Locating multiple information sources in social networks based on the naming game

Highlights

• •A method is proposed to locate multiple information sources based on naming game.
• •The method can estimate the number and location of actual sources accurately.
• •The number of observations is set on demand rather than predetermined.
• •Label propagation is used to divide the observations into different sets.

Abstract

Identifying the source of information in a network plays a key role in controlling the impact of information. Herein, we study the problem of multiple source localization in the context of information propagation in social networks. We use the theory of the naming game to conduct observations. Moreover, we divide the observations into different sets based on the information provided by them and then estimate the source of each set. Finally, we combine the source of each observation set to obtain all the estimated information sources. The proposed method can locate sources without knowing the number of information sources. Simulations on four real data sets are provided to verify the performance of our method.

1. Introduction

In January 2020, we witnessed the outbreak of the Coronavirus Disease 2019 (COVID-19) [1]. The outbreak and propagation of the virus threatened health worldwide and changed people's lives. However, there are other kinds of information spreading, such as viruses on social networking sites, self-Media and WeChat Moments. There are “goodwill reminders”, “good wishes” and sensational “revelations” in this information. Some of the false information has bad social influences and even worse consequences. In essence, this situation can be modeled as information spreading through social networks. Thus, effectively locating the sources of information propagation in a network is significant for forecasting the range of information propagation, controlling the propagation process, etc.

Generally, information propagation in a social network originates from a few information sources and then spreads quickly through the underlying social network [2], [3], [4]. According to the number of information sources, existing studies can be divided into single source location and multiple source location. Zhu et al. [5] provide a greedy optimization algorithm for selecting the minimum number of observers in cyber-physical systems and combine the propagation delay and back-diffusion methods to locate the source node of information diffusion. Louni and Subbalakshmi [6] propose a two-stage source localization algorithm, and the experimental results show that the algorithm decreases the percentage of sensors significantly. Spinelli et al. [7] propose the first online approach for source localization. This approach allows for source localization with an extremely small number of sensors.

The works listed above accurately identify the propagation source when there is only one source. In many applications, there may be multiple information sources in the network, such as rumors that often originate from more than one person to propagate quickly and widely [8]. In social networks, there is some uncertainty in the propagation paths of the sources of information propagation, and the influence ranges of multiple sources are mixed with each other, which makes it difficult to locate multiple sources compared to a single source. Some researchers use infection timestamps to investigate this problem. Such methods use the observations of the infection times of a subset of nodes in the network [9], [10], [11]. Fu et al. [12] propose a backward diffusion-based source localization method and find that multiple sources can be located with high accuracy even when the fraction of observers is small and the time delay along the links is not known exactly. Tang et al. [13] introduce a new heuristic involving an optimization over a parametrized family of Gromov matrices to develop an estimation algorithm for both a single source and multiple sources. However, the above studies assume that the propagation delays along each edge and the number of sources are known. Such assumptions can be restrictive for practical applications of multiple source localization.

Many of the works on information source localization are based on the knowledge of the network topology, the infection status and the infection time of a portion of nodes in the network. However, information propagation usually occurs in a large scale social network, and it is challenging to obtain the infection status and the infection time of a large number of nodes. Thus, Yang et al. [14] introduce a novel deployment scheme based on the naming game theory [15], which does not need to obtain the infection states or infection times of the nodes. The experimental results show that the method can achieve better localization accuracy with a small number of observations for single source information propagation. To improve the speed and scope of information propagation, more than one source is selected at the beginning of information propagation. However, the multiple source localization problem is much more complicated than the single source localization problem.

In this paper, our goal is to propose a strategy to solve the problem of multiple information source localization when the number of information sources is unknown a priori. We hope to achieve a high location accuracy with as few observations as possible. We first need to know the topology of the network. Then, we select some nodes as observations. Next, we need to know the local sources of each observation. Here, ‘local sources’ are defined as the nodes that first spread the information to the observation, as in [14]. On these bases, this paper proposes a method to locate multiple information sources in social networks based on naming game theory. According to the information provided by the observations, we modify the network topology by removing edges that are not needed for information propagation. Moreover, label propagation is used to divide the observations into different sets [16]. Then, a maximum likelihood estimation function is used to calculate the estimated sources of each set of observations. Finally, all the expected sources of the information propagation are obtained by filtering the results. In this method, we place the selection of observations in the information source localization process, and the number of observations is set on demand rather than predetermined. Moreover, the number of sources is not required beforehand. The simulation results demonstrate that our approach is able to infer multiple sources from a portion of the observations.

