Accurate ranking of influential spreaders in networks based on dynamically asymmetric link-impact

TL;DR

This paper introduces a novel asymmetric link-impact measure and s-shell decomposition for accurately ranking influential spreaders in networks, outperforming traditional degree and k-shell methods.

Contribution

The paper presents a new asymmetric edge-weighting scheme and a weighted coreness measure that improve the identification of influential spreaders in complex networks.

Findings

The proposed measures outperform degree and k-shell rankings in real-world networks.

The methods maintain low computational complexity.

Effective in applications like disease and information spread control.

Abstract

We propose an efficient and accurate measure for ranking spreaders and identifying the influential ones in spreading processes in networks. While the edges determine the connections among the nodes, their specific role in spreading should be considered explicitly. An edge connecting nodes i and j may differ in its importance for spreading from i to j and from j to i. The key issue is whether node j, after infected by i through the edge, would reach out to other nodes that i itself could not reach directly. It becomes necessary to invoke two unequal weights wij and wji characterizing the importance of an edge according to the neighborhoods of nodes i and j. The total asymmetric directional weights originating from a node leads to a novel measure si which quantifies the impact of the node in spreading processes. A s-shell decomposition scheme further assigns a s-shell index or weighted coreness to the nodes. The effectiveness and accuracy of rankings based on si and the weighted coreness are demonstrated by applying them to nine real-world networks. Results show that they generally outperform rankings based on the nodes' degree and k-shell index, while maintaining a low computational complexity. Our work represents a crucial step towards understanding and controlling the spread of diseases, rumors, information, trends, and innovations in networks.

Tables

Table 1. Table 1: Properties of the real-world networks studied in this work. Structural properties of the number of nodes ( N 𝑁 N ), number of edges ( E 𝐸 E ), average degree ( ⟨ k ⟩ delimited-⟨⟩ 𝑘 \langle k\rangle ), degree assortativity ( r 𝑟 r ), clustering coefficient ( C 𝐶 C ), epidemic threshold ( λ c subscript 𝜆 𝑐 \lambda_{c} ), infection probability ( λ 𝜆 \lambda ) used in the SIR dynamics, and the optimal value of a 𝑎 a as given by the Kendall’s tau correlation coefficient ( a o ​ p ​ t subscript 𝑎 𝑜 𝑝 𝑡 a_{opt} ).

Network	$N$	$E$	$⟨k⟩$	$r$	$C$	${\lambda }_c$	$\lambda$	$a_{opt}$
CA-Hep	8638	24806	5.7	0.239	0.482	0.08	0.12	1.0
Astro	14845	119652	16.1	0.228	0.670	0.02	0.05	0.9
Emailcontact	12625	20362	3.2	-0.387	0.109	0.01	0.10	0.3
PGP	10680	24340	4.6	0.240	0.266	0.06	0.19	1.0
Blog	3982	6803	3.4	-0.133	0.284	0.08	0.27	0.9
AS	22963	48436	4.2	-0.198	0.230	0.004	0.13	0.2
Router	5022	6258	2.5	-0.138	0.012	0.08	0.27	0.7
Hamster	2000	16097	16.1	0.023	0.540	0.02	0.04	0.8
Netsci	379	914	4.8	-0.082	0.741	0.14	0.30	0.8

