Linear inelastic kinetic equations modelling the spread of fake news and its interplay with personal awareness

TL;DR

We develop two linear inelastic Boltzmann-type kinetic descriptions of fake-news dynamics: (i) an awareness evolution model where individuals with awareness $x\in[0,1]$ learn from information with reliability $y$ through a Boltzmann interaction, and (ii) a popularity model where content popularity $v$ interacts with a fixed reliability $y$ and a connectivity distribution $C(c)$; The analysis yields well-posedness and trend-to-equilibrium results, with explicit Maxwellian equilibria characterized in Fourier space: $\hat{f}^\infty(\xi)=\prod_{k=0}^{\infty}H(m_X^\infty,(1-\alpha)^k\xi)$ and $\hat{p}_y^\infty(\xi)=\prod_{k=0}^{\infty}K_y((1-\mu)^k\xi)$, where $H$ and $K_y$ encode the dependence on $G(m_X^\infty)$ and the connectivity transform $\hat{C}$; the tail of $p_y^\infty$ is shown to mirror the tail of $C$, so fat tails in $C$ induce fat tails in popularity. Numerical Monte Carlo simulations validate the analytical predictions, revealing emergent clustering of awareness at discrete points (and full concentration at endpoints when $\alpha=1$) and the influence of network topology on long-run popularity. The work also outlines a data-driven method to infer the news reliability distribution from real data via topic clustering. Overall, the paper provides a rigorous kinetic framework linking learning-driven awareness to topology-driven information diffusion, with clear avenues for extending to coupled and co-evolving networks.

Abstract

In this paper, we introduce a kinetic model which describes a learning process leading individuals to build personal awareness about fake news. Next, we embed the results of this model into another kinetic model, which describes the popularity gained by news on social media conditioned to the reliability of the disseminated information. Both models are formulated in terms of linear inelastic Boltzmann-type equations, of which we investigate the main analytical properties - existence and uniqueness of solutions, trend to equilibrium, identification of the equilibrium distributions - by employing extensively Fourier methods for kinetic equations. We also provide evidence of the analytical results by means of Monte Carlo numerical simulations.

Figures (6)

• Figure 1: Cumulative distribution function of $Y\sim\operatorname{Beta}(a,\,b)$ for different values of the parameters $a,\,b$. Panel (a) shows three sets of parameters for which an asymptotically stable value of $m_X^\infty\in (0,\,1)$ exists. Panel (b) shows instead a set of parameters for which $m_X^\infty\in (0,\,1)$ is not asymptotically stable
• Figure 2: Asymptotic awareness distribution for $Y\sim\operatorname{Beta}(0.5,\,0.3)$ when $\alpha$ satisfies \ref{['ass:alpha']}. In particular: (a) $\alpha<1$, (b) $\alpha=1$
• Figure 3: Asymptotic awareness distribution for (a) $Y\sim\operatorname{Beta}(0.3,\,0.5)$ ($m_X^\infty=0.785$, cf. Figure \ref{['fig:betacdf_stable']}) and (b) $Y\sim\operatorname{Beta}(0.5,\,0.5)$ ($m_X^\infty=0.5$, cf. Figure \ref{['fig:betacdf_stable']})
• Figure 4: Asymptotic awareness distribution for $Y\sim\operatorname{Beta}(2,\,2)$ ($m_X^\infty=0.5$ unstable, cf. Figure \ref{['fig:betacdf_no_stable']}) and: (a) $m_X^0<0.5$; (b) $m_X^0>0.5$
• Figure 5: Popularity tails for $\alpha=1$ and: (a) $C\sim\operatorname{Inv-Gamma}(10,\,0.5)$, log scale; (b) $C\sim\operatorname{Exp}(5)$, log-linear scale

Theorems & Definitions (26)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• Remark 2.4
• proof : Proof of Theorem \ref{['theo:ds']}
• Proposition 2.5
• proof
• Corollary 2.6