Manipulating Collective Opinion through Social Network Intervention

TL;DR

Analytically investigates a mathematical model that captures the complex interplay of polarization, radicalization, and consensus within networked societies, and identifies critical thresholds marking phase transitions in collective behavior, interpreted via a stability landscape.

Abstract

Social media platforms have transformed the dynamics of collective opinion formation, enabling rapid, large-scale interactions while simultaneously exposing online discourse to polarization and manipulation. Traditional models of opinion dynamics often predict convergence to a consensus, yet empirical evidence consistently highlights persistent polarization and radicalization, especially on contentious issues. This paper analytically investigates a mathematical model that captures the complex interplay of polarization, radicalization, and consensus within networked societies. By analyzing the emergence and stability of opinion clusters, we identify critical thresholds marking phase transitions in collective behavior, interpreted via a stability landscape. We further explore network-based interventions to manipulate the collective opinion, revealing that reducing inter-agent interactions can lead to unintended, irreversible shifts in opinion distributions. Our results underscore the dual-edged nature of intervention strategies, offering theoretical insight into the fragility and manipulability of public opinion in digital environments.

Figures (16)

• Figure 1: Schematic illustration of this study. This study investigates a scenario in which users on social media discuss a particular issue (right). The dynamics of their opinions, i.e., the evolution of all users' views, can be interpreted using a stability landscape: the collective state of opinions is represented as a ball within a potential (center). This study explores how the intervention of the social network may influence the formation of collective opinions (left and right).
• Figure 2: Schematic illustration of the opinion dynamics model (Eq. \ref{['eq_OD']}). The color of each node represents the value of the opinion variable, $x_i$, for each user: red, blue, and gray represent that a user is agree, disagree, and neutral, respectively. The influence of the $j$-th user on the opinion of $i$-th user depends on the opinion difference between the users, $|x_i - x_j|$, which is represented by the link width. The opinions $\{ x_i \}_{i=1, 2, ..., N}$ thus evolve with the effective coupling strengths between the users.
• Figure 3: Three typical opinion states and the phase diagram of the opinion dynamics model. (a)-(c) Time series of the individual users' opinions $\{ x_i \}$ ($i= 1, 2, \cdots , N$) (top) and the stability landscape (bottom) approaching three states: (a) Neutral Consensus (NC): All opinions $x_i$ converge to zero: $x_i= 0$, (b) RAdicalization (RA): All opinions are distributed with the same sign, and (c) POlarization (PO): The opinions split into two groups which localize in different signs. The interaction network $A_{ij}$ is given by the Erdös-Rényi random graph with the number of nodes $N= 400$ and the average degree $\langle k \rangle= 20$. The parameters are set as $\alpha = 1.0$ and $d=1.0$. Coupling strength is set as (a)$K=10$, (b) $K=30$ and (c) $K=50$. Opinions $x_i$ are initially distributed uniformly in $x_i \in [-2, 2]$. (d) and (e) Phase diagrams for three states. Green and blue diamonds represent NC and RA, respectively. Red diamonds represents parameter region where both PO and RA states are observed depending on the initial state of $\{x_i\}$. Black curves are theoretical predictions of boundaries for the three phases. Crosses in panel (d) represent parameter values used in panels (a)-(c). Parameters are set as $\alpha= 1$ for (d) and $K= 20$ for (e).
• Figure 4: Manipulating the opinion state of users. (a) Stability landscape interpretation to induce the opinion state via social network intervention: Polarization (PO) to Radicalization (RA). Left: Initially, the opinions are in the PO state, while the RA state is also stably attainable. Center: An intervention in the social network has the effect of altering the stability landscape, thereby allowing only the RA state. Right: The opinion state remains RA even after the intervention. (b), (c): Two types of interventions in the social network: the decrease in the coupling strength $K$ (b) and the removal of the links (c). Left: Time series of individual opinions $x_i$. Right: Histograms of the opinions before (top: $t = 0$) and after (bottom: $t= 150$) the intervention. The shaded areas represent the intervention period ($t \in [50, 100]$). Note that users with positive (negative) opinions before the intervention are shown in red (blue). The coupling strength and the links were reduced in 20 %. (d) Stability landscape interpretation to induce the opinion state by external driving forces (e.g., propaganda): Radicalization (RA) to polarization (PO). Left: Initially, opinions are in the RA state. Center: External driving forces cause opinions to surpass the barrier and move closer to the PO state. Right: The opinion state remains PO after the intervention. (e), (f): Induction from the RA state, in which most users' opinions are either positive (e) or negative (f), to the PO state. Note that users with positive (negative) opinions at time $t=100$ are shown in red (blue). All numerical simulations are done on Erdös-Rényi network of size $N= 400$ and mean degree $\langle k \rangle= 20$. Parameters are set as $K = 20$, $\alpha= 3$ and $d= 0.7$.
• Figure 5: Manipulation toward a targeted opinion state through a selective intervention in a social network. (a) Stability landscape interpretation to manipulate the opinion state to a targeted state via social network intervention: Polarization (PO) to Radicalization with negative opinions (RA$-$). Left: Initially, the opinions are in the PO state, while the RA$-$ state is also stably attainable. Center: A selective intervention in the social network has the effect of breaking the symmetry in the stability landscape, thereby allowing that the RA$-$ state is more realizable than the RA$+$ state. Right: The opinion state remains RA$-$ even after the intervention. (b) A radicalized state of users with negative opinions (RA$-$ state) can be induced by $20\%$ of the coupling strengths between the users with positive opinions ($x_i > 0$) and $10\%$ of the coupling strengths of the other links. (c) Stability landscape to manipulate the opinion state to a targeted state: Polarization (PO) to Radicalization with positive opinions (RA$+$). Left: Initially, opinions are in the RA state. Center: A selective intervention in the social network has the effect of breaking the symmetry in the stability landscape, thereby allowing that the RA$+$ state is more realizable than the RA$-$ state. Right: The opinion state remains RA$+$ even after the intervention. (d) A radicalized state of users with positive opinions (RA$+$ state) can be induced by randomly removing $20\%$ of the links between the users with negative opinions ($x_i < 0$) and $10\%$ of the coupling strengths of the other links. Note that the shaded area in panels (b) and (d) indicates the intervention period ($50<t<100$). The users with positive (negative) opinions before the intervention are shown in red (blue).