^{1,2,a)}, Jinde Cao

^{1,b)}, Jianquan Lu

^{1,3,c)}and Jürgen Kurths

^{3,4,5,d)}

### Abstract

In this paper, we present an efficient opinion control strategy for complex networks, in particular, for social networks. The proposed adaptive bridge control (ABC) strategy calls for controlling a special kind of nodes named bridge and requires no knowledge of the node degrees or any other global or local knowledge, which are necessary for some other immunization strategies including targeted immunization and acquaintance immunization. We study the efficiency of the proposed ABC strategy on random networks, small-world networks, scale-free networks, and the random networks adjusted by the edge exchanging method. Our results show that the proposed ABC strategy is efficient for all of these four kinds of networks. Through an adjusting clustering coefficient by the edge exchanging method, it is found out that the efficiency of our ABC strategy is closely related with the clustering coefficient. The main contributions of this paper can be listed as follows: (1) A new high-order social network is proposed to describe opinion dynamic. (2) An algorithm, which does not require the knowledge of the nodes’ degree and other global/local network structure information, is proposed to control the “bridges” more accurately and further control the opinion dynamics of the social networks. The efficiency of our ABC strategy is illustrated by numerical examples. (3) The numerical results indicate that our ABC strategy is more efficient for networks with higher clustering coefficient.

The social network refers to the relatively stable relation system grounded upon the interactions between social individual members. This realm is concerning with the interactions and relations among social individuals as well as its impact exerted upon human social behaviors. In the actual social network, some insignificant gossip or thoughts, if not properly controlled, would eventually erupt on a large scale or even headline the whole network. If some control strategies against the gossip spread on the internet could be carried out based upon further understanding of its internal mechanism, its potential application value would be enormous. In this paper, we bring forward the adaptive bridge control (ABC) strategy that could control the opinion evolution without the overall or even partial information and perform very well in the numerical experiment. It is believed that this strategy is feasible, economic, and highly effective in real-world applications, especially for the real social networks bearing high clustering coefficient.

The work of J. Cao was supported by the National Natural Science Foundation of China under Grant Nos. 11072059 and 60874088, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20070286003, and the Natural Science Foundation of Jiangsu Province of China under Grant No. BK2009271. The work of J. Q. Lu was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11026182, the Natural Science Foundation of Jiangsu Province of China under Grant No. BK2010408, the Innovation Fund of Basic Scientific Research Operating Expenses of Southeast University under Grant No. 3207010501, Program for New Century Excellent Talents in University (NCET-10-0329), and the Alexander von Humboldt Foundation of Germany. The work of J. Kurths was supported by SUMO (EU) and ECONS (WGL).

I. INTRODUCTION

II. ABC STRATEGY FOR OPINION EVOLUTION

A. Traditional control strategies

B. Our proposed control strategies

III. COMPARISON AMONG RANDOM NETWORK, SCALE-FREE NETWORK AND SMALL-WORLD NETWORK

IV. APPLICATION SCOPE OF OUR STRATEGY

V. CONCLUSION

### Key Topics

- Networks
- 40.0
- Cluster analysis
- 28.0
- Network topology
- 24.0
- Social networks
- 15.0
- Topology
- 5.0

## Figures

(Color online) A schematic representation of a network with bridge structure. Six star points connect two tight networks with larger intensity, and the two big red nodes denote the nodes with high degree.

(Color online) A schematic representation of a network with bridge structure. Six star points connect two tight networks with larger intensity, and the two big red nodes denote the nodes with high degree.

(Color online) Comparison of the controlling effect. The changes of average win percentage of opinion 1, as one increases the probability *p* that *a* vacillating node is controlled. The three curves are, respectively, for ER random networks, BA scale free networks, and WS small world networks. The network size is 1000, the average degree of all the networks is 6, and all the data are the average of 1000 independent experiments.

(Color online) Comparison of the controlling effect. The changes of average win percentage of opinion 1, as one increases the probability *p* that *a* vacillating node is controlled. The three curves are, respectively, for ER random networks, BA scale free networks, and WS small world networks. The network size is 1000, the average degree of all the networks is 6, and all the data are the average of 1000 independent experiments.

(Color online) Comparison of the number of controlled nodes. In x-axis, *p* is the control probability, y-axis denotes the controlled proportion of all nodes. The network size is 1000, the average degree of all the networks is 6, and all the data are the average of 1000 independent experiments.

(Color online) Comparison of the number of controlled nodes. In x-axis, *p* is the control probability, y-axis denotes the controlled proportion of all nodes. The network size is 1000, the average degree of all the networks is 6, and all the data are the average of 1000 independent experiments.

(Color online) Visualization-based simulation results. The network has already been occupied by opinion 1, and the green nodes are the nodes which have been controlled during the evolution. We have emphasized them by blue circles.

(Color online) Visualization-based simulation results. The network has already been occupied by opinion 1, and the green nodes are the nodes which have been controlled during the evolution. We have emphasized them by blue circles.

(Color online) Comparison of cost and effect. The figure reflects the performance price ratio of our ABC strategy in different kinds of networks. The network size is 1000, the average degree of all networks is 6, and all the data are the average of 1000 independent experiments.

(Color online) Comparison of cost and effect. The figure reflects the performance price ratio of our ABC strategy in different kinds of networks. The network size is 1000, the average degree of all networks is 6, and all the data are the average of 1000 independent experiments.

(Color online) Edge exchanging method: for any network, randomly pick a pair of edge (AB and CD in graph (a), for example) then rewire to have different end nodes (AC and BD as in (b) and AD and BC is also ok). This edge exchanging method can keep each nodes unchanged.

(Color online) Edge exchanging method: for any network, randomly pick a pair of edge (AB and CD in graph (a), for example) then rewire to have different end nodes (AC and BD as in (b) and AD and BC is also ok). This edge exchanging method can keep each nodes unchanged.

(Color online) The relationship between the clustering coefficient and the controlling efficiency. The clustering coefficient here is adjusted by the edge exchanging method, so that the degree of each node will remain the same. The network size is 200, the average degree of all networks is 6, and all the data are the average of 250 independent numerical simulations.

(Color online) The relationship between the clustering coefficient and the controlling efficiency. The clustering coefficient here is adjusted by the edge exchanging method, so that the degree of each node will remain the same. The network size is 200, the average degree of all networks is 6, and all the data are the average of 250 independent numerical simulations.

## Tables

Comparison of clustering coefficient.

Comparison of clustering coefficient.

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