Controlling COVID-19 transmission with isolation of influential nodes.

To understand the transmission dynamics of any infectious disease outbreak, identification of influential nodes plays a crucial role in a complex network. In most infectious disease outbreaks, activities of some key nodes can trigger rapid disease transmission in the population. Identification and immediate isolation of those influential nodes can impede the disease transmission effectively. In this paper, the technique for order of preference by similarity to ideal solution (TOPSIS) method with a novel formula has been proposed to detect the influential and top ranked nodes in a complex social network, which involves analyzing and studying of structural organization of a network. In the proposed TOPSIS method, several centrality measures have been used as multi-attributes of a complex social network. A new formula has been designed for calculating the transmission probability of an epidemic disease to identify the impact of isolating influential nodes. To verify the robustness of the proposed method, we present a comprehensive comparison with five node-ranking methods, which are being used currently for assessing the importance of nodes. The key nodes can be considered as a person, community, cluster or a particular area. The Susceptible-infected-recovered (SIR) epidemic model is exploited in two real networks to examine the spreading ability of the nodes, and the results illustrate the effectiveness of the proposed method. Our findings have unearthed that quarantine or isolation of influential nodes following proper health protocols can play a pivotal role in curbing the transmission rate of COVID-19.

Introduction

Spreading process is a natural phenomenon and it is often found in biological, electrical, social, chemical, and many other fields. Complex networks have been used comprehensively in determining the most effective nodes and spreading ability-based ranking [1], [2]. In comparison to other nodes, influential nodes play an important role in complex networks in many applications like viral marketing, rumour spreading, disease outbreak and so on. However, selecting a set of influential nodes may demand taking the mutual influence into account between the nodes instead of simply selecting nodes with substantial influence. In addition, it is seriously challenging to select the initial node-set for maximizing spreading scale after the propagation process. To analyze and control the information dissemination process, complex networks require identifying influential network spreads [3], [4], [5]. Influential spreaders are the critical nodes running as a controller or a maximizer of a spread process in a complex network. For instance, an epidemiological network manages the virus propagation by controlling such influential nodes behavior, which expands information propagation in a social network using them as a maximizer [6]. In the literature, multiple indexing methods have been proposed to determine the influential network spreaders [7], [8], [9]. The real world consists of many complex networks, including social networks, biological networks, research networks, and transport networks. Real-world complex networks are unconstrained and often can be categorized as unweighted and undirected. Identifying and controlling the information propagation process in such networks significantly contributes to decision-making activities. It has many functions, such as market advertising, rumor control, disease control, and innovation propagation. As influencers are critical actors for information spread to help the decision-making process, it is necessary to identify influencers in complex networks [10], [11], [12], [13], [14], [15].

In recent times, analysis of complex networks has been widely used and developed due to its high correlation with various research issues. Recently, various relevant studies have been carried out to study the significant properties of the complex networks and how they become useful in modeling evidence theory and different decision-making scenarios [16], [17], [18], [19], [20]. Ye et al. [21] studied group behavior adaptability and welfare dynamics adaptability in multi-agent dynamics by adopting coevolution of game dynamics and network structure. Ranking the influential nodes in a complex network and knowledge-based system has been really popular in recent times, as this concept is really insightful in various fields. Liu et al. [22] proposed a generalized mechanical model incorporating both local and global information to identify the influential nodes in a complex network. In this work, ten real networks have been used to verify the effectiveness of the proposed method. To address the limitations of several centrality measures, Wen et al. [23] proposed a multi-local dimension (MLD) based on fractal property to identify the vital spreaders in complex networks. To show the effectiveness and reasonableness of the proposed method, some real-world and theoretical complex networks are exploited in the study. There are numerous real-life applications of complex networks in evidence theory. To visualize the relationship between individuals, Zhao and Deng [24] proposed a complex network model of evidence theory based on the similarity of evidence. In a recent work [25], structural similarity of nodes has been considered and original transition matrix has been replaced by similarity matrix to incorporate the fact that the transition probability between nodes and their neighbors is not same. Based on node ranking methods, Yu et al. [26] introduced a novel re-ranking algorithm through information spread function to determine a set of influential nodes for complex networks. Yang et al. [27] studied how to identify influential spreaders in complex networks based on node-local centrality and network embedding. Several approaches have been proposed from different perspectives to identify critical nodes in complex networks. Among these, the gravity model can effectively find significant nodes based on local and path information. In this regard, Li et al. [28] used a generalized gravity model for identifying influential spreaders in complex networks. Namtirtha et al. [6] provided the best method of identifying influential spreaders based on network global structural properties. Gupta and Mishra [29] studied information dissemination in complex networks and identified a set of top-N influential nodes using network structure.

