0.02$ in this experiment. Subgraphs (A–L) respectively represent the experimental results of 12 datasets in Table 4.

In addition to using the SIR model, we also use the LT model to evaluate the propagation ability of nodes. Figure 5 shows the change curve of the number of activated nodes in the stable stage with different initial node proportions. The experimental results are the average of 1,000 simulations. As can be seen from the figure, except in Netscience, the DILVoteRank algorithm outperforms other algorithms in 11 other network datasets. The simulation results in the LT model are consistent with the evaluation results of the SIR model in Figure 2, which once again verifies that the nodes identified by the DILVoteRank algorithm have stronger propagation ability.

FIGURE 5

FIGURE 5. The change curve of the number of activated nodes in the stable stage with different initial node proportions, the experimental results are the average of 1,000 simulations. Subgraphs (A–L) respectively represent the experimental results of 12 datasets in Table 4.

By the way, in the two propagation models, the ratio of the number of propagated nodes to the total number of nodes in the network is the most important indicator to measure the propagation ability of the node. The larger the ratio, the stronger the propagation ability of the node. Figures 3, 5 respectively describe the variation curve of the propagation scale with the proportion of initial nodes based on two different propagation models. As can be seen from both sets of graphs, the algorithms performance ranking changes continuously with the proportion of initial nodes, and there are also differences in different networks. This may be because each network has differences in community distribution, aggregation degree, degree distribution, etc., and different algorithms have different ideas and optimization indicators, therefore, the performance of the algorithm varies with the change of the initial node ratio and the network. In general, under the influence of the above two factors, the DILVoteRank algorithm can maintain better performance in most cases than other algorithms, which shows stronger applicability and stability.

Figure 6 shows the shortest paths between spreaders selected by different algorithms with different initial proportions. The average shortest path of the initial spreaders considerably influences network propagation, and we want the selected initial nodes to be distributed as widely as possible in different communities so as to reach the maximum coverage area of the network. This can be evaluated by the average shortest path of the initial node set. From Figure 6, we can see that the set of nodes selected by the DILVoteRank algorithm always achieves a larger average shortest path in different networks. In CEnew, Email, Hamster, Facebook, and Cond-Mat, our proposed algorithm has an obvious advantage over other algorithms, which means that the node set selected by the DILVoteRank algorithm has a wider distribution in the network. This further indicates that the node set selected by the algorithm has superior propagation ability.

FIGURE 6

FIGURE 6. The shortest paths between spreaders selected by different algorithms, with different initial proportions. Subgraphs (A–L) respectively represent the experimental results of 12 datasets in Table 4.

The propagation ability of nodes depends on its importance and dispersion in the network. Therefore, the $Ls$ cannot be used as a direct indicator for evaluating the propagation ability of the identified node. The larger $Ls$ value, does not necessarily correspond to a greater propagation ability, and the influence of nodes importance needs to be considered. The idea of the DILVoteRank algorithm is to make the identified nodes as widely distributed in the network as possible under the premise of ensuring the importance of the identified nodes. Taking the Power network in Figure 6 as an example, the average shortest distance of nodes identified by the DILVoteRank is smaller than DC, VoteRank and EnRenew, but as can be seen from Figures 2–5 the propagation ability of the nodes identified by the DILVoteRank is better than the other three algorithms. This shows that the nodes identified by our method can achieve greater propagation ability under the smaller $Ls$ value, which also point out that the nodes identified by our method have greater importance in the network.

Conclusion

In this paper, we proposed an algorithm called DILVoteRank that selects spreaders with stronger propagation ability in complex networks. The algorithm uses a voting mechanism to determine the influence of nodes in the order in which they are selected from voting. In this algorithm, we believe that the voting ability of nodes in the voting process should reflect the local importance of the nodes to some extent, rather than treating all nodes the same. For this reason, we selected the DIL as the initial metric to evaluate the importance of nodes. We compared the DIL method with other methods and found that the DIL reflects the importance of nodes in the network more accurately and with less computational complexity than other algorithms. Then, we optimized the node voting ability based on the local importance of nodes calculated by the DIL method. The propagation was simulated in 12 real network datasets based on the SIR model and LT model. The experimental results showed that the proposed DILVoteRank algorithm outperformed other algorithms in terms of the propagation rate, propagation scale, and algorithm stability in different propagation conditions and datasets. Furthermore, we confirmed that the initial spreaders selected by the DILVoteRank algorithm have excellent propagation ability. As such, our algorithm has application value for finding influential nodes in networks, preventing the spread of diseases and rumors, and improving the anti-destructive properties of network systems.

Data availability statement

Publicly available datasets were analyzed in this study. This data can be found here: https://networkrepository.com/index.php.

Author contributions

YL and XY performed the analysis. XY validated the analysis and drafted the manuscript. XZ reviewed the manuscript. MX and XL designed the research. All authors have read and approved the content of the manuscript.

Funding

This work was financially supported by regional foundation of the National Natural Science Foundation of China (No.12171481).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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