A New K-Shell Decomposition Method for Identifying Influential Spreaders of Epidemics on Community Networks

Kai GONG; Li KANG

doi:10.21078/JSSI-2018-366-10

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Journal of Systems Science and Information ›› 2018, Vol. 6 ›› Issue (4) : 366-375. DOI: 10.21078/JSSI-2018-366-10

A New K-Shell Decomposition Method for Identifying Influential Spreaders of Epidemics on Community Networks

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Abstract

An efficient method for the identification of influential spreaders that could be used to control epidemics within populations would be of considerable importance. Generally, populations are characterized by its community structures and by the heterogeneous distributions of out-leaving links among nodes bridging over communities. A new method for community networks capable of identifying influential spreaders that accelerate the spread of disease is here proposed. In this method, influential spreaders serve as target nodes. This is based on the idea that, in k-shell decomposition method, out-leaving links and inner links are processed separately. The method was used on empirical networks constructed from online social networks, and results indicated that this method is more accurate. Its effectiveness stems from the patterns of connectivity among neighbors, and it successfully identified the important nodes. In addition, the performance of the method remained robust even when there were errors in the structure of the network.

Key words

complex networks / community structure / k-shell decomposition / influential spreaders

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Kai GONG , Li KANG. A New K-Shell Decomposition Method for Identifying Influential Spreaders of Epidemics on Community Networks. Journal of Systems Science and Information, 2018, 6(4): 366-375 https://doi.org/10.21078/JSSI-2018-366-10

1 Introduction

Epidemics can lead to serious loss of life and they have huge an impact on the economy^[1], as witnessed during the 2003 outbreak of severe acute respiratory syndrome (SARS)^{[2, 3]}, the 2009 outbreak of H1N1 influenza A virus^[4], and the 2013 outbreak of H7N9 Influenza A virus^[5]. Knowledge regarding the pathways by which diseases spreading through networks and how this network might be used to prevent epidemics is of great importance. This issue has attracted a great deal of attention from researchers across various fields^{[6, 7]}. Identifying the influential spreaders that can hinder the spread of disease effectively so as to suppress outbreaks remains an open issue^[8].

Hubs, individuals who have high centrality in networks, are commonly believed to be the most influential nodes in the spreading process because they can affect many neighbors^[9–11]. In the case of networks with broad-degree distribution^[12], the degree for the well-connected individuals has been shown to be an efficient method of identifying efficient spreaders^{[9, 10]}, Betweenness is another centrality. It involves measuring the number of shortest paths that cross the current node. It has been used to determine who has the most influence on others in networks^{[11, 13]}. However, Kitsak, et al. pointed out that the most efficient spreaders are those located within the core of a network as targeted by the k-shell decomposition method^[6]. This method is based on iterative pruning of nodes with degree smaller than or equal to the

k_{c o r e}

index of the current layer until each node is associated with

k_{c o r e}

index that reflects the core or periphery layer in network^[14].

Community structure^[15] is ubiquitous in complex networks^[16], such as Facebook^[17] and Twitter^[18]. It serves an important function in the dynamics of epidemic^[19–21]. In the presence of community structures, heterogeneous distribution of out-leaving links was observed among real networks, such as air traffic networks^[22], social networks^[23], and communication networks^[24]. The out-leaving links connecting a pair of nodes belonging to different communities have been found to provide shortcuts from one community to another^[25–27]. These links have been proven to be more efficient in diffusing diseases through network^[28]. Identifying the influential spreaders in community networks is quite challenging because the mesoscopic features of the community structure are complex. Here, a k-shell with community method is proposed. This new method, based on the idea of the k-shell decomposition process, involves identifying the influential spreaders using two types links. Results demonstrated that the proposed method performs better in empirical networks than degree, betweenness, and k-shell decomposition strategies do. Simulation also shows that this method has the merit of being significantly robust against noise.

2 Data and Model

The present paper compares results based on the current method and identifying efficient spreaders in empirical networks within epidemiological models. First, details are given with respect to the following issues: Network construction and dynamic model.

2.1 Network Construction

Empirical networks are constructed using online collegiate social data from Facebook (https://code.google.com/p/socialnetworksimulation). Data from universities in U.S. was studied here. It includes anonymous data from students from Caltech, Princeton, Georgetown, and the University of Oklahoma. The data concern the dormitories, majors, and year for each individual, and dormitories were found to be key elements in the social organization of large universities^[17]. Based on these data, the networks were constructed by linking up pairs of individuals who (i) were online friends and lived in the same dormitory, or (ii) they lived in different dormitories but had the same major and year. The giant network was then used for the present study. Basic statistical properties of networks are given in Table 1. In here, degree heterogeneity was calculated using the equation proposed by Hu^[29], and, for every network, community structure was detected by Louvain algorithm^[30]. All networks exhibited small-world characteristics with high clustering coefficient and short average path length. They also had high modularity.