In an epidemiological network, virus propagation can be controlled by analysing the behavior of influential nodes. From any topological position, spreading process can be controlled by calculating each node's spreading capability. As several mass vaccination programs of COVID-19 are going on in different countries, several virulent variants of COVID-19 have started posing threats of new surges of cases. From the beginning of COVID-19 pandemic, numerous mathematical modeling approaches have contributed a lot in terms of designing resilient public health policies to control the disease transmission [30], [31], [32], [33], [42]. However, effective identification of influential spreaders is still a puzzle to be solved. In this paper, the technique for order of preference by similarity to ideal solution (TOPSIS) method with a novel formula has been proposed to identify the influential spreaders efficiently in a complex social network, which involves analyzing and studying of structural organization of a network. Though numerous node-ranking methods are available in the literature, experimental results highlight that our proposed method outperforms all the contemporary methodologies over a large variety of network connectivity structures. Our proposed method is competitive with more sophisticated approaches that are computationally costly. It has been found in our experimental results that our proposed method can identify the most influential spreaders efficiently from complex networks. As the real disease transmission rate varies from network to network, infection probability is an insightful parameter to understand the real disease spreading scenario. Even though it is difficult to estimate real infection rate of a network, we have calculated disease spread probability pertinent to network epidemic threshold. Two real COVID-19 networks have been used to validate the robustness of the proposed method in identifying and ranking influential nodes. Eventually, a comprehensive comparison with the contemporary methods is presented to illustrate the effectiveness of the proposed method.

Materials and methods

Effective and influential identification of influential people or organizations can be helpful in controlling COVID-19 spread. In this section, several methods are presented to identify important nodes in random networks. Identifying influential propagators in the social or complex networks is a major challenge. To address this problem, several centrality measures such as eigenvector centrality (EC), degree centrality (DC), betweenness centrality (BC), and closeness centrality (CC) are proposed here. In the following, TOPSIS along with new modified weights are proposed to identify important and influential nodes.

Node ranking using centrality measures

In this section, centrality measures such as DC [14], CC [15], BC [16], and EC [12] are proposed to identify the influential and important nodes within a complex or social network. Inverse sum of the shortest distances from a central node to all other nodes denotes closeness centrality. Therefore, the higher the score of closeness centrality is, the closer the node to others. Betweenness measures number of the shortest paths through a node. Eigenvector centrality, which is another method to detect important and influential nodes, was first presented by Bonacich [17]. In the eigenvector centrality method, importance and influence of a node is based on the number of neighbors and their quality. The DC, CC, BC, and EC methods are defined as follows:

Degree centrality

The degree centrality (DC) of node i, denoted as C (i), is defined as:where, N is the total number of nodes, i is the focal node, j represents all other nodes, and x represents the connection between node i and node j. If node i is connected to node j, the value of x is 1 and otherwise, it is 0. By minimizing the number of mediator nodes, the shortest path is found in a binary network, and the minimum number of ties grafting the two nodes either directly or indirectly define its length [14]. The binary shortest distance is defined as follows [18],Here, the number of mediator nodes on the path between nodes i and j is defined by h.

Closeness centrality method

The closeness centrality C (i) for node i is defined as follows:

Where, d is the distance between node i and node j.

Betweenness centrality method

The betweenness centrality C (i) for node i is defined as:

With the following definitions for parameters p , p (i) and

p: number of binary shortest paths between node k and node l,

p(i): number of paths that go through node i.

: used to normalize the betweenness centrality value.

Eigenvector centrality method

If A be an n × n similarity matrix, then the eigenvector centrality x of node i is defined as follows:where, x is the ith entry in the normalized eigenvector belonging to the largest eigenvalue of matrix A, a is array of similarity matrix A, λ is the largest eigenvalue and n is the number of vertices.

