Figure 1

Time-constrained information cascade. Time-constrained cascades: nodes are disposed in concentric circles indicating the time when they received a specific tweet. Links between them represent the follower/friend relationship: an arrow from i to j indicates that j follows i, as any tweet posted by i is automatically received by j. Red nodes are those who posted a new message at the corresponding time, whereas gray nodes only listened to their friends. In this particular example, user 0 acts as the initial seed, emitting a message at time $t_0$ which is instantaneously sent to its nearest neighbors, laying on the first dashed circle, who are counted as part of the cascade. Some of them (nodes 1, 2, 3 and 4) decide to participate at the following time step, $t_1=t_0+\mathrm{\Delta }\tau$, posting a new message and becoming intermediate spreaders of the cascade. If any of their followers show activity at $t_2=t_0+2\mathrm{\Delta }\tau$ the process continues and the cascade grows in size as new users listen to the message. The process finally ends when no additional users showed activity (as it happens in the cases of users 3 and 4), or when an intermediate spreader does not have any followers (users 7, 8 and 9).