Table 1 Structural properties of networks

Network	N	E	< k >	k_max	< d >	C	r	H	G	Q
Caltech	571	7127	24.96	96	2.96	0.57	0.01	0.36	9	0.79
Princeton	3975	23457	11.80	129	4.721	0.32	0.41	0.43	30	0.74
Georgetown	6309	73022	23.14	311	4.21	0.25	0.24	0.45	16	0.68
Oklahoma	6850	152985	44.66	247	4.36	0.52	0.48	0.51	42	0.92

Structural properties including the network size (N), number of edges (E), average degree (

< k >

), max degree (

k_{max}

), average shortest path length (

< d >

), clustering coefficient (C), degree assortativity (r), degree heterogeneity (H), the number of communities (G), and modularity (Q) are tabulated for each networks. These networks include data from the California Institute of Technology, Princeton University, Georgetown University, and the University of Oklahoma.

2.2 Dynamic Model

In the susceptible-infected-recovered (SIR) model, each node in each network represented an individual who could be in one of three states: susceptible, infected, or recovered, and each link between nodes represented one connection that could spread an infection. Initially, all nodes were susceptible. To initiate an infection, one node was randomly chosen and considered infected. In every time step, each infected individual randomly contacts two neighbors, and

β

= 0.08 is the probability of a susceptible neighbor would be infected. The probability that an infected node would recover was

γ

= 0.2. Once an individual was recovered, there would be no further change. In the simulation, states of every node were updated synchronously. The dynamics ended when all infected recovered. The average size of the infected, M, and the fraction of the population ever infected at the end of the epidemic were recorded, allowing quantification of the influence of given node on spreading process.

3 Methods

3.1 Identification Methods

The ideas behind degree (

k

), betweenness (

c b

), and k-shell decomposition method (

k_{c o r e}

) are outlined, and the current method is discussed. Briefly, degree was based on the idea that most influence nodes would be those with the largest number of connections, and it is one measure of local influence: only the structure around the node has to be considered^[31]. Betweenness measures the number of shortest paths from all nodes to others that cross through that node. Kitsak, et al. argued that the structure of network organization serves an important function such that there are plausible circumstances under which the most degree or highest betweenness as influential spreaders have the least pronounced impact on the spreading process^[6]. K-shell decomposition is one method based on iteratively pruning of nodes with degree no more than

k_{c o r e}

of the current layer. The highest

k_{c o r e}

index is closely related to the concept of most influential nodes on spreading process. They used the method by identifying the core and periphery of given node in real network to identify key spreaders and found that k-shell decomposition method is more accurate.

3.2 A New k-Shell Decomposition Method

The k-shell with community method is here proposed as a means of more effective identification. It is based on the idea that k-shell decomposition involves both out-leaving links and inner links. This method starts with successive pruning of the network by removing nodes with two types links separately. This process has three main components: (ⅰ) Removal of nodes with out-leaving links, (ⅱ) removal of nodes with inner links, and (ⅲ) assignment of values.

(ⅰ) After initialization of networks by removal of all nodes with inner links, all nodes with out-leaving links

k^{o}

= 1 among nodes bridging over communities were then removed, and some nodes with out-leaving links may have remained, so we continued pruning the network repeatedly until there was no node left with

k^{o}

= 1 in the network. These nodes are associated with an index

k_{c o r e}^{o}

= 1. In a similar step in the original work, the next level,

k^{o}

= 2, was iteratively removed and the system continued removing higher

k^{o}

until all nodes were associated with an index of

k_{c o r e}^{o}

(ⅱ) After initialization of networks by removal of all out-leaving links, a procedure analogous to previous component was repeated by removing nodes with inner links

k^{i}

= 1 until there were no nodes left with

k^{i}

= 1. These nodes are associated with an index of

k_{c o r e}^{i}

= 1. The procedure was repeated until each node was associated with an index

k_{c o r e}^{i}

(ⅲ) Finally, the values

k_{c o r e}^{m}

was assigned to each node. It can be defined using the following equation:

\begin{array}{rcl} k_{c o r e}^{m} (j) = \frac{k_{c o r e}^{i} (j) + k_{c o r e}^{o} (j)}{2} . \end{array}

(1)

4 Results

4.1 Heterogeneity in Empirical Networks

To illustrate the necessity of an identification method that focuses on the most efficient spreaders in networks using heterogeneous distributions, the distributions of the cumulative probability density were analyzed are by out-leaving links and inner links in real networks are denoted (Figure 1). The phenomenon of heterogeneity indicates that nodes have significantly different according to their pattern. This phenomenon also indicates the striking influence that heterogeneity has on the spreading process.