Weighted TOPSIS method

The TOPSIS method [19] was developed by Hwang and Yoon in 1981 [20]. It is a ranking conception and application method in a social or complex network. The standard TOPSIS method is for the selection of a solution that simultaneously has the farthest distance from the negative ideal solution and shortest distance from the positive ideal solution. Maximizing the benefit criteria and minimizing the cost criteria is positive ideal solution, whereas minimizing the benefit criteria and maximizing the cost criteria is negative ideal solution.

Suppose D = (xmn) be a decision matrix, which consists of criteria and alternatives. The decision matrix R = (rij) can be normalized as follows,

Weighted decision matrix A = (v ) can be derived by multiplying the related weights in the columns of the normalized decision matrix as follows,

Where w is the weight for j criterion. In this paper, the new weights are proposed as follows:

The positive-ideal solution and the negative-ideal solution are defined as follows:

Where, K and K are the set of benefit criteria and the set of cost criteria, respectively. From the solutions A + and A −, we have,

Solutions S + and S − are the separation measures based on Euclidean distance. Using the positive ideal and negative ideal solutions S + and S −, the relative closeness can be calculated as,All nodes are ranked based on the relative closeness to the ideal solution. It means that the influential and higher priority nodes have higher C .

Analysis of time complexity

The time complexity is involved in calculating the weighted decision matrix. The complexity of this step is based on each iteration. One matrix multiplication with a vector is needed where time complexity is O(n 2), where n is the length of the vector. However, in case of sparse matrix with on average m non-zero elemets on every row, the matrix multiplication with a vector can instead be done in O(mn) time.

Testing of the proposed method

In this Section, four graph networks as showed in the Fig. 1, Fig. 2, Fig. 3, Fig. 4 , from previous published papers [1], [34], [35], [36] are presented to illustrate the application of centrality measures and proposed method based on relative closeness to the ideal solution. First network containing 12 nodes and 13 edges is presented from reference [34]. Fig. 1 shows the proposed network with four top ranking nodes highlighted with red color.

Fig. 1

A network containing 12 nodes and 13 edges [34]. Four top ranking nodes highlighted with red color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2

A network containing 13 nodes and 19 edges [35]. Four top ranking nodes highlighted with red color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3

An example network consisted of 23 nodes and 40 edges [1]. Although node 23 has lower degree than node 1, its influence may be even higher. Four top ranking nodes highlighted with red color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4

An example network consisted of 24 nodes and 29 edges [36]. Although node 23 has lower degree than node 1, its influence may be even higher. Four top ranking nodes highlighted with red color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Based on the connected and unconnected nodes for the first graph as showed in the Fig. 1, the adjacency matrix for this graph is as follows,

Therefore, the decision matrix D = (x ) for the graph 1 be obtained as follows,

Three centrality measures of DC, BC, and EC as shown in Table 1 are employed to derive the decision matrix. Eq. (6) can be used to normalize the decision matrix, and then the normalized decision matrix R = (r ) is derived as follows, Here, T is transpose of matrix R. Eq. (8) is used to obtain weighted decision matrix A = (v ) as, w = {0.3493, 0.4044, 0.2462}, a 25 = 0.4044 × 0.2214 = 0.0896 ≃ 0.089,The new weights shown in the Eq. (8) were used to obtain the weighted decision matrix. The positive ideal solution A + and the negative ideal solution A − are derived using Eqs. (9), (10),

Table 1

Decision matrix D = (x).

Degrees centrality	Betweenness centrality	Eigenvector centrality
2	68	0.2866
1	60	0.1316
1	39	0.1316
4	39	0.3622
3	13	0.4468
3	3	0.4267
3	0	0.4408
5	0	0.3394
1	0	0.1233
1	0	0.1233
1	0	0.1233
1	0	0.1233

A+ = {0.1978, 0.1441, 0.1565}; A − = {0.0396, 0.0818, 0.0000.}

The separation measures S + and S − are calculated using Eqs. (11), (12), respectively as follows,

The result of node ranking based on relative closeness to the ideal solution for the graph 1 (Fig. 1) is presented in the Table 2 . According to Eq. (13), the values of C were calculated using the values of S − and S +. The largest value of C belongs to the node with the highest ranking in the network. As can be seen in the Table 2, and . Therefore, node 8 is ranked first and node 9 is ranked last in the twelve ranked nodes of graph 1.