Equations

$$\tau =\frac{{\sum }_{i<j}sgn[(x_i-x_j)(y_i-y_j)]}{\frac{1}{2}N(N-1)},$$

$${\epsilon }_{\theta }(p)=1-\frac{M_{\theta }(p)}{M_{eff}(p)}$$

$$w_{ij}=1+(k_i{k}_j^{out}{)}^a$$

$$s_i=\sum _{j\in {\Gamma }_i}{w}_{ij},$$

$$w_{ij}^{\prime }=\frac{1}{2}(w_{ij}+w_{ji}),$$

Full text

I Introduction

The structural properties of complex networks and the intricate interplay between the structure and spreading dynamics lead to highly diversified spreading capabilities among individual nodes. From the perspective of the severity of a spreading process, the most influential spreaders are those resulting in a much larger final infected proportion of the whole system when the spread of a disease or a piece of information originates from them than from other nodes. Centrality measures such as the degree freeman1978 , betweenness freeman1977 , closeness sabi1966 , eigenvector centrality bonacich2001 and $k$-shell coreness bolobas1984 have been used to identify the most influential spreaders. The degree is the simplest measure. In social networks, for a example, an individual with a large degree has more direct contacts to other people and is thus likely to be more influential than one with a small degree for transmitting disease or information. Subsequent research indicates that the core nodes as identified by the $k$-shell decomposition are the most influential spreaders kitsak2010identification . Algorithms based on other centrality measures have been proposed to improve the accuracy of identifying influential spreaders pei2013spreading ; chen2013identifying ; ren2014iterative ; lv2016vital . They include the neighborhood coreness bae2014identifying , improved eigenvector centrality martin2014localization ; alvarez2015eigen , H-index hirsch2005index ; lv2016hindex and nonbacktracking centrality radicchi2016leveraging .

Methods other than centrality-based algorithms have also been proposed for predicting how influential a node can be in a spread. For example, by counting the number of possible infection paths of various lengths, the final infection range can be estimated for a spread originated from any node bauer2012identifying . The degree distribution of clusters of infected nodes after certain transmission events leads to a node property called the expected force, which can be applied to predict the spreading influence of all nodes under different epidemiological models lawyer2015understanding . The dynamic-sensitive centrality is able to locate influential nodes from both topological features and the dynamical parameters, such as the infection and recovery rates in a susceptible-infected-recovered (SIR) spreading model liu2016locating . In the $k$-truss decomposition, which is a triangle-based extension of $k$-shell decomposition, the maximal $k$-truss subgraph contains the most influential spreaders malli2016locating .

In most studies on identifying influential spreaders so far, the networks are taken to be unweighted and undirected. Each edge is treated to be equivalent in its function, as in the centralities and ranking methods lv2016vital . However, edges in a network could be quite different grady2012robust . In weighted networks, the weight of an edge reflects the strength of the interaction between the connected nodes, as in situations concerning the number of communications, size of trade, intimacy of friendship and frequency of cooperation, etc. opsahl2010node ; garas2012kshell ; eidsaa2013score . In addition, edges may not be equally important in keeping the network robust wu2011onion . An example is the small influence on network robustness in food web networks when redundant links are removed allesina2009functional . In terms of network functionality, differences among edges are also observed. For example, removing redundant links has no effect on network synchronization zhang2014network but closing specific routes in air transportation networks can minimize the spreading of a disease chung2012impact . To quantify the weight of an edge, a class of measures relying on its importance in the network structure have been defined brockmann2013hidden . For example, the edge betweeness counts the number of shortest paths between any two nodes that go through the edge and it can be regarded as the weight of an edge brandes2001faster . Immunizing edges of high betweenness was found to be effective in suppressing epidemics holme2002attack , but deleting such edges in scale-free networks would enhance the transmission efficiency dramatically zhang2007enhancing . In global air transportation network, the strength of an edge that reflects the volume of passengers travelling between two airports was found to correlate positively with the product of the degree of the connected nodes. Thus, a measure $w_{ij}=(k_{i}k_{j})^{\theta}$ barrat2004arch was introduced as the weight of an edge. This measure has been adopted in many works for distinguishing the importance among edges in unweighted networks wang2005general ; tang2011efficient ; zhang2014suppression .

In the present work, we propose a better measure to quantify the importance of an edge when spreading processes are concerned. As spreading is necessarily directional, e.g. only an infected node would spread a disease to a neighboring susceptible node but not the other way round, the new measure stresses the importance of an edge in the spreading dynamics in the vicinity of the two nodes connected by the edge and it has the general property of $w_{ij}\neq w_{ji}$. The sum of asymmetric weights of links originated from a node defines a new measure of the strength $s_{i}$ of a node $i$, which is shown to be an efficient quantity for identifying influential spreaders with a low computational complexity. Based on the node strength, a $s$-shell decomposition scheme is proposed for assigning a $s$-shell index to every node, which provides a more accurate ranking of the nodes in their influence in spreading processes.

