Network segregation in a model of misinformation and fact-checking

Abstract

Misinformation under the form of rumor, hoaxes, and conspiracy theories spreads on social media at alarming rates. One hypothesis is that, since social media are shaped by homophily, belief in misinformation may be more likely to thrive on those social circles that are segregated from the rest of the network. One possible antidote to misinformation is fact checking which, however, does not always stop rumors from spreading further, owing to selective exposure and our limited attention. What are the conditions under which factual verification are effective at containing the spreading of misinformation? Here we take into account the combination of selective exposure due to network segregation, forgetting (i.e., finite memory), and fact-checking. We consider a compartmental model of two interacting epidemic processes over a network that is segregated between gullible and skeptic users. Extensive simulation and mean-field analysis show that a more segregated network facilitates the spread of a hoax only at low forgetting rates, but has no effect when agents forget at faster rates. This finding may inform the development of mitigation techniques and raise awareness on the risks of uncontrolled misinformation online.

Appendix: Mean-field computations

In previous work we showed mean-field analysis for our model on a homogeneous network [44]. Similarly, here we perform a similar analysis for our model on a network segregated in two groups, skeptic and gullible: for each group we have three equations (see Eq. 5) representing the spreading process.

$$\begin{aligned} {p_{i_\mathrm{g}}^{B}(t+1)}&= f_{i_\mathrm{g}}(t) s_{i_\mathrm{g}}^{S}(t) + (1 - p_\mathrm{f})(1 - p_\mathrm{v}) s_{i_\mathrm{g}}^{B}(t)\\ {p_{i_\mathrm{g}}^{F}(t+1)}&= g_{i_\mathrm{g}}(t) s_{i_\mathrm{g}}^{S}(t) + p_\mathrm{v} (1 - p_\mathrm{f}) s_{i_\mathrm{g}}^{B}(t) + (1 - p_\mathrm{f}) s_{i_\mathrm{g}}^{F}(t)\\ {p_{i_\mathrm{g}}^{S}(t+1)}&= p_\mathrm{f} \left[ s_{i_\mathrm{g}}^{B}(t) + s_{i_\mathrm{g}}^{F}(t)\right] + \left[ 1 - f_{i_\mathrm{g}}(t) - g_{i_\mathrm{g}}(t)\right] s_{i_\mathrm{g}}^{S}(t)\\ {p_{i_\mathrm{sk}}^{B}(t+1)}&= f_{i_\mathrm{sk}}(t) s_{i_\mathrm{sk}}^{S}(t) + (1 - p_\mathrm{f})(1 - p_\mathrm{v}) s_{i_\mathrm{sk}}^{B}(t)\\ {p_{i_\mathrm{sk}}^{F}(t+1)}&= g_{i_\mathrm{sk}}(t) s_{i_\mathrm{sk}}^{S}(t) + p_\mathrm{v} (1 - p_\mathrm{f}) s_{i_\mathrm{sk}}^{B}(t) + (1 - p_\mathrm{f}) s_{i_\mathrm{sk}}^{F}(t)\\ {p_{i_\mathrm{sk}}^{S}(t+1)}&= p_\mathrm{f} \left[ s_{i_\mathrm{sk}}^{B}(t) + s_{i_\mathrm{sk}}^{F}(t)\right] + \left[ 1 - f_{i_\mathrm{sk}}(t) - g_{i_\mathrm{sk}}(t)\right] s_{i_\mathrm{sk}}^{S}(t) \end{aligned}$$

In these equations we can substitute $s_i(t)$ with $p_i(t)$ and when $t \rightarrow \infty$ we can assume $p_i(t)=p_i(t+1)=p_i(\infty )$ for all $i \in N$. Hereafter we simplify the notation using $p_\mathrm{g}^B(\infty )=p_\mathrm{g}^B$ (and analogously for the other cases). Now, let us consider the spreading functions for the gullible agents. Similar equations can be written for the case of skeptic agents. The spreading functions are:

$$\begin{aligned} f_{i_\mathrm{g}}(t) = \beta \,\frac{{n_{i_\mathrm{g}}^B}(1 + \alpha )}{{n_{i_\mathrm{g}}^B}(1 + \alpha ) + {n_{i_\mathrm{g}}i^F}(1 - \alpha )}\\ g_{i_\mathrm{g}}(t) = \beta \,\frac{{n_{i_\mathrm{g}}^F}(1 - \alpha )}{{n_{i_\mathrm{g}}^B}(1 + \alpha ) + {n_{i_\mathrm{g}}^F}(1 - \alpha )} \end{aligned}$$

Assuming that all vertices have the same number of neighbors $\langle k \rangle$, and that these neighbors are chosen randomly, we can write $n_i^B= s\cdot n^B{i_\mathrm{g}} + (1-s) \cdot n^B_{i_\mathrm{sk}}$, where $n^B_{i_\mathrm{g}}= \gamma \cdot \langle k \rangle p^B_\mathrm{g}$ and $n^B_{i_\mathrm{sk}}= (1-\gamma ) \cdot \langle k \rangle p^B_\mathrm{sk}$. Similarly, for $n_i^F$, we can obtain an expression that is not dependent on i. This simplifies the equations and lets us to simulate the process iterating the application of them until the values of $p^S_\mathrm{sk},p^B_\mathrm{sk}, p^F_\mathrm{sk}, p^S_{g}, p^B_{g},p^F_{g}$ have reached stability.