Table 2

Nodes ranking based on relative closeness to the ideal solution for the graph showed in the Fig. 1[34].

Nodes	Si+	Si−	Ci	Ranking
1	0.1935	0.0508	0.2080	8
2	0.2226	0.0623	0.2187	7
3	0.2226	0.0623	0.2187	6
4	0.0868	0.1506	0.6344	2
5	0.1170	0.1199	0.5062	4
6	0.1564	0.0860	0.3547	5
7	0.1024	0.1592	0.6086	3
8	0.0545	0.2227	0.8034	1
9	0.2231	0.0467	0.1732	12
10	0.2231	0.0467	0.1732	11
11	0.2231	0.0467	0.1732	10
12	0.2231	0.0467	0.1732	9

Node ranking based on relative closeness to the ideal solution for the graphs 2–4 (Fig. 2, Fig. 3, Fig. 4) are showed in the Table 3, Table 4, Table 5 . According to Table 3, and , then, node 4 is ranked first and node 2 is ranked last in the thirteen ranked nodes of graph 2. As can be seen in the Table 4, and , therefore node 1 is ranked first and node 5 is ranked last in the twenty-three ranked nodes of graph 3. The results of Table 5 show, and . Therefore, node 1 is ranked first and node 22 is ranked last in the twenty-four ranked nodes of graph 4.

Table 3

Nodes ranking based on relative closeness to the ideal solution for the graph showed in the Fig. 2[35].

Nodes	Si+	Si−	Ci	Ranking
1	0.2155	0.0686	0.2414	9
2	0.2177	0.0444	0.1693	13
3	0.1280	0.1078	0.4572	4
4	0.0766	0.2153	0.7375	1
5	0.1672	0.0935	0.3587	6
6	0.1807	0.0702	0.2798	7
7	0.2177	0.0444	0.1693	12
8	0.1457	0.0986	0.4036	5
9	0.0789	0.1898	0.7062	2
10	0.1990	0.0503	0.2017	11
11	0.1990	0.0503	0.2017	10
12	0.1268	0.1088	0.4619	2
13	0.2153	0.0766	0.2625	8

Table 4

Nodes ranking based on relative closeness to the ideal solution for the graph showed in the Fig. 3[1].

Nodes	Si+	Si−	Ci	Ranking
1	0.0253	0.1924	0.8836	1
2	0.1759	0.0412	0.1899	22
3	0.1606	0.0547	0.2540	16
4	0.1759	0.0412	0.1899	21
5	0.1919	0.0364	0.1593	23
6	0.1326	0.1137	0.4616	4
7	0.1759	0.0412	0.1899	20
8	0.1606	0.0547	0.2540	15
9	0.1759	0.0412	0.1899	19
10	0.1136	0.1240	0.5221	3
11	0.1171	0.0807	0.4078	5
12	0.1393	0.0669	0.3246	11
13	0.1464	0.0697	0.3226	12
14	0.1403	0.0667	0.3222	13
15	0.1427	0.0688	0.3254	10
16	0.1367	0.0694	0.3366	8
17	0.1408	0.0691	0.3293	9
18	0.1585	0.0576	0.2666	14
19	0.1570	0.0517	0.2479	17
20	0.1339	0.0685	0.3385	7
21	0.1340	0.0686	0.3388	6
22	0.1489	0.0489	0.2472	18
23	0.0859	0.1157	0.5737	2

Table 5

Nodes ranking based on relative closeness to the ideal solution for the graph showed in the Fig. 4[36].

