Discovering critical nodes in social networks has many important applications. For finding out the critical nodes and considering the widespread community structure in social networks, we obtain each node’s marginal contribution by Owen value. And then we can give a method for the solution of the critical node problem. We validate the feasibility and effectiveness of our method on two synthetic datasets and six real datasets. At the same time, the result obtained by using our method to analyze the terrorist network is in line with the actual situation.
It is a basic process that happened in the network for the spread, diffusion, and cascade behavior of information. Considering that we plan to introduce new products, we can use the network feature which is called “wordofmouth” or “viral marketing.” That is, we may find out some individuals with influence and let them recommend the product to their friends so that such a cascade spreads by the greatest extent in the people. How to choose these influential individuals is called critical node problem (CNP). An effective solution for the problem has an important practical value [
This paper gives a solution for the CNP, which assigns a marginal contribution for every node in a community of social networks using the solution concept and union concept of cooperative games. Then we sort all nodes by their contribution and obtain critical nodes according to some rules. The rest of the paper is organized as follows. Section
The models for information propagation on networks have been widely studied [
A network is modeled as a graph
In this model, a propagation probability
In this model, vertex
The difference between ICM and LTM is that each attempt of activation is independent of the attempts by all the other active individuals while in the later model each inactive individual is influenced by the aggregated weight of all its active neighbors.
Given
Many statistical properties for social network analysis have been presented in the complex network theory, such as degree, clustering coefficient, and betweenness [
Domingos and Richardson firstly studied the CNP as an algorithmic problem [
(1)
(2) for
(3)
(4)
(5) end for
There is a key problem how to compute the value of
However, the method’s efficiency severely restricts its scalability. Leskovec et al. proposed an optimized method referred to as the costeffective lazy forward (CELF), which can speed up the above greedy algorithm [
Given a finite set of players
However, the Shapely value does not consider the impact of coalition structure and Owen extends it [
Assume that
The game
The idea of the greedy algorithm is to find a node with the greatest influence during an iteration based on the diffusion model of social networks. In nature, the node with the maximum marginal contribution is chosen. Because the community structure is prevalent in the social networks [
A network is a list of which pairs of players are linked to each other. The network structure is the key determinant of the level of productivity or utility to the society of players involved. A network game consists of a set of players and a value function. The value function assigns a real value to each possible network on all players. An allocation rule is a way to allocate the real value generated by a set of players and has to take into account the marginal value of a player. If we take the value function as the characteristic function, then network game can be seen as the cooperative game with transferable utility [
We define the cooperative game
So, we can use the Owen value to obtain the marginal contribution of every node. Because the Owen value can be seen as a twostep procedure in which the Shapely value is applied twice, we firstly compute a node’s Shapely value.
Given a node
Note that this method must work with
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for
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end
end
for
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end
Top
AsceSort(
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while
end
According to the description in Section
We use the CNM algorithm [
We consider the computational complexity for
We validate our method on two synthetic datasets and six real network datasets. All experiments are executed in the PC with 3.2 GHz CPU, 4 G memory, and Windows 7. The development tools are MATLAB 2009 and Microsoft Visual Studio 2010.
We compare our method (the Ov algorithm) with the Shapley valuebased algorithm (the Sv algorithm), the greedy algorithm (the greedy algorithm), and the degreeheuristic algorithm (the degree algorithm). The greedy algorithm is as a benchmark for measuring other algorithms; the degree algorithm selects
The
Datasets 


BA  500 
FF  500 
DBLP  10000 
5000  
Enron  4000 
YouTube  3500 
AS  1000 
Power grid  500 
We consider the influence of community structure on the Ov algorithm.
We use BA model [
We, respectively, compute the Shapely value and Owen value of every node in the BA and FF datasets and obtain the initial set and the number of activated nodes after running the ICM. The process is repeated 100 times and the average number of the nodes activated by the initial set with different size is drawn in Figure
The influence of community structure on our method.
BA
FF
The real datasets used by the paper include DBLP, Facebook, Enron, Youtube, AS, and PG, where the former four datasets have obvious community structure and the latter two datasets have not. The DBLP dataset [
We, respectively, use the greedy, Ov, Sv, and degree algorithms to find out the initial sets from above six real datasets. Figure
Performance analysis.
DBLP
Enron
YouTube
AS
PG
We discuss our method’s time efficiency.
We generate three datasets FF_{0.35,0.2} (sparse graph), FF_{0.37,0.32} (densifying graph), and FF_{0.38,0.35} (dense graph) with 1000 nodes using the FF model with three groups of parameters. Then we find out the top20 critical nodes on these datasets by the Ov and greedy algorithms and plot the running times in Figure
Time efficiency analysis.
We give an example of the Ov algorithm.
Krebs studied the terrorist network in the event of September 11, 2001. Figure
The hijackers on different airlines.
Airlines  Hijacker’s name 

American Airlines 11  Mohamed Atta, Waleed M. Alshehri, Wail Alshahri, Satam alSuqami, 


American Airlines 77  Khalid alMidhar, Majed Moqed, Salem Alhamzi, 


United Airlines 93 



United Airlines 175  Marwan alShehhi, Fayez Ahmed, Ahmed Alqhamdi, Hamza Alghamdi, 
The terrorist network.
For solving the CNP, this paper presents a method based on the Owen value from cooperative game, which considers the widespread community structure in social networks. We validate the proposed method on two synthetic datasets and the results show that our method is more suitable for the networks with community structure. Compared with other algorithms on six real datasets, our method is more effective. How to further improve the time efficiency of the Ov algorithm needs to be studied.
The author declares that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Social Science Foundation of China (nos. 11BFX125 and 13CFX049).