Figure 1 Cumulative distribution of out-leaving links and inner links in empirical networks

Full size|PPT slide

In Figure 1, for every network, community structure was detected by Louvain algorithm in ^[29]. The out-leaving links and inner links were then identified, and the number of out-leaving links

k_{o}

and inner links

k_{i}

emanating from each node were recorded to give the cumulative distribution. Empirical results are shown for Caltech (squares), Princeton (circles), Georgetown (upward-facing triangles), and the University of Oklahoma (downward-facing triangles).

4.2 Comparison of Spreading Efficiency

To compare the efficiency of

k

c b

, and

k_{c o r e}

with that of our method, the SIR was performed on empirical networks. The imprecision function

ε

, in the previous work^[6], quantifies the difference between the average infected between the fN nodes (0

<

<

1) with the highest

k

c b

k_{c o r e}

k_{c o r e}^{m}

, and the average infected of the fN most efficient spreaders. For a given fraction f the set of the fN most efficient spreaders was first identified as measured by

M_{e f f}

. (designated

F_{e f f}

). Similarly, the fN nodes with the highest

k_{c o r e}^{m}

(

F_{k_{c o r e}^{m}}

) were identified. In this way, the imprecision of

k_{c o r e}^{m}

can be defined as follows:

\begin{array}{rcl} ε_{k_{c o r e}^{m}} = 1 - \frac{M_{k_{c o r e}^{m}}}{M_{e f f}} . \end{array}

(2)

Here,

M_{k_{c o r e}^{m}}

and

M_{e f f}

are the average infected percentages over the

F_{k_{c o r e}^{m}}

and

F_{e f f}

sets of nodes, respectively.

ε_{k_{c o r e}^{m}}

approaches 0, an indication that the highest nodes are chosen by the strategy and are usually those that contribute the most to epidemics, and vice versa.

ε_{k}

ε_{c b}

, and

ε_{k_{c o r e}}

are defined similarly to

ε_{k_{c o r e}^{m}}

. Here, the differences

Δ ε_{x}

(

x

denoting

k

c b

k_{c o r e}

) would indicate the spreading efficiency of strategy relative to the proposed method in the fN set can be defined using the following equation:

\begin{array}{rcl} Δ ε_{x} = \frac{ε_{x} - ε_{k_{c o r e}^{m}}}{ε_{x}} . \end{array}

(3)

In most cases, the differences of

Δ

are positive over almost all of the set of different strategies (Figure 2). For example,

k_{c o r e}^{m}

was on average 7.66% (40.43%, 55.06%) higher than

k_{c o r e}

(

k

c b

) for the Caltech data set, 5.28% (12.91%, 42.19%) higher for the Princeton set, 5.66% (17.88%, 41.65%) higher for the Georgetown set, and 4.93% (15.48%, 40.34%) higher for the University of Oklahoma set.

Figure 2 Comparison of spreading efficiency of identification in empirical networks

Full size|PPT slide

In Figure 2, the difference in imprecision

Δ ε_{x}

, x denoting

k

(left panel),

c b

(middle panel), and

k_{c o r e}

(right panel), are shown for each networks as function of f. The positive percentages shows that

k_{c o r e}^{m}

is more accurate. Results are obtained by averaging over 2000 for each node.

4.3 Comparison of Average Out-Leaving Links Among Neighbors

As spreaders, whether it has influence on spreading process is closely related to its pattern of connections in speeding up the transmission of epidemics. The effectiveness of

k_{c o r e}^{m}

is illustrated in Figure 3, which compares the out-leaving links among neighboring nodes identified using different strategies in networks. Here, the set

Γ

is the union of neighbors of nodes those identified by method. Let

< k^{o} >

(

Γ_{x}

), with x denoting method, be the average out-leaving links within the union

Γ

after applying said method. The difference

Δ k^{o}

is the measure of how effective

k_{c o r e}^{m}

identify bridge hubs within networks. This can be defined using the following two equations:

\begin{array}{rcl} Δ k^{o} =< k^{o} > (Γ_{k_{c o r e}^{m}}) - < k^{o} > (Γ_{x}) . \end{array}

(4)

Figure 3 Comparison of average out-leaving links among neighbors of nodes identified by $k_{c o r e}^{m}$ with those others

Full size|PPT slide

As shown in Figure 3,

Δ k^{o}

in almost all the cases indicates that nodes identified by

k_{c o r e}^{m}

have more extensive connections of neighbors with well-connected than other strategies do, improving the effectiveness of identification for efficient spreaders. But, in Oklahoma case also shows that nodes identified by

k_{c o r e}^{m}

have comparable out-leaving links of neighbors with

k_{c o r e}

, that make the curves are fluctuated in the right rows of Figure 2, even

k_{c o r e}^{m}

is still better than

k_{c o r e}

in most cases.