The rest of this paper is organized as follows. Section 2 introduces the model and methods for multiple information source location in social networks using the naming game. Section 3 describes the simulation results with analyses and comparisons. Section 4 is the conclusion section.

2. Model and methods

In this section, we first describe our social network model, the fundamentals of the Naming Game, and the assumptions.

Generally, in a social network, a node is an individual and an edge exists between any two individuals if they have any kind of social relationship, such as friendship or followership. In most social networks, information propagation is directed. In particular, if the connection of edges in a network is undirected, we treat each undirected edge as two directed edges. Thus, the information propagation network is modeled by a directed graph $G=(V,E)$, where V and E are the node set and edge set of G, respectively. An edge from node u to v is noted by $e_{u,v}$. We then define $N=|V|$ and $M=|E|$ to represent the number of nodes and the number of edges in the network. When information propagates in a social network, there must be a small number of nodes that first spread the information to their neighbors, and they are called information propagation sources. Subsequently, the nodes that receive the information spread it to their neighboring nodes. This repetition of the process allows information to spread quickly across the network. Many propagation models can well represent the propagation process, such as the classical susceptible-infected (SI) [17], susceptible-infected-susceptible (SIS) [18], and susceptible-infected-removed (SIR) [19] models. Many information source localization methods are designed for a certain type of propagation model. However, it is impossible to know the type of propagation and its parameters in the actual localization of information propagation sources. Thus, our work does not focus on modeling the information propagation process. For a better practical application, we should estimate the propagation sources of the information without knowing anything about the propagation process, including the propagation type, the parameters and the number of sources.

The naming game studies the emergence of shared lexicons in a population of agents about some objects they observed, such as naming a new object [20]. The rules of a naming game model [21], [22], [23] are as follows. Initially, every agent has a memory. In each iteration, randomly pick a node as the speaker and a node as the hearer that are neighbors to each other in the network. Then, the speaker says a word to the hearer. The word is randomly picked from the speaker's memory or, if it has an empty memory, from external lexicons. If, by coincidence, the hearer has the same word in its memory, which means that the speaker and hearer reach a consensus, then both of them will clear their memories except for the common word; otherwise, the hearer will add the new word to its own memory. This process iterates until the whole population reaches the global consensus for which all the agents have one and only one common word in their memories.

2.1. Methods for selecting observations

It is essential to obtain some specific information in the propagation process to accurately locate the information sources. Many works identify the most likely information sources based on the knowledge of the infection status of all the nodes in the network. However, it is impractical to track the infection status of each node in a large scale social network. Thus, we obtain information from a specific set of a limited number of nodes in a network. We denote a node in this specific set as an observation. Different from the diffusion of an epidemic, information propagation has specialized characteristics, such as the decision to spread information is determined by the personal will of individuals, and each individual clearly knows which one is its local source (the node that first spreads information to the individual). Inspired by the naming game, we adopt a similar method developed in [14] for selecting observations. Specially, we add an extra node $n^{⁎}$ that is connected to all the nodes in the underlying network and represents an individual or organization that detects the information sources. Thus, $n^{⁎}$ could obtain the information provided by each node in G. We perform the edge removal operation on the network according to the local source information provided by the observations. Then, we iteratively determine the observations until a certain number of edges have been removed. In each iteration t, we randomly chose an observation $o_t$ as the speaker and let $n^{⁎}$ be the hearer. The speaker $o_t$ tells its local sources to the hearer $n^{⁎}$. We denote $L{S}_{o_t}=\{l{s}_{o_t,1},...,l{s}_{o_t,k}\}\subset V$, which contains the local sources of observation $o_t$. We assume that all the speakers are honest. This method has good practicability because it makes it possible to obtain the local sources of each observation. Table 1 clarifies the main symbols used in the model developed in this paper and their specific meanings.

Table 1.

Model notations and definitions.