The paper is organized as follows. In Sec. II, the degree centrality, $k$-shell index, the spreading model used in the study and the methods of evaluating the performance of measures for identifying influential spreaders are introduced for completeness. In Sec. III, we propose a new measure that focuses on the importance of an edge in the dynamics of a spreading process. The measure is then applied to define a node strength for every node. A $s$-shell decomposition method that emphasizes the importance of a node in the spreading dynamics is proposed. In Sec. IV we apply the node strength and $s$-shell index to rank and identify influential spreaders in nine real-world networks and demonstrate their effectiveness. A conclusion is given in Sec. V.

II Centralities, spreading model and evaluation methods

We review briefly the degree centrality and the $k$-shell index for completeness. They are efficient measures for identify influential spreaders castellano2012competing ; liu2015core ; liu2015improving . We will compare the performance of our newly defined node strength and $s$-shell index with these methods. The Susceptible-Infected-Recovered (SIR) model is adopted to simulate the spreading dynamics on networks. To quantify the performance of our measures in predicting the influence of the nodes and identifying influential spreaders, the Kendall’s correlation and the imprecision function are introduced.

II.1 The degree and $k$-shell centrality

In a graph $G=(V,E)$, where $V$ is the set of nodes and $E$ is the set of edges, the degree $k_{i}$ of a node $i$ is the number of links it carries. It is given by $k_{i}=\sum_{j}a_{ij}$, where $a_{ij}$ is an element of the adjacent matrix, with $a_{ij}=1$ if there is a link between nodes $i$ and $j$ and $a_{ij}=0$ otherwise. The $k$-shell decomposition method decomposes the network into hierarchical shells in a progreesive process. Initially, nodes with degree $k=1$ are removed from the network together with their links. After the process, there may appear nodes with only one link left. These nodes and their links are then removed and the process is repeated until there is no nodes left in the network with only one link. The removed nodes and links form the $1$-shell, and these nodes are assigned with an index $k_{S}=1$. Next, nodes with degree $k\leq 2$ are removed in a similar way and the set of removed nodes are assigned an index $k_{S}=2$. This pruning process is continued until all nodes are removed and assigned a $k_{S}$ index. This index is called the $k$-shell index or coreness of a node. It represents the core position of a node in the network. Nodes with a large $k_{S}$ are considered as to be at the core of the network, while nodes with a small $k_{S}$ form the peripheral part of the network.

Nodes with large degree and large coreness are considered the most influential spreaders in networks. These measures have a low computational complexity of order $O(E)$ and $O(N+E)$, where $N$ and $E$ are the number of nodes and edges in the network respectively.

II.2 SIR model

The SIR model is chosen to simulate spreading on complex networks. In the model, the nodes have three possible states: S (susceptible), I (infected) and R (recovered). At each time step, the infected nodes infect their susceptible neighbors with a probability $\lambda$ and then recover with a probability $\beta$. To quantify the influence of each node on spreading, we let one node, node $i$ say, be infected and all the other nodes being susceptible initially. The SIR dynamics proceeds from the seed infected node to other nodes until there is no infected node in the network. The recovered nodes at the end are those once infected and the fraction of recovered nodes gives the final infected range of the initial seed. For an initially infected node $i$, the spreading dynamics is repeated for 100 times. The average infected range $M_{i}$ of node $i$ is recorded and taken to reflect the influence or the spreading efficiency of the node $i$. This quantity can be obtained for any node $i$ in the network and used as a measure to rank the nodes on their importance in the spreading dynamics. This dynamics-based list is taken to be the exact ranking that gauges the accuracy of other topology-based measures.

While the final infected ranges for the nodes vary with the parameters $\lambda$ and $\beta$ in the SIR model, the relative ranking of spreading efficiency of the nodes remains unchanged in a wide range of infection probabilities liu2015core . Thus, we take the recovered probability to be $\beta=1$ for simplicity. The infection probability $\lambda$ should be chosen more carefully. Too large an infection probability gives spreading efficiencies of the nodes that are too close to each other to distinguish their relative importance clearly. In the results that follow, we choose an infection probability $\lambda$ above the epidemic threshold that gives a final infected range that amounts to $1\%$ to $20\%$ of the system for most nodes kitsak2010identification .

