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the total number of republishing nodes and impede the information from further spreading in the network. As to the case of selecting strong ties preferentially, although it will tend to select the nodes with higher degrees to republish, the local trapping [19] will limit the scope of selected nodes into some local areas and make it harder to propagate the information further in the network. 

To validate the analysis above, we also observe the frac- tion of the nodes that have published I during the diffu- sion, denoted by f$_{pub}$ . As shown in Fig. 4, f$_{pub}$ increases more slowly when α = − 1, and the time-varying proper- ties of f$_{pub}$ are similar to those of C in Fig. 3 for different α values, respectively. We also monitor the fraction of the nodes that have published I in each hop away from the source node, denoted by f$_{local}$ . As shown in Fig. 5, when α = − 1, f$_{local}$ decreases faster than other cases, in particular the α = 0 case. It means when α = − 1, the number of republishing nodes selected from the neigh- borhood decreases sharply as the information spreading far away from the source, which agrees with our former analysis. As for the case of α = 1, f$_{pub}$ increases more and more slowly during the diffusion, because the nodes selected to republish are trapped in some local clusters. In other words, it is hard to find some new nodes to re- publish the information to the outer space. 

Based on the above results, we can conclude that se- lecting weak ties preferentially as the path to republish information cannot make it diffuse faster. However, this does not mean that weak ties play a trivial role in the information diffusion in online social networks, especially when we recall its special role in the network structure in Section III. Let α = 0 in ID ( α, β ), we compare the variation of C under the situation of removing weak ties first with that of removing strong ties first. As shown in Fig. 6, for the case of removing weak ties first, the cover- age of the information decreases rapidly, e.g., from 0.8 to 0.4 in Facebook when the fraction of removed weak ties reaches about 0.4. This implies that weak ties are indeed crucial for the coverage of information diffusion in online social networks. 

To further study the effect of β , we conduct experi- ments with different β values, as shown in Fig. 7. As can be seen, no matter what the β value is, random selection ( α = 0) is still the fastest mode for the information dif- fusion, although the gap tends to shrink with higher β values. It is also shown that when β grows, C will also rise for all α values. That is, the greater the strength of the information is, the more nodes will be attracted to republish it, and the wider it will spread in the network. 

Until now we can conclude that weak ties play a subtle role in the information diffusion in online social networks. On one hand, they are bridges that connect isolated com- munities and break through the trapping of information in local areas [19]. On the other hand, selecting weak ties preferentially as the path of republishing cannot make the information diffuse faster and wider. 

The growing popularity of the online social networks does not mean that it is safe and reliable. On the con- trary, the virus spread and the private information diffu- sion have made it become a massive headache for IT ad- ministrators and users [31, 32]. For example, “KooFace” is a Trojan Worm on Facebook, which spreads by leaving a comment on profile pages of the victim’s friends to trap a click on the malicious link [33]. About 63% of system administrators worry that their employees will share too much private information online [34]. So as time goes by, it becomes more and more important and urgent to control the virus spread and the private information dif- fusion in online social networks. 

In the light of this, we can make use of the weak ties for the information diffusion control. That is, in the real- world practices, we can assume that the behavior of re- publishing information is random, i.e., α = 0. Then ac- cording to the results in Fig. 6, we can make the virus or the private information trapped in local communities by removing weak ties and stop them from diffusing further 

## V. DIFFUSION CONTROL 

