Fig. 4

Typical example of reply case focused on user \(u_2\). The hashtags posted by \(u_2\) are \(h_1\) and \(h_2\). The probabilities resulting from the random walk UH starting from \(u_2\) are then \([1/2,1/2,0,0]^T\). For meta-path of length 2, a walker starting from \(u_2\) following meta-path RP-UH has to go, with probability 1, to \(u_1\) and then to \(h_1\), \(h_2\) and \(h_3\). The resulting probabilities are \([1/3,1/3,1/3,0]^T\). Now, for meta-path of length 3, the walker can not return to \(u_2\) after being on \(u_1\): he has to go to \(u_3\) or \(u_4\). But since these latter are not in connection with \(u_2\) via the reply action, their hashtags are more different. This time, the probabilities are \([0,0,1/2,1/2]^T\), which is far from those obtained with UH: \([1/2,1/2,0,0]^T\). Consequently, the \(r^2\) is really low (in this case, it is null). However, for meta-path of length 4, the walker can return to \(u_2\) after being on \(u_1\) so in the next step (the third step), the walker can only jump to \(u_1\) who is a direct neighbor of \(u_2\). The rationale is the same for longer meta-paths: for even lengths, the walker is not affected by the restriction on the penultimate step of the walk while for odd lengths, it has huge importance