Nodes	Si+	Si−	Ci	Ranking
1	0.0696	0.2066	0.7479	1
2	0.1545	0.0649	0.2957	9
3	0.1545	0.0649	0.2957	8
4	0.1273	0.1267	0.4990	2
5	0.2124	0.0220	0.0938	23
6	0.2124	0.0220	0.0938	22
7	0.2124	0.0220	0.0938	21
8	0.2124	0.0220	0.0938	20
9	0.2092	0.0385	0.1554	18
10	0.2092	0.0385	0.1554	17
11	0.2092	0.0385	0.1554	16
12	0.2092	0.0385	0.1554	15
13	0.1164	0.1090	0.4837	13
14	0.1635	0.0585	0.2637	10
15	0.1711	0.0569	0.2495	13
16	0.1435	0.0844	0.3704	7
17	0.1393	0.0865	0.3832	6
18	0.1335	0.0911	0.4057	5
19	0.1909	0.0463	0.1953	14
20	0.1766	0.0601	0.2540	12
21	0.2098	0.0348	0.1424	19
22	0.2128	0.0202	0.0865	24
23	0.1280	0.1188	0.4814	4
24	0.2066	0.0715	0.2571	11

A new formula for disease spread probability in a complex network

In this Section, we study the disease spread probability in a complex network with quarantine of infected influential nodes. Supposing G as proposed graph, N as number of total nodes and E as number of influential nodes with top ranking, the pandemic spread probability P(G ) for the graph G can be defined as follows,to illustrate the application of disease spread probability in a complex network, we calculate the spread probability after removing 2 first top ranking nodes as follows,therefore, the pandemic disease spread probability in the complex network containing 12 nodes and 13 edges (see Fig. 1) with deleting the 2 first top-ranking nodes, will decrease P(G 0) − P(G 2) = 1 − 0.7404 = 0.2596.

Fig. 5, Fig. 6, Fig. 7, Fig. 8 illustrate the decreasing of pandemic disease spread probabilities for the graphs 1–4 shown in the Fig. 1, Fig. 2, Fig. 3, Fig. 4, by removing step-by-step nine infected top-ranking nodes. It can be seen that by removing the first few influential nodes with top ranking, the probability of spreading the pandemic disease decreases more rapidly. It is assumed that disease spread probability is equal 1, for the proposed graph network without removing of any node. The results show the power of effective nodes in spreading pandemic disease in a complex network. Therefore, influential nodes quarantine can be very effective in controlling the spread of pandemic diseases, including COVID-19.

Fig. 5

Probability of disease spreading for graph 1.

Fig. 6

Probability of disease spreading in graph 2.

Fig. 7

Probability of disease spreading for graph 3.

Fig. 8

Probability of disease spreading for graph 4.

Node ranking performance evaluation with SIR model

The SIR (Susceptible–Infected-Recovered) model [37], [38] has been applied for the evaluation of node ranking performance [39]. The spread power of top-ranked nodes can be evaluated using SIR model. Fig. 9 shows an SIR model for epidemic disease transmission in a population of high-risk individuals.

Fig. 9

A schematic of SIR model.

The dynamic system of SIR model can be mathematically shown as follows,where, γ is the rate of recovered people from the infected people, λ is the infection rate and depends on the transmission probability per partner (β > 0), the number of partners per individual per unit time (r > 0), and the proportion of infected individuals to sexually active individuals as follows,

In the proposed model, the susceptible population or healthy nodes in class S will be infected with the transmission rate λ. Using the initial conditions S (0) = S 0, I (0) = I 0 and R (0) = R 0 analytical solutions of systems (15) are as follows,In the SIR model, S (t) represents the number of susceptible nodes or healthy individuals, I (t) denotes the number of infected nodes or infected individuals that are capable to propagate the disease to healthy individuals or susceptible individuals, and R (t) is the recovered people from infected people. One randomly susceptible neighbor gets infected by each infected node with the probability of λ at each step. In this paper, infection rate was assumed to be and recovered ratio was assumed to be γ = 0.05 for uniformity purposes. Transmission rate can specify the influence range of a node in the epidemic spread on networks. Disease can spread to its higher order neighbors through the intermediaries not only to its immediate neighbors by an infected node. Therefore, detection of the influential nodes plays an important role in the spread of an epidemic disease on networks.

