This paper introduces an extended model of opinion dynamics based on pairwise dialogues within networked communities. Building upon an existing binary dialogue model, we propose a more realistic framework that allows agents to adopt, reject, or choose alternative opinions when