The Impact of External Sources on the Friedkin-Johnsen Model

TL;DR

The paper generalizes the Friedkin--Johnsen opinion dynamics model by incorporating stubborn external media sources to analyze how biased information shifts public opinion. It derives analytic bounds on the influence of one or two external sources, with tight results on regular graphs, and investigates multi-period bias updates. Through experiments on real social networks and synthetic graphs, the authors demonstrate that the theoretical bounds closely predict observed convergence and radicalization times, highlighting conditions under which external bias can dominate or preserve collective opinion. The work provides a theoretical foundation for understanding media-driven polarization and informs discussions on the role of scrutiny and accountability of external information sources in shaping public discourse.

Abstract

To obtain a foundational understanding of timeline algorithms and viral content in shaping public opinions, computer scientists started to study augmented versions of opinion formation models from sociology. In this paper, we generalize the popular Friedkin--Johnsen model to include the effects of external media sources on opinion formation. Our goal is to mathematically analyze the influence of biased media, arising from factors such as manipulated news reporting or the phenomenon of false balance. Within our framework, we examine the scenario of two opposing media sources, which do not adapt their opinions like ordinary nodes, and analyze the conditions and the number of periods required for radicalizing the opinions in the network. When both media sources possess equal influence, we theoretically characterize the final opinion configuration. In the special case where there is only a single media source present, we prove that media sources which do not adapt their opinions are significantly more powerful than those which do. Lastly, we conduct the experiments on real-world and synthetic datasets, showing that our theoretical guarantees closely align with experimental simulations.

Figures (3)

• Figure 1: Two equivalent ways to present the influence of a stubborn media source $M$ and its neighbors on node $1$, at each time step. The nodes represent the innate or expressed opinions; we use circles to present nodes' expressed opinions and use boxes to annotate fixed innate opinions. We assume that $N(1) = \{2,3,4\}$ and $w_{1,M} = \beta(1 + \sum_{i=1}^3 w_{1,i})$.
• Figure 2: In $(a), (b), (c), (d)$ and $(e)$ we consider the FB SN and the BA and DREG graph with comparable parameters to FB. In $(f),(g),(h),(i)$ and $(j)$ we consider the WK SN and the BA and DREG graph with comparable parameters to WK. In $(a)$ and $(f)$ the normalized sum of expressed opinions over multiple periods is depicted. The logarithmic number of periods to reach normalized average opinion $1/(1 + \gamma)$ is depicted in $(b)$ and $(g)$ for different values of $\gamma$ and in $(c)$ and $(h)$ for different values of $\beta$. In $(d)$ and $(i)$ the logarithmic number of periods to reach normalized average opinion $1/(1+\gamma)$ or $\epsilon := 10/n$ for different values of $\alpha$ are shown. Lastly, in $(e)$ and $(j)$ the final normalized sum of opinions at the end of the process for different values of $\alpha$ are given. The graph TB depicts the theoretical bound from \ref{['cor:twomediaoneroundreg']} in $(a)$ and $(f)$ and \ref{['lem:numroundsphase1multimedia']} for $(b),(g),(c),(h)$ and $(d)$ and $(i)$.
• Figure 3: In all the above figures, we plot the normalized sum of opinions when $\alpha =0.5$ or $\alpha\approx 0.5$. Each one of the $20$ repetitions is plotted individually. In $(a)$ and $(e)$ we consider the FB and WK SN respectively with a randomly chosen node removed. Figures $(b)$ and $(f)$ correspond to DREG graphs with $n=4038$, $d=44$ and $n=7114$, $d=30$ respectively (note that these are comparable parameters to the FB and TW SN with a single node removed, which is the same across repetitions). In $(c)$ and $(g)$ we depict the FB and WK SN respectively. Lastly, in $(d)$ and $(h)$ DREG graphs with comparable parameters to FB and WK SN respectively are shown.

Theorems & Definitions (14)

• Lemma 2.1
• Definition 1
• Lemma 2.2
• Definition 2: $M$-matrix
• Lemma 2.3
• Theorem 3.1
• Lemma 3.2
• Theorem 3.3
• Corollary 3.4
• Proposition 3.5