Results and discussion

This Section study about two network graphs for the pandemic COVID-19 from references [85, 86]. Fig. 10 shows the flow chart of the proposed method. First graph shows six regions of Portugal and some of their connections [40]. Three top ranking nodes for this graph are highlighted with the red color (Fig. 11 ). Second graph presents Greek COVID-19 infection visibility [41]. Nine top ranking nodes are highlighted with red color (Fig. 12 ). According to Fig. 11, the regions Centro, Lisboa e Vale do Tejo and Alentejo are influential areas. Quarantine each of these areas will decrease the outbreak of pandemic COVID-19, and probability of pandemic disease spread decreases more rapidly.

Fig. 10

Flow chart of proposed method.

Fig. 11

Six regions of Portugal and some of their connections: (1) Norte; (2) Centro; (3) Lisboa e Vale do Tejo; (4) Alentejo; (5) Algarve; (6) Pinhal Litoral[40]. Three top ranking nodes highlighted with red color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 12

Probability of disease spreading in the network of six regions of Portugal [40].

In this paper, many of centrality measures and their applications were used for detecting influential nodes. All those methods have some restrictions and drawbacks and all have only focused on one centrality measure. The information flows are pivotal to identify the influential nodes in a network. As pointed by several authors, it is necessary to employ different types of centrality measures in different networks with different information [12], [14], [15], [16].

Analysis of Portugal regions

Fig. 11 shows Portugal's graph of six regions: (1) Norte; (2) Centro; (3) Lisboa e Vale do Tejo; (4) Alentejo; (5) Algarve; (6) Pinhal Litoral [40]. The graph consists of 6 nodes and 7 edges and three top-ranking nodes are highlighted with the red color. Table 6 highlights the top ranked nodes by DC measure, CC measure, BC measure, EC measure, basic TOPSIS method and proposed method for the network of six Portugal regions. Table 3 shows that we get same results for the proposed method, basic TOPSIS method and BC measure method. The disease spreading probability for Portugal's six regions has been illustrated in Fig. 11. It can be seen from Fig. 12, if the three effective nodes are removed, the probability of COVID-19 transmission decreases more rapidly (decreases from P(G 0) = 1 to P(G 3) = 0.2562).

Table 6

The top six ranked nodes by DC measure, CC measure, BC measure, EC measure, basic TOPSIS method and proposed method for the network of six Portugal regions [40].

Rank	DC	CC	BC	EC	Basic TOPSIS method	Proposed method
1	2	5	4	5	4	4
2	6	1	2	1	2	2
3	4	6	3	4	3	3
4	3	4	6	6	6	6
5	1	3	5	3	5	5
6	5	2	1	2	1	1
Running time (s)	0.0448	0.0496	0.0462	0.0452	0.3262	0.3349

Fig. 13. shows the infected cumulative number versus time between node 4 and node 2 for the six regions network of Portugal. According to the Fig. 13, node 4 infects the entire network nodes more quickly, so node 4 ranks higher than node 2. Therefore, the proposed method works better than the DC method. The number of cumulative infected cases versus time between node 4 and node 5 for the above network is shown in the Fig. 14. According to the analysis, node 4 performs better than node 5 in terms of infecting all network nodes. Hence, the proposed method works better than the CC and EC methods. Table 6 summarizes the order of running time using the DC measure, CC measure, BC measure, EC measure, basic TOPSIS method and proposed method for the six Portugal regions network is as RT < RT < RT < RT < RT < RT (RT: Running Time).

Fig. 13

Infected cumulative number versus time between node 4 and node 2 for the Portugal six regions network.

Fig. 14

Infected cumulative number versus time between node 4 and node 5 for the Portugal six regions network.

Analysis of Greece

Fig. 15 illustrates the network overview of Greek's COVID-19 transmission [41]. The network graph consists of 43 nodes and 107 edges and nine top ranking nodes are highlighted with the red color. As shown in the Fig. 15, the proposed network graph is divided into five clusters namely Q 1 to Q 5. Out of the top nine nodes, four nodes as 9, 11, 17 and 19 with rank of 5, 8, 3 and 1 respectively, are ranked in the first cluster Q 1. None of the nine effective nodes are located in the second cluster Q 2. Influential nodes 26 and 32 with rank of 5 and 7 are located in the third and fourth clusters Q 3 and Q 4 respectively, whereas three nodes 35, 37 and 42 with rank of 4, 2 and 9 respectively belong to the fifth cluster Q 5.