II.3 The Kendall’s tau correlation and imprecision

function

Two figures of merit are used to quantify the performance of different topology-based measures for predicting the spreading efficiency of the nodes. The Kendall’s tau correlation coefficient measures the ranking consistency of two lists that rank the same set of objects. By referring to the number of concordant ranking pairs and the number of discordant ranking pairs in two ranking lists of $N$ objects, the correlation coefficient is evaluated by

[TABLE]

where sgn(x) is the sign function which returns $1$ if $x>0$, $-1$ if $x<0$, and 0 if $x=0$, and the summation is over all distinguished pairs $i$ and $j$. Here, $x_{i}$ is the rank of node $i$ in ranking list 1, while $y_{i}$ is the rank of node $i$ in ranking list 2. In the present context, list 1 is a topology-based ranking and list 2 is the SIR dynamics-based ranking. If $(x_{i}-x_{j})$ has the same sign as $(y_{i}-y_{j})$, the two lists give the same relative ranking of node $i$ and node $j$. Therefore, a large $\tau$ implies a more concordant relation between two methods of ranking the nodes.

For spreading processes, it is also important to quantify the accuracy in pinpointing the most influential spreaders. For a topology-based measure $\theta$, e.g. some kind of node centrality, let $M_{\theta}(p)$ be the average spreading efficiency of the $pN$ nodes carrying the highest measure $\theta$. Similarly, let $M_{eff}(p)$ be the average spreading efficiency of $pN$ nodes carrying the highest actual spreading efficiency according to the SIR dynamics. The imprecision function kitsak2010identification

[TABLE]

quantifies how close is the average spreading of the $pN$ nodes based on centrality ranking to the actual spreading. A smaller $\varepsilon_{\theta}$ represents a higher accuracy of $\theta$ in identifying the most influential spreaders.

III Dynamical importance of edges and weighted node centrality

The dynamical importance of an edge is analyzed by focusing on the spreading dynamics and the edge’s local structure. This leads to the necessity of assigning bi-directional and asymmetric weights to an edge. A novel node strength $s$ can then be defined to quantify the impact of a node on spreading. A $s$-shell decomposition method is proposed to be a reliable way of ranking the nodes for spreading processes.

III.1 Dynamical importance of edges

Figure 1 shows part of a network. When a disease originates from node $i$ and spreads along the edge $e_{ij}$, node $j$ will be infected first. Once node $j$ is infected, it could spread to other parts of the network through node $j$’s “out-reaching” edges, which are edges that connect node $j$ to nodes that are not in $i$’s neighborhood. The number of out-reaching edges from $j$ is denoted by $k_{j}^{out}$ and it is three in the example of Fig. 1. Note that $k_{j}^{out}$ should depend on the node $i$, as $j$ must be a neighboring node of $i$. In contrast, the edge $e_{ik}$ has zero out-reaching edge after it is infected by node $i$. Therefore, the edge $e_{ij}$ is expected to be more important in that it is more likely to lead to a larger infected area than confining the infection to node $i$’s neighborhood as $e_{ik}$ does  lawyer2015understanding ; liu2015improving . We are, therefore, motivated to introduce a new measure to distinguish the different importance of edges in a spreading process, even though the links may be unweighted in the construction of the network.

For our purpose, we define a weight $w_{ij}$ for an edge $e_{ij}$ by

[TABLE]

to represent its importance in a spreading process from node $i$ to node $j$. The first term stands for a basic and symmetric effect of an edge. The factor $k_{j}^{out}$ is included to reflect the potential impact of node $i$ through infecting its neighbor $j$. The product $k_{i}k_{j}^{out}$ include the degree of node $i$ into consideration. The idea can be illustrated by considering a leaf node (node of degree 1) connected to a hub (a node of large degree), the number of out-reaching links is very large for this leaf node. However, its impact is not necessarily high because only when itself and its neighboring hub are infected that the infection could spread to the other part of the network. The parameter $a$ serves to tune the contribution of $k_{i}k_{j}^{out}$ to the importance of edge $e_{ij}$.

