Reversible bootstrap percolation: Fake news and fact checking

Bootstrap percolation has been used to describe opinion formation in society and other social and natural phenomena. The formal equation of the bootstrap percolation may have more than one solution, corresponding to several stable fixed points of the corresponding iteration process. We construct a reversible bootstrap percolation process, which converges to these extra solutions displaying a hysteresis typical of discontinuous phase transitions. This process provides a reasonable model for fake news spreading and the effectiveness of fact checking. We show that sometimes it is not sufficient to discard all the sources of fake news in order to reverse the belief of a population that formed under the influence of these sources.

Figure 1

Dynamic updates during reversible bootstrap percolation. Stage (0): Two seed nodes are activated. The static links are shown by dashed lines. Each node has its activation threshold on its head, and its identification number on its body. Active nodes are shown by bold circles around their heads. Seed nodes are active by definition. $Z_0=0$, $S_0=2/5$. Stage (1): The influence links from the seeds are created, forming set $I_1$. $Z_1=4/12$. Nodes 3 and 4 become active because the number of influence links they have received is greater than or equal to their thresholds. $S_1=4/5$. Stage (2): Old influence links ($I_1$) are shown by bold arrows. New influence links ($I_2$) obtained by the rules of the model from $I_1$ are shown by thin arrows. Node 3 receives influence from node 4, because node 4 does not depend on node 3 to stay at the threshold. However, seed 1 does not receive the influence from node 4 because the latter does depend on the former to stay at the threshold. In contrast, node 3 influence all of its neighbors, because, from the point of view of any of its neighbors, node 3 is active without its influence. Node 5 now receives a sufficient number of influence links to become activated. $Z_2=9/12$, $S_2=1$. Stage (3): Seed 1 receives influence from node 4, because now node 4 has additional influence from node 3 and stays at the threshold without the influence of seed 1. Node 5 cannot send influence links to any of its neighbors, because it would be inactive without the support of each particular neighbor. $Z_3=10/12$, $S_3=1$. Stage (4): The influence network comes to a steady state, but both seeds spontaneously lose their seed properties. $Z_4=10/12$, $S_4=1$. Stage (5) Node 2 is deactivated, because its number of influence links is below the threshold. The influence links that come from node 2 are removed. Node 1 stays active at the threshold and influences all of its neighbors, because it stays at the threshold without the support of each of them. Node 5 falls below the threshold and is deactivated. The network comes to a new steady state. $Z_5=8/12$, $S_5=3/5$.

Figure 2

Comparison of the theory and simulations for the reversible bootstrap simulation model. In both panels the degree distribution $P\left(k\right)$ is Poisson with $\langle k\rangle =10$ and $r(j,k)=F_{\gamma }(j/k)$ with $\gamma =0.53$ (a) and $\gamma =0.48$ (b). In the inset we show the function $F_{\gamma }\left(x\right)$ for $\gamma =1/3$, $\gamma =1/2$, and $\gamma =2/3$. In both cases the size of the network is $N={10}^7$ nodes. In (a) the simulations follow a reversible bootstrap process based on the influence network updates described in the text for a sequence of $f_t$, marked on the graph by numbers $t=0,1,2,\cdots ,10$. For (b), the simulations are done for increasing $f$ from $f=0$ to $f=0.125$ giving the low solution and then in reverse order from $f=0.125$ to $f=0$ giving the high solution. It can be observed that in panel (b) the fraction of active nodes once it undergoes transition to a high activated phase at $f=f_t=0.114$ never returns to a low activated phase even for $f=0$. One can see that a small increase in susceptibility of the population to the fake news, modeled by a small decrease of the inflection point in the distribution $r(j,k)$, may lead to the irreversible opinion change of the population. We obtained similar results for many other types of distributions $P\left(k\right)$ and $r(j,k)$.

Figure 3

Phase diagram of the reversible bootstrap model in the $(f,S)$ plane, which is completely analogous the Van der Waals phase diagram in the $(P,\rho )$ plane. The inflection point, $\gamma$, of the threshold distribution is analogous to temperature. The black lines correspond to isochores (lines of equal $\gamma$ with steps of 0.01 from $\gamma =0.48$ to $\gamma =0.65$) following the high stable point of Eq. (5), physically analogous to the superheated metastable liquid crossing the liquid-gas spinodal. The dashed lines follow the low stable point, physically corresponding to the metastable supercooled gas phase crossing the gas-liquid spinodal. Above the critical point $\gamma =0.6035$, $f=0.2292$, $S=0.754$ the second stable fixed point of Eq. (5) disappears.