Fig. 15

Greek's COVID-19 infection visibility [41]. Nine top ranking nodes highlighted with red color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 7 presents the summary of the top nine ranked nodes by DC measure, BC measure, EC measure, basic TOPSIS method and proposed method for the Greek COVID-19 infection visibility network [41]. The order of running time using the DC measure, CC measure, BC measure, EC measure, basic TOPSIS method and proposed method for the nine Greek COVID-19 transmission network is as RT < RT < RT < RT < RT < RT (RT: Running Time). The disease spreading probability for the Greek COVID-19 infection visibility network with removal of nine influential nodes is shown in the Fig. 16 . It can be seen from Fig. 16, the transmission probability of COVID-19 decreases more rapidly (decreases from P(G 0) = 1 to P(G 3) = 0.8137) as the five effective nodes are removed.

Table 7

The top nine ranked nodes by DC measure, BC measure, EC measure, basic TOPSIS method and proposed method in the network of Greek Covid-19 infection visibility [41].

Rank	DC	CC	BC	EC	Basic TOPSIS method	Proposed method
1	19	43	19	43	19	19
2	17	28	37	28	37	37
3	37	42	35	27	17	17
4	9	31	32	31	35	35
5	35	27	17	29	32	9
6	26	29	26	42	9	26
7	11	7	9	38	26	32
8	32	8	42	41	11	11
9	23	14	11	39	42	42
Running time (s)	0.0493	0.2043	0.0479	0.0483	2.4996	2.5622

Fig. 16

Probability of disease spreading in the network of Greek's COVID-19 infection visibility [41].

Fig. 17 shows the infected cumulative number versus time graph between node 19 and node 37 for the Greek's network. It highlights the fact that node 19 infects all healthy nodes of network faster than node 37, hence node 19 ranks higher than node 37 in terms of ranking. Infected cumulative number versus time between node 9 and node 32 for the Greek Covid-19 infection network is presented in the Fig. 18 . From the results we have found that node 9 can infect all healthy nodes of network faster than node 32, therefore node 9 ranks higher than node 37 in terms of ranking.

Fig. 17

Infected cumulative number versus time between node 19 and node 37 in the network of Greek [41].

Fig. 18

Infected cumulative number versus time between node 9 and node 32 in the network of Greek [41].

Fig. 19 shows the infected cumulative number versus the time graph between node 42 and node 23 for the Greek's network. As can be seen, node 42 infects all network healthy nodes much faster than node 23, so node 42 ranks higher than node 23. Infected cumulative number versus time between node 37 and node 17 for the Greek's network is illustrated in Fig. 20 . Results depict that node 37 and node 17 can infect all healthy nodes of the network with almost the same speed and power. It can be seen that between times 1 to 9, node 17 works better than node 37, but between times 9 to 21, opposite phenomenon is evident. Fig. 21 shows the number of cumulative cases versus the time between node 19 and node 43 for the Greek's network. It is also evident that node 19 infects all network healthy nodes much faster than node 43.

Fig. 19

Infected cumulative number versus time between node 42 and node 23 in the network of Greek [41].

Fig. 20

Infected cumulative number versus time between node 37 and node 17 in the network of Greek [41].

Fig. 21

Infected cumulative number versus time between node 19 and node 43 in the network of Greek [41].

Conclusion

To detect the influential nodes with high accuracy in a complex social network, a modified TOPSIS method has been proposed to obtain more efficient weights. This study shows how to detect influential nodes in a complex network more effectively in controlling COVID-19 transmission in a population. The spreading probabilities of influential nodes have also been calculated using a novel proposed formula, which also illustrate the rank wise importance of key nodes. Experimental results highlight that removal of the influential nodes can decrease the transmission probability of COVID-19. Therefore, detection of influential nodes in a COVID-19 complex network could be really insightful in terms of slowing down the rate of disease transmission. The SIR epidemic model was used in two real COVID-19 transmission networks to verify the efficiency of the proposed method. The results show that the proposed method is more successful in finding effective nodes with higher accuracy than traditional methods.

CRediT authorship contribution statement

All authors have contributed equally to produce this study.

Declaration of competing interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.