The presence of $(k_{i}k_{j}^{out})^{a}$ emphasizes the asymmetric importance of an edge. The weight $w_{ij}$ is different from $w_{ji}$ for the same link connecting node $i$ and node $j$. From Eq. (3), $w_{ji}=1+(k_{j}k_{i}^{out})^{a}$, which measures the importance of the edge $e_{ji}$ when a spread goes from node $j$ to node $i$ along the link $e_{ji}$ and then move on to other parts of the network. Note that $w_{ij}\neq w_{ji}$ generally as they are defined by considering the neighborhoods of the neighbors of node $i$ and node $j$, respectively. Given a network, $w_{ij}$ and $w_{ji}$ can be evaluated entirely based on the network topology and they label every edge to better reflect the bidirectional and yet asymmetric contributions of the edge in spreading processes.

III.2 A novel node strength and $s$-shell decomposition

It will be advantageous to introduce a node-level quantity analogous to the degree to quantify the importance of a node in spreading dynamics. This will put the computational complexity at the same level as those based on the degree and $k$-shell decomposition. Motivated by the idea of weighted degree opsahl2010node that the strength of a node in a weighted graph is the sum of the weights of its edges, we define the strength $s_{i}$ of a node $i$ by

[TABLE]

where the summation is over the nodes $j$ belonging to the neighborhood $\Gamma_{i}$ of node $i$. Invoking $w_{ij}$ in the definition of $s_{i}$ makes it a better measure in quantifying a node’s importance in spreading dynamics.

We propose a $s$-shell decomposition method as an extension of the $k$-shell decomposition. The algorithm is as follows. With the strengths $s_{i}$ evaluated for all nodes, the algorithm starts with removing the nodes with the smallest strength $s_{m}$ and the links associated with the nodes. Let node $i$ be removed. The strength of its neighboring node $j$ is then updated to $s_{j}-w_{ji}$ as the edge $e_{ij}$ is removed. The network is then checked and the removal of nodes with $s_{m}$ continues until no nodes of strength less than or equal to $s_{m}$ remains. The deleted nodes are assigned a $s$-shell index of $s_{s}=1$, where the symbol emphasizes that the decomposition is based on the nodes’ strength and the subscript represents a shell. The trimming process is repeated for the nodes with the smallest strength among the remaining nodes and the nodes so removed are assigned the index $s_{s}=2$. This pruning process is continued until all nodes are removed and assigned a $s_{s}$ index. The $s$-shell index of a node can be regarded a weighted coreness of the node emphasizing its importance in spreading dynamics.

IV Performance in identifying influential spreaders in real-world networks

To examine the effectiveness of using the node strength and weighted coreness in identifying influential spreaders, we apply the measures to nine real-world networks as listed in Table 1.

The real networks studied are: (1) CA-Hep (giant connected component of collaboration network of arXiv in high-energy physics theory) leskovec2012 ; (2) Astro physics (collaboration network of astrophysics scientists) newman2001 ; (3) Emailcontact (email contacts at Computer Science Department of University College London) kitsak2010identification ; (4) PGP (an encrypted communication network) boguna2004 ; (5) Blog (the communication relationships between owners of blogs on the MSN (Windows Live) Spaces website) xie2006 ; (6) AS (Internet at the autonomous system level) newmandataas ; (7) Router (the router level topology of the Internet, collected by the Rocketfuel Project) spring2004 ; (8) Hamster (friendships and family links between users of the website hamsterster.com) hamster2014 ; and (9) Netsci (collaboration network of network scientists) newman2006 . The new measures are found to outperform predictions based on the degree centrality and $k$-shell decomposition, as we now show.

IV.1 Performance of node strength

From the structure of each network, every node carries a degree $k_{i}$ and a node strength $s_{i}$. Using the SIR dynamics, the spreading efficiency $M_{i}$ of each node can be obtained by simulations. Fig. 2 compares the correlations between the spreading efficiency with the node strength and with the degree in nine real-world networks. Here, we take $a=0.5$ in Eq. (3) in determining $w_{ij}$ for the edges. The sensitivity to the parameter $a$ will be discussed later.

The strength and the degree are both positively correlated with the spreading efficiency. The merit of using the strength over the degree as a measure is that their values cover a much wider range and they can distinguish the spreading efficiency more specifically. This advantage is built into the definition of the node strength as it captures the key elements in spreading dynamics.