In Figure 3, the difference

Δ k^{o}

in out-leaving links with x denoting

k

(circles),

c b

(upward-facing triangles) and

k_{c o r e}

(squares) of neighbor set

Γ

are shown for different f in empirical networks.

4.4 Robustness of the Method

Another important aspect of an effective method is the robustness to missing or noise information. Such errors are common in real networks due to, for example, the inconsistency with which two individuals describe their relationship^[32].

Using empirical networks, the incomplete information by randomly removing the percentage of connections, then have tested the robustness by using Kendall rank correlation coefficient

τ

(

- 1 \leq τ \leq

1)^[33], which measures the similarity of the ordering of nodes when ranked by on the incomplete and original network. Using empirical networks, the

k_{c o r e}^{m}

values re-calcuted after removing the percentage of connection, then have tested the robustness by using Kendall correlation coefficient. This can be defined using the following equation:

\begin{array}{rcl} τ = \frac{n u m b e r o f c o n c o r d a n t p a i r s - n u m b e r o f d i s c o r d a n t p a i r s}{N (N - 1) / 2} . \end{array}

(5)

τ

similar to 1 denotes an exact agreement between two ranks for both elements, and vise-verse. The robustness of the proposed method is indicated by the relatively higher

τ

values in Figure 4. In most cases, both the proposed method and

k_{c o r e}

yield comparable robust results, and

k

and

c b

both perform poor.

Figure 4 Robustness of the methods in networks with noise as modeled by random removal of connections

Full size|PPT slide

In Figure 4, results are shown for

k_{c o r e}^{m}

(squares),

k_{c o r e}

(circles),

k

(upward-facing triangles) and

c b

(downward-facing triangles), by averaging over 2000 realizations.

5 Conclusions

The heterogeneity distribution of out-leaving links and inner links under real-world conditions is important to identifying those nodes which are most influential spreaders in the transmission of diseases, but these nodes are difficult to identify in community networks. In summary, the new k-shell decomposition method proposed here was studied for effectiveness. There is one method for identifying nodes. This method is based on the idea that k-shell decomposition involves both out-leaving links and inner links. The current method was used to empirically tested networks among students in U.S. universities constructed by using online social networks. In most cases, the current method were found to be more effective in identifying influential spreaders than degree, betweenness, and k-shell for the range of the set, and results were more accurate. Its effectiveness can be attributed to the neighbors identified by our method because the pattern of connections among neighbors was more striking than those produced using other strategies. For this reason, it can spread through the entire network more quickly and effectively. The current method kept its stability well throughout the missing processes, and has also been strongly performed robustness.

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Funding

Fundamental Research Funds for the Central Universities(JBK170133)

Natural Science Foundation of Sichuan Province of China(17ZB0434)

Ministry of Education of Humanities and Social Science Foundation of China(11XJCZH002)

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Received	Accepted	Published
2016-09-23	2018-03-06	2018-08-25
Issue Date
2018-08-25

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Abstract

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1 Introduction

2 Data and Model

2.1 Network Construction

Table 1 Structural properties of networks

2.2 Dynamic Model

3 Methods

3.1 Identification Methods

3.2 A New k-Shell Decomposition Method

4 Results

4.1 Heterogeneity in Empirical Networks

Figure 1 Cumulative distribution of out-leaving links and inner links in empirical networks

4.2 Comparison of Spreading Efficiency

Figure 2 Comparison of spreading efficiency of identification in empirical networks

4.3 Comparison of Average Out-Leaving Links Among Neighbors

Figure 3 Comparison of average out-leaving links among neighbors of nodes identified by $k_{c o r e}^{m}$ with those others

4.4 Robustness of the Method

Figure 4 Robustness of the methods in networks with noise as modeled by random removal of connections

5 Conclusions

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References

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Abstract

Key words

Cite this article

1 Introduction

2 Data and Model

2.1 Network Construction

Table 1 Structural properties of networks

2.2 Dynamic Model

3 Methods

3.1 Identification Methods

3.2 A New k-Shell Decomposition Method

4 Results

4.1 Heterogeneity in Empirical Networks

Figure 1 Cumulative distribution of out-leaving links and inner links in empirical networks

4.2 Comparison of Spreading Efficiency

Figure 2 Comparison of spreading efficiency of identification in empirical networks

4.3 Comparison of Average Out-Leaving Links Among Neighbors

Figure 3 Comparison of average out-leaving links among neighbors of nodes identified by kcorem with those others

4.4 Robustness of the Method

Figure 4 Robustness of the methods in networks with noise as modeled by random removal of connections

5 Conclusions

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References

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Footnotes

Funding

Figure 3 Comparison of average out-leaving links among neighbors of nodes identified by $k_{c o r e}^{m}$ with those others