Fig 1.
Virality as a function of network polarization.
For parameters, see Table 1. This figure shows the effects of the echo chamber with Po = 0, i.e. without opinion polarization (implying no difference in activation threshold between the cluster and the overall network.) The results are shown for different average threshold levels. As can be seen, the cluster increases the virality until the network polarization passes 0.6, from which it starts having a negative impact on virality. The lower graph shows the effects of varying the number of nodes in the network. The lower graph shows the variance for network polarization with θ = 0.27 for varying node counts, to show that the results are robust for varying network sizes. As can be seen, the virality falls with larger network sizes, however, the effects of having a cluster present seems to possibly increase with network size. The lower graph was averaged over 300 iterations, with degree 8, and 20% of nodes in cluster. (Runs performed with opinion polarization showed the same result).
Table 1.
Model parameters.
Fig 2.
Virality as a function of threshold.
Shows virality (V) as a function of the average threshold value (θ), for different levels of opinion polarization (Po). The upper, filled line for each Po shows virality at the critical level of network polarization (i.e the Pn for which the effects of the presence of a cluster, V(θc, Pn, Po) − V(θc, Pn, 0) = ΔV(θc, Pn, Po), peaks for some given Po, denoted ) and the lower, dashed line for an Erdős-Rényi network. The upper figure shows the case (the nodes in the echo chamber are opinion polarized), while the lower shows the control (opinion polarization is assigned to random nodes in the network.) Two things should be noted: (i) The relationship between threshold and virality is far from linear, and (ii) as can be seen, the distance between the dashed and filled lines becomes significantly wider in the case than in the control: this tentatively suggests the presence of a synergic interaction between network and opinion polarization.
Fig 3.
Virality and opinion polarization.
In these graphs, we look V(θc, Pn, Po) (i.e. at the critical threshold level) for each level of Po (opinion polarization) (note: since θc varies with Po, the lines in the graphs cannot be compared with regard to the absolute level of virality.) The different lines denote different level of opinion polarization (Po). The upper graph shows the case (when the echo chamber is polarized), and the lower the control (when opinion polarization is assigned to random nodes in the network.) As can be seen, in the case, Po (the level of opinion polarization) affects , i.e. at what level of network polarization that virality peaks. In the control case, there is no such effect.
Fig 4.
Critical network polarization as a function of opinion polarization.
This graph shows how (the network polarization for which the highest virality is reached) depends on Po (the opinion polarization of the cluster). Comparing to Fig 3, this shows the network and opinion polarization at each point at which virality peaks. As the graph shows, the optimal echo chamber has more external connections and fewer internal connections as Po increases. In the control (dashed line), there is no interaction effect between network and opinion polarization.
Fig 5.
Illustrates the order in which the node activation occurs in successful cascades, by showing the fraction of echo chamber and non-echo chamber nodes to be activated at each time step. The left graph has no difference in activation threshold between inside and outside the cluster, while the right has Po = 0.2. As can be seen, the cluster is activated the first few steps, while the bulk of the non-cluster nodes are activated later, peaking around step 6 or 7. The runs are averaged over 1000 iterations, Pn = {0.7, 0.75, 0.85}, with θ = 0.27.
Fig 6.
The graph shows the effect of the presence of the echo chamber for the network polarization level with highest effect, as a function of opinion polarization (i.e. as a function of Po). In other words, looking at Fig 3, for each level of opinion polarization, it substracts the virality without an echo chamber from peak virality,
. The dashed line represents the control case where the opinion polarization has been randomly assigned. As can be seen, the presence of an opinion polarized echo chamber has more impact than the presence of one that is not opinion polarized. In the control case, there is little positive interaction between network and opinion polarization. The lower graph shows the same curve for different network sizes, to show that the results are robust for varying network sizes. As can be seen, the effects of having a cluster present actually increases with the network size. Lower graph runs were averaged over 300 iterations, with average degree 8, and 20% of nodes were assigned to be part of echo chamber.
Fig 7.
Virality as a function of network polarization in empirical networks.
These figures correspond to Fig 1, and show the effects of network polarization on virality for different activation thresholds. As can be seen, virality (V) is affected by network and opinion polarization. As opinion polarization (Po) increases, the effect of network polarization (Pn) becomes stronger, and we see a familiar peak at around Pn = 0.55 to 0.6. For higher values of network polarization, virality starts to fall.
Fig 8.
Critical virality on empirical networks.
The left figure corresponds to Fig 4, showing the data from Fig 7 at critical threshold, i.e. the threshold where the impact of the cluster is the highest. (As in Fig 4, comparison between the lines should be done with care, as they represent different values of Θ.) The right figure shows the difference between the case and the control for these critical threshold levels. The figure shows that there is indeed a strong interaction effect between political and network polarization, also when the model is run on empirical networks.