Notation	Definition
n⁎	The individual or organization that detects the information sources
ot	The observation in iteration t
LSot	The set that contains the local sources of observation ot
S	The set that contains the actual information sources
G2	The propagation path graph
Fot	The set that contains the neighbors of observation ot
CFot	The set that contains the common neighbors of the observation node ot and its local sources
O	The set of observations
ALS	The set of local sources for all observations
NLot	The set that contains the neighbor of the observation ot that is not a local source
ANL	The set that contains the nonlocal source neighbors of all observations
$\overrightarrow{p_k^{(t)}}$	The probability vector $\overrightarrow{p_k^{(t)}}=(p_{k,1}^{(t)},...p_{k,N}^{(t)})$ where $p_{k,i}^{(t)}$ represents the probability that node i is the propagation source of the k-th observation set when considering the information provided by the first t observations
d(u,v)	The length of the shortest path between u and v in G
Ω	The set of the estimated information sources
〈d〉	The network diameter
〈δ〉	The average degree
〈c〉	The average clustering coefficient

Through the information provided by each observation, we can obtain a propagation path graph $G_2$ and it can be simplified by G. Initially, $G_2=G$. For each $o_t$, we remove some of the edges in $G_2$ according to the contents of set $L{S}_{o_t}$. We let $F_{o_t}=\{f_{o_t,1},\dots ,f_{o_t,k}\}\subset V$ denote the set that contains the neighbors of observation $o_t$. In addition, we denote $N{L}_{o_t}=F_{o_t}-L{S}_{o_t}$, which contains the neighbors of observation $o_t$ that are not regarded as local sources. We let $C{F}_{o_t}=\{c{f}_{o_t,1},...,c{f}_{o_t,k}\}\subset V$ denote the set that contains the common neighbors of observation node $o_t$ and its local sources. It is obvious that there are some edges in the network that information does not propagate through. Thus, we identify and remove these edges in $G_2$ as accurately as possible. If $L{S}_{o_t}=\varnothing$, all the neighbors of $o_t$ do not spread the information. Thus, $G_2$ must not contain the edges starting from the neighbor of $o_t$, and they should be removed. Similarly, if $L{S}_{o_t}\ne \varnothing$, then there are three types of edges that can be removed according to the information transmission order. (1) The edges whose start node is in the $N{L}_{o_t}$ set and whose target node is in the $L{S}_{o_t}$ should be removed. (2) The edges whose start node is in $N{L}_{o_t}$ and whose target node is $o_t$ should be removed. (3) The edges whose start and target nodes are all in $L{S}_{o_t}$ should be removed. Our method does not determine all the observations beforehand, and it only confirms the existence of an observation at each iteration. We denote NE as the percentage of the removed edges. We stop selecting the observations when NE reaches a certain value. Then, we obtain the simplified network graph $G_2$, the set of observations $O=\{o_1,...,o_t\}\subset V$, and the set of local sources for all the observations $ALS=L{S}_{o_1}\cup ...\cup L{S}_{o_t}$. We also denote ANL as the set that contains the nonlocal source neighbors of all the observations. It is clear that the nodes in set ANL must not be the information source.

2.2. Method for multiple source localization based on the pre-set observations

We denote $S=\{s_1,...,s_k\}\subset V$, which contains the actual information sources. Our goal is to identify set S from the network topology and the local sources provided by the observations. In the previous Section, we introduced a method for selecting observation nodes. In this Section, we divide the observations into sets according to the simplified network $G_2$. Then, we find the source of each observation set according to the information provided by observations.

The localization of multiple information sources is more complicated than identifying a single source. This is because different observations may receive information from different sources in the case of multiple sources. In this section, we discuss how to divide observations into several sets. Compared with the observations in different set, the observations in the same set have higher probability of receiving information from the same source. We overcome this challenge through label propagation. We propose the observation division algorithm in Algorithm 1 to find some suitable observation sets. First, we take all the observations as the sources to carry out reverse label propagation and find the nodes that can spread the information to each observation in a certain number of steps. In a network with diameter $〈d〉$, the longest distance between any node and its source is $〈d〉$. Without any loss of generality, we choose $〈d〉/2$ steps to find the nodes that can spread the information to each observation. Thus, if a node receives labels from multiple observations, these observations are considered to be propagated by the same source, that is, they belong to the same set. We denote an $N\times N$ matrix OD to record the observation labels received by each node. Initially, if $j\in O$ and $i=j$, $\mathbf{OD}(i,j)=1$, else $\mathbf{OD}(i,j)=0$, where $1<i,j<N$. It means that label propagation starts at each observation. Subsequently, the value of $\mathbf{OD}(i,j)$ changes from 0 to 1, indicating that node i propagates the label of observation j. After $〈d〉/2$ rounds of tag propagation, we obtain the matrix OD for the result of the reverse label propagation, and the i-th row of this matrix contains the observations that can be spread by node i in $〈d〉/2$ steps. It is straightforward that if a node can spread the information to an observation, then it also can spread the information to the nodes that the observation can reach. Thus, we reorganize the result matrix in an iterative way. Then, we eliminate all-zero rows and the rows that can be contained by others. Finally, we obtain the matrix OD in which the non-zero items in each row indicate the observations getting information from the same source node. A parameter $r_{\mathbf{OD}}$ is defined as the number of rows of matrix OD, and it also represents the number of observation sets.