The node strength provide a ranking of the nodes. This list can be compared with the list based on the actual spreading efficiency by calculating the Kendall’s tau correlation coefficient. We calculate the ranking correlation of nodes’ spreading efficiency and their strength for different values of $a$ and obtained $\tau(a)$, as shown in Fig. 3 (squares).

For $a=0$ (see Eq. (4)), $s_{i}$ reduces to the degree $k_{i}$ and thus $\tau(a=0)$ measures the correlation between the rankings based on the degree and the spreading efficiency. Note that $\tau$ is significantly enhanced for $a>0$, implying that the node strength, which includes the bi-directional and asymmetric weights of the edges, ranks the nodes more accurately. Results in Fig. 3 further show that there exists an optimal value of $a$ for each network at which $\tau$ is a maximum. The optimal value of each network is given in Table 1, together with the other network properties. Fig. 3 also shows the $\tau(a)$ obtained by ranking the nodes according to the $s$-shell index. The results will be discussed later.

Fig. 4 shows the imprecision function of the ranking based on the node strength, together with the results based on the degree and $k$-shell index for comparison. Recall that a lower imprecision implies a higher accuracy in identifying the influential spreaders. The node strength (triangles) give an imprecision which is less than 0.1 for all $p$ in nearly all cases. Only in the network Netsci, the imprecision is slightly larger than 0.1 for a few values of $p$. The results show that the node strength outperforms the degree (squares) in accuracy in almost all networks. Only in the network Hamster, the imprecisions based on node strength and on the degree become comparable but they are both small. The node strength is, therefore, a better index for pinpointing the influential spreaders than the degree. More noticeably is that the node strength performs even better than the $k$-shell index in most cases, except at some small values of $p$ in the AS and Netsci networks. The $k$-shell index is regarded as an efficient measure for identifying influential spreaders and it is widely used in ranking algorithms. However, the assignment of $k$-shell index requires a higher computational complexity and complete network structure than index relying solely on node-level quantities such as the degree or the node strength. The node strength introduced here does not only provide a more accurate measure, but also a computationally efficient method in handling large-scale networks.

IV.2 Performance of weighted coreness

The $k$-shell index works better than the degree in identifying influential spreaders liu2015improving . Here, we investigate how the $s$-shell index $s_{s}$ or weighted coreness works in comparison to the other measures. The results of the Kendall’s tau correlation of $s_{s}$ ranking in Fig. 3 suggest that it is a better measure than using the node strength in eight systems out of nine. In the Emailcontact network, $s_{s}$ and $s$ rankings work equally well. In fact, the $s_{s}$ and $s$ rankings approach the same value of $\tau$ as $a$ increases. Given that the optimal values of $a$ in the networks are less than or equal to $1$, the weighted coreness gives a better ranking. Note that the $a=0$ case gives the value of $\tau$ corresponding to the $k$-shell index $k_{S}$. Using $s_{s}$ to rank the nodes always gives a higher $\tau$ than the $a=0$ value, implying $s$-shell index is also a better measure than the $k$-shell index.

The imprecision functions of rankings using $s_{s}$ and $s$ are compared in Fig. 5. Their performances are comparable and they both work better than measures based on the degree alone (see Fig. 4). Looking closer, the lower imprecision of $s_{s}$ ranking in six (CA-Hep, PGP, Blog, AS, Router and Netsci) out of nine cases suggests that the $s$-shell decomposition method is more accurate in identifying influential spreaders in real-world networks. Even in the networks of Astro, Emailcontact and Hamster that $s_{s}$ and $s$ work almost equally well, the imprecision of $s_{s}$ is slightly lower or equal to that of $s$. Only in the Hamster network that $s$ works slightly better than $s_{s}$ at $p=0.01$, even so the imprecision functions are small (under 0.05) on the absolute scale.

IV.3 Robustness of proposed weighted centrality

So far, we have used the optimal value of $a$ to evaluate $w_{ij}$ and $s_{i}$, and compared results with other measures. However, the optimal value is not often known precisely in real applications. It will be useful to examine the performance of the node strength $s_{i}$ for some arbitrarily chosen value of $a$. Let us set $a=1/2$ so that the term $(k_{i}k_{j}^{out})^{a}$ in $w_{ij}$ represents a geometric mean. The comparison in Fig. 6 of the imprecision function shows that $s_{i}$ ranking gives a lower imprecision than the degree and $k$-shell index. An interesting point is that the imprecision of node strength evaluated at $a=1/2$ is even lower than that evaluated at the optimal value of $a$ in the AS network for $p<0.1$. The result indicates that although the best performing overall ranking correlation coefficient occurs at some optimal $a$, the same value does not necessarily give the best identification of the most influential spreaders.

