Fig. 1

Probabilistic Meeting Rule—illustrative example. Top row: we depict an interaction network with five users, the social status of users (\(s_1\) to \(s_5\)) and the adjacency matrix \(\varvec{A}.\) All edge weights in \(\varvec{A}\) are 1, indicating that the corresponding users interacted only once with each other in the past. If we restrict meetings to the edges of the interaction network, the meeting probabilities are symmetric and equal to the entries of \(\varvec{A}.\) Thus, the users 1 and 4 cannot participate in a meeting since \(p_{14}=p_{41}=0\) (elements marked in red in \(\varvec{A}\)). The average meeting probability \(p_m\) corresponds to the network density and evaluates to 0.5. Middle row: we calculate the regular equivalence matrix \(\varvec{\sigma }\) and normalize it with the degrees and the minimal neighbor similarity (normalization results in asymmetric similarities). We set closeness factor \(\gamma =1/2\) (modular society) and calculate the matrix of meeting probabilities \(\varvec{P_{\sigma }}\) (we set zeros on the diagonal since each meeting requires two users). The rows correspond to the meeting probabilities of a user acting as the speaker. We observe now non-zero probabilities between users who are not connected by an edge. For example, for users 1 and 4 (cf. red-marked elements in \(\varvec{P_{\sigma }}\)), the meeting probability is \(p_{14}=0.31\) (user 1 acts as the speaker) and \(p_{41}=0.54\) (user 4 acts as the speaker). In this setting, the average meeting probability is significantly higher than previously \(p_m=0.69.\) Bottom row: the matrix \(\varvec{S}\) keeps the (asymmetric) social status differences between users. Again, the rows correspond to users acting as the speaker in a meeting. We set stratification factor \(\beta =1/2\) (ranked society) and calculate the matrix of the meeting probabilities \(\varvec{P_S}.\) The social status mechanism results in non-zero probabilities between all pairs of users. For example, for users 1 and 4 (cf. red-marked elements in \(\varvec{P_S}\)), the meeting probability is \(p_{14}=0.22\) (user 1 is the speaker) and \(p_{41}=1\) (user 4 is the speaker). The average meeting probability for this configuration is \(p_m=0.71.\) Finally, if similarity as well as social status rules apply, the final meeting probabilities are calculated by element-wise multiplication of \(\varvec{P_\sigma }\) and \(\varvec{P_S}\)