Algorithm 1. Observation Division
01 for m starts from 1 to 〈d〉/2 do
02 OD1 = OD
03 for each node i ∈ V do
04 for each node j ∈ V do
05 if ei,j is an edge of graph G2
06 for each node ot ∈ O do
07 if OD1 (j, ot)=1 and OD(i, ot)=0 then
08 OD(i,ot)=1
09 end if
10 end for
11 end if
12 end for
13 end for
14 end for
15 flag=1
16 while flag=1
17 OD2 = OD
18 for each node u ∈ V do
19 for each node ot ∈ O do
20 if OD(u,ot)==1 then
21 if u ∈ O and u ∈ ALS then
22 OD(u,:)=OD(u,:)∨OD(ot,:)
23 end if
24 end if
25 end for
26 end for
27 if OD = OD2 then
28 flag=0
29 break the while loop
30 end if
31 end while
32 delete the rows of all zeros in OD.
33 delete the rows in OD that are subsets of others.

After dividing the set of observations, another problem to be solved is to find the source of each set. It is obvious that the observations receive information later than their local source. However, it is impossible to set all the nodes as observations. In general, the longer the propagation path from the source to the node, later the node receives the information. Hence, in each observation set, for each observation $o_t$, we update the probability that a node is the source of the observation set with the difference between the distance of the node to $o_t$ and the distance of the node to its local source set. When all the observations are considered, the node with the highest probability is the source of the underlying set. In detail, in set $k(1<k<r_{\mathbf{OD}})$, We denote $\overrightarrow{p_k^{(t)}}=(p_{k,1}^{(t)},...p_{k,N}^{(t)})$ as a probability vector, where $p_{k,i}^{(t)}$ represents the probability that node i is the source of the k-th observation set when considering the information provided by the first t observations. Initially, any node in set V is equally likely to be the source, i.e. for $\forall i\in 1,2,\dots ,N$, $p_{k,i},{}^{(0)}$ equals $1/N$. Then, for each observation $o_t$, we quantify the effect of the information provided by $o_t$ on the value of $p_k^{(t)}$. If $\mathbf{OD}(k,o_t)=0$, it means that $o_t$ does not belong to the k-th observation set. Thus, we have $p_k^{(t)}=p_k^{(t-1)}$. If $\mathbf{OD}(k,o_t)=1$, it means that $o_t$ belongs to the k-th set. It is obvious that the node in set $NL{S}_{o_t}$ must not be the source. We denote $[u,v]$ as the shortest path between nodes u and v. Let $d(u,v)$ be the length of the shortest path between u and v in $G_2$. When $d(i,o_t)$=inf, node i can not propagate the information to the observation $o_t$. Thus, node i must also not be the propagation source. We denote Q as the set of all nodes that satisfy $d(i,o_t)$=inf $(i\in V)$. For the sake of calculation, we denote $T=L{S}_{o_t}\cup Q$ and update the value of each $p_k^{(t)}$ according to (1).

$$p_{k,i}^{(t)}=\{\begin{array}{cc}{p}_{k,i}^{(t)}+\frac{{\sum }_{r\in T}{p}_{k,i}^{(t-1)}}{n_z},\hfill & i\notin Tand{p}_{k,i}^{(t-1)}\ne 0\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}$$ (1)

where $n_z$ represents the number of nodes which are not in T.