Fig. 7 compares the effectiveness of the node strength and the $s$-shell decomposition for $a=1/2$. Again, the $s$-shell index works better in most cases. In fact, the results resemble those in Fig. 5 when the optimal $a$ is used. These results further support the assertion that the node strength and the corresponding $s$-shell index are better measures for spreading processes than methods based on the degree. Between them, the $s$-shell index performs slightly better, but evaluating the index requires more computing effort than the node strength alone.

V Conclusion

The roles of nodes and edges in deciding the structural properties of a network should be carefully distinguished from their roles in determining the extent of spreading processes. Although an edge between nodes $i$ and $j$ certainly helps spread a disease, its role may be different when the infection goes from $i$ to $j$ than in the other direction. It is because what matters is whether the node $j$, after infected by $i$, would reach out to other nodes that node $i$ itself could not reach. If so, the link carries a greater importance for infection from $i$ to $j$ which is quantified by a higher weight $w_{ij}$ for the link. It is, therefore, necessary to invoke asymmetric and bidirectional weights with $w_{ij}\neq w_{ji}$ for a link so as to capture the dynamics in spreading processes. Here, we introduced a form of $w_{ij}$ (see Eq. (3)) and showed that it facilitates more accuracy ranking in the node’s importance. Pictorially, the network is better described by the nodes connected by links with asymmetric weights in different directions when spreading dynamics is concerned.

To establish the effectiveness of our method, the weights of the links were used to construct a node strength $s$ that predicts the importance of a node in spreading processes. A $s$-shell decomposition scheme based on the node strength was then introduced. The $s$-shell index $s_{s}$ of the nodes provide another way of ranking them. Applying $s$ and $s_{s}$ rankings to nine real-world networks, it was found that our novel measures generally outperform the standard rankings based on the degree of the nodes and the $k$-shell decomposition method. Superiority is shown in both the overall performance of the ranking as indicated by Kendall’s tau correction coefficient and in identifying the influential spreaders as indicated by the imprecision.

The success of our measure relies on the asymmetry in the weights contained in $w_{ij}$ and $w_{ji}$. To stress the point, we constructed a related network with weighted links but the weights are symmetric by assigning a weight $w_{ij}^{\prime}$ to a link according to

[TABLE]

with $w_{ij}$ given by Eq. (3). The weights $w_{ij}^{\prime}$ can then be used to assign a strength $s^{\prime}$ to the nodes and a corresponding $s$-shell decomposition based on $s^{\prime}$ can be carried out to assign an index $s_{s}^{\prime}$ to each node. Fig. 8 compares the Kendall’s tau correlation of rankings based on $s_{s}$ and $s_{s}^{\prime}$ with the actual SIR spreading efficiency for different values of the parameter $a$. In all cases, the measure with asymmetric weights $s_{s}$ works better than that without the asymmetry. In the Emailcontact and Hamster networks, the two measures are equally accurate. The results confirm that it is important to include the different roles of a link in spreading a disease between nodes $i$ and $j$ in two different directions into the construction of a reliable measure.

In summary, we proposed a node strength as an alternative centrality measure for efficient and accurate identification of influential spreaders. The idea of examining the functionality of a link in spreading in either directions is a general one and thus could be further developed for ranking a set of objects. We used the SIR model as the spreading dynamics. However, the idea of invoking asymmetric weights $w_{ij}\neq w_{ji}$ for a link remains valid for other processes such as rumor spreading and information diffusion, although the exact form of the weights may depend on the details of the process under consideration.

Acknowledgements.

This work was jointly funded by the National Natural Science Foundation of China (Grant No. 11575041, 61672238), the Scientific Research Starting Program of Southwest Petroleum University (No. 2014QHZ024), the Data Intelligence Academic Innovation Team of Southwest Petroleum University (2015CXTD06) and the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2015J153).