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Figure 1. Illustration of the status of our botnet at the time of the interventions. The bots had accumulated a large number (~ 25 000) of followers (A) at the time of the interventions (shaded region), and many of the target users followed several distinct bots (B).

Figure 1 Illustration of the status of our botnet at the time of the interventions. The bots had accumulated a large number (~ 25 000) of followers (A) at the time of the interventions (shaded region), and many of the target users followed several distinct bots (B).

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Table 1. List of interventions
Table 1. List of interventions
Figure 2. Bots in a botnet can work together to provide users with multiple exposures to an intervention. (A) user U only follows bot B;. Bot (B;) acts as a proxy and exposes the user not only to its own content, but also to content from two other bots (Bz and B3), that the user does not follow. (B) Twitter feed from the perspective of user U.
Figure 2. Bots in a botnet can work together to provide users with multiple exposures to an intervention. (A) user U only follows bot B;. Bot (B;) acts as a proxy and exposes the user not only to its own content, but also to content from two other bots (Bz and B3), that the user does not follow. (B) Twitter feed from the perspective of user U.
Figure 3. Simple contagion (SC, left) does not adequately describe the contagion dynamics: the best fit underestimates the probability of retweeting after a low number of exposures and overestimates the probability with a large number of exposures. The best fit of complex contagion (CC, right) dynamics correctly estimates the probability of retweeting across the number of sources of exposure. A. Percentage of tweets that were retweeted after k successful exposures (SC) or after exposures from kK sources (CC). B. Number of tweets retweeted following k successful exposures (SC), or after exposures from kK sources (CC). Best fit of SC model (equation 4) and CC model (equation 10) to the data using g = 0.20, plotted up to k =7 (and k = 8) to avoid plotting noisy data for large values of k (and k).
Figure 3. Simple contagion (SC, left) does not adequately describe the contagion dynamics: the best fit underestimates the probability of retweeting after a low number of exposures and overestimates the probability with a large number of exposures. The best fit of complex contagion (CC, right) dynamics correctly estimates the probability of retweeting across the number of sources of exposure. A. Percentage of tweets that were retweeted after k successful exposures (SC) or after exposures from kK sources (CC). B. Number of tweets retweeted following k successful exposures (SC), or after exposures from kK sources (CC). Best fit of SC model (equation 4) and CC model (equation 10) to the data using g = 0.20, plotted up to k =7 (and k = 8) to avoid plotting noisy data for large values of k (and k).
Figure 4. BIC scores for both SC and CC models for a range of values of q, the lower the score the better. Across the values of the g parameter, complex contagion model achieves lower BIC scores than simple contagion. The thick lines are the mean values of the simulations, and the shaded regions are the percentiles corresponding to one standard deviation, i.e. they contain 68% of the simulation results.
Figure 4. BIC scores for both SC and CC models for a range of values of q, the lower the score the better. Across the values of the g parameter, complex contagion model achieves lower BIC scores than simple contagion. The thick lines are the mean values of the simulations, and the shaded regions are the percentiles corresponding to one standard deviation, i.e. they contain 68% of the simulation results.
Figure 5. Time needed to compute P(k|A) using a brute force approach (solid red line) and our unique subset generation approach (dashed blue line). For each data point, we generated a simulated activity list with length |A| with elements drawn with probabilities based on their frequency in our data, and computed P(xk|A) using the two methods. The runtime is then averaged over 100 such computations in order to minimize noise. Our approach allows us to compute Poisson binomial distributions for activity lists of lengths that would otherwise be infeasible.
Figure 5. Time needed to compute P(k|A) using a brute force approach (solid red line) and our unique subset generation approach (dashed blue line). For each data point, we generated a simulated activity list with length |A| with elements drawn with probabilities based on their frequency in our data, and computed P(xk|A) using the two methods. The runtime is then averaged over 100 such computations in order to minimize noise. Our approach allows us to compute Poisson binomial distributions for activity lists of lengths that would otherwise be infeasible.
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