Moreover, we denote $d(i,L{S}_{o_t})=\{d(i,l{s}_{o_t,r})|\underset{l{s}_{o_t,r}\in L{S}_{o_t}}{\max }(d(i,l{s}_{o_t,r},)),d(i,l{s}_{o_t,r}\ne \inf \}$ as the distance from node i to set $L{S}_{o_t}$. Then, we denote $\eta =d(i,o_t)-d(i,L{S}_{o_t})$ as the deduction between $d(i,o_t)$ and $d(i,L{S}_{o_t})$, and η is larger than zero. The node in $L{S}_{o_t}$ would receive the information earlier than node $o_t$. Thus, the larger the value of η, the greater the probability that node i is the propagation source of the k-th observation set. In order to avoid negative distance difference, we define $ϵ=\underset{l{s}_{o_t,r}\in L{S}_{o_t}}{\min }\eta$. Based on this, we further update the probability that node i is the source of the k-th observation set.

$$p_{k,i}^{(t)}=\{\begin{array}{cc}{p}_{k,i}^{(t)}\times \frac{\alpha }{\alpha +1}+\frac{\eta -ϵ-1}{{\sum }_{r=1}^N(\eta -ϵ-1)\times (\alpha +1)},\hfill & p_{k,i}^{(t-1)}\ne 0\hfill \\ 0,\hfill & p_{k,i}^{(t-1)}=0\hfill \end{array}$$ (2)

where $\alpha ={\sum }_{r=1}^t\mathbf{OD}(k,r)$, it means that $o_t$ is the α-th observation in the k-th set.

When t equals $|O|$, the information provided by each observation is evaluated in the estimation process of the source node, that is, the probability update process stopped. The node with the biggest probability of (3) is considered as the source of the k-th observation set.

$${\beta }_k=\arg \underset{i\in V}{\max }\overrightarrow{p_{k,i}^{(|O|)}},k=1,2...,r_{\mathbf{OD}}$$ (3)

Finally, a set of the estimated multiple information sources is obtained by merging the source of each observation set, which we denote as $\mathrm{\Omega }=\{{\beta }_1,...,{\beta }_{r_{\mathbf{OD}}}\}\subset V$.

3. Experimental results and analysis

To evaluate the performance of the proposed algorithm, we considered four real social networks, as in [14]. The network characteristics of these social networks are listed in Table 2 , where $〈\delta 〉=\frac{1}{N}\sum _{i=1}^N{\delta }_i$ is the average degree, and ${\delta }_i$ denotes the degree of node i. $〈c〉=\frac{1}{N}\sum _{i=1}^N\frac{2\times m_i}{{\delta }_i\times ({\delta }_i-1)}$ is the average clustering coefficient and $m_i$ denotes the number of links between i's friends. Additionally, $〈d〉=\{d|\underset{i,j\in V}{\max }d(i,j)\}$ is the network diameter, and $d(i,j)$ denotes the number of edges on the shortest path connecting nodes i and j. Then, we chose the SIS propagation model for the simulation experiment of the four social networks. We perform 300 simulation runs. For each simulation, we randomly choose a few nodes as the sources to propagate information and we let the number of sources be uniformly chosen from 2, 3, 4.

Table 2.

Model notations and definitions.

Networks	Type	N	M	〈δ〉	〈c〉	〈d〉
Enron	directed	143	2492	8.713	0.453	8
Simmons81	directed	1510	131936	43.687	0.325	7
Hamilton46	directed	2312	385572	83.385	0.302	6
Wake73	directed	5366	1116744	104.057	0.279	9

3.1. Accuracy of the information source location

The difference between the number of estimated sources $|\mathrm{\Omega }|$ and the number of actual sources $|S|$ can be used to indicate the accuracy of the algorithm. We define $\mathrm{\Delta }=|\mathrm{\Omega }|-|S|$ as the error quantity between $|\mathrm{\Omega }|$ and $|S|$. Fig. 1 shows the distribution of the error quantity Δ in different datasets when NE is different. The closer the value of Δ is to 0, the more accurate the number of estimated sources is. In Fig. 1, we can see that with the increase of NE, the proportion of Δ approaching 0 is gradually increasing. It indicates that the larger NE, the more accurate our method achieves. Furthermore, when NE=50%, our method accurately estimates the number of propagation sources by 39% in the case of the Enron network, 31% in the case of the Simmons81 network, 26% in the case of the Hamilton46 network, and 25% in the case of the Wake73 network. In addition, most values of Δ are close to 0, which indicates that our method can accurately estimate the number of information sources.

Fig. 1.

Distribution of the error quantity (Δ) in different datasets. (a) Enron, (b) Simmons81, (c) Hamilton46, and (d) Wake73.

Then, we match the estimated source nodes in Ω with the actual sources in S so that the sum of the error distances between each estimated source and its match is minimized. We define a matching function ${\hat{\beta }}_i=\pi (i)=\arg \underset{{\beta }_j\in \mathrm{\Omega }}{\min }({\sum }_{s_i\in S}d(s_i,{\beta }_j))$, which matches each actual source $s_i$ to the estimated source node ${\hat{\beta }}_i$. We then define the average error distance as

$$\varphi =\frac{1}{\min (|\mathrm{\Omega }|,|S|)}\times (\sum _{i=1}^{\min (|\mathrm{\Omega }|,|S|)}d(s_i,{\hat{\beta }}_i))$$ (4)

Then, we use the average error distance to quantify the performance of our algorithm. The distribution of the error distance ϕ in four real social networks is shown in Fig. 2 . When the parameter NE=10%, our method achieves an accuracy of 16% in the case of the Enron network, 19% in the case of the Simmons81 network, 10.33% in the case of the Hamilton46 network, and 15.33% in the case of the Wake73 network. As the value of NE increases, the accuracy of the algorithm at estimating the position improves gradually.

Fig. 2.

Distribution of the error distance (ϕ) in different datasets. (a) Enron, (b) Simmons81, (c) Hamilton46, and (d) Wake73.

From Fig. 1 and Fig. 2, it can be observed that the range of the error quantity Δ is larger and the range of the error distance ϕ is smaller in the network that has a lower network diameter. It is because the smaller the network diameter is, the smaller the distance between the nodes in the observation set is. This results in a large number of observation sets. The more the number of sets is, the more estimated sources are and the smaller the distance between the estimated sources and the actual sources is.

3.2. Efficiency of the proposed method

In our method, the more observations there are, the more iterations and time required. Thus, the number of observations should be used to evaluate the effectiveness of our information source location method. We expect to locate the sources accurately by selecting as few observations as possible. The smaller the number of observations, the better the efficiency that can be obtained. Corresponding to the accuracy in Fig. 1 and Fig. 2, Fig. 3 shows the average percentage of observations versus different NE in the four social networks. As NE increases, the accuracy of the algorithm improves, and we can see that the average number of observations increases. Furthermore, when NE=50%, the average percentage of observations we need is less than 49.53% in the Enron network, 43.5% in the Simmons81 network, 42.78% in the Hamilton46 network, and 40.13% in Wake73 network.

Fig. 3.

Average percentage of observations versus different NE in the four social networks.

3.3. Comparison with other source location methods

The experimental results in Sections 3.1 and 3.2 show that our proposed method can achieve high-localization accuracies. Fig. 4 shows a comparison of the average error distance between our method and the method proposed in [13]. Our proposed method outperforms the method proposed in [13] in terms of accuracy when setting up a similar number of observations.

Fig. 4.

Comparison of the average error distance between our method and the method in [13].

4. Conclusions and future work

In this study, we develop a method to estimate multiple information sources using a subset of observations when the number of sources is unknown. Our method uses the theory of the naming game and label propagation flexibly. The method we proposed can estimate the number and location of actual sources accurately without knowing the type and parameters of the propagation model. Moreover, the number of observations is determined in the localization process rather than artificially determined at the beginning of localization. Experimental evaluations with real-world data reveal that our method can locate the information sources within a small number of hops from the actual sources. Although our experiment is only based on SIS propagation model, our method focuses on the sequence of time when the neighbors first propagate messages, rather than the state transition time of each node. Therefore, different propagation models seldom affect the results. Because of this reason, our method can not only be used in other propagation models, but also do not need to know the content of the propagation model in the work of source location. However, generally, with the same fraction of observations, the multiple source estimation performance is not as good as that of single source estimation. In the future, we will focus on designing a better observations selection algorithm to improve the localization accuracy of multiple information source.

CRediT authorship contribution statement

Xue Yang: Conceptualization, Data curation, Methodology, Software, Writing - original draft, Writing - review & editing. Zhiliang Zhu: Conceptualization, Methodology, Supervision. Hai Yu: Conceptualization, Investigation, Methodology, Visualization. Yuli Zhao: Conceptualization, Methodology, Writing - review & editing. Ying Wang: Conceptualization, Methodology, Software, Validation.

Declaration of Competing Interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

Communicated by M. Perc