3 FIG. 2: (Color online) The variations of ¯ S and f$_{GCC}$ during the removal of weak ties first and strong ties first  respectively. f$_{r}$ is the fraction of removed ties. formation diffusion in online social networks is different from traditional models  such as SIS  SIR and random walk. We start by discussing the procedure of informa- tion diffusion in online social networks. nected cluster (GCC)  denoted by f$_{GCC}$   shows differ- ent dynamics between the removals of weak ties first and strong ties first. We denote the critical fractions of the re- moved ties at the phase transition point by f$_{c}$ . It is inter- esting to note that f$_{c}$ = 0 . 753 for YouTube and f$_{c}$ = 0 . 890 for Facebook when ¯ S reaches the submit  which are very close to the case when f$_{GCC}$ ≈ 0. In the percolation theory  the existence of the above phase transition means that the network is collapsed  while the network is just shrinking if there is no phase transition when removing the ties [19]. So the above ex- periments tell us that weak ties play a special role in the structure of online social networks  which is different from the one strong ties play. In fact  they act as the important bridges that connect isolated communities. In what follows  we build a model that associates the weak ties with the information diffusion  to discuss the coupled dynamics of the structure and the information diffusion. The procedure of the diffusion in online social networks can be briefly described as follows: The information diffusing in online social networks in- cludes blogs  photos  messages  comments  multimedia files  states  etc. Because of the privacy control and other features of online social sites  the mechanism of the in- It is easy to find that the key feature of the informa- tion diffusion in online social networks is that the infor- mation is pushed actively by the site and only part of • The user i publishes the information I   which may be a photo  a blog  etc. • Friends of i will know I when they access the profile page of i or get some direct notifications from the online social site. We call this scheme as push . • Some friends of i   may be one  many or none  will comment  cite or reprint I   because they think that it is interesting  funny or important. We call this behavior as republish . • The above steps will be repeated with i replaced by each of those who have republished I . A. The Procedure of Information Diffusion IV. DIFFUSING ROLE OF WEAK TIES 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 f$_{r}$ ¯ S Remove Weak Ties First Remove Strong Ties First f$_{c}$ (a) Facebook 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f$_{r}$ f GCC Remove Strong Ties First Remove Weak Ties First f$_{c}$ (b) Facebook 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 f$_{r}$ ¯ S Remove Strong Ties First Remove Weak Ties First f$_{c}$ (c) YouTube 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f$_{r}$ f GCC Remove Strong Ties First Remove Weak Ties First f$_{c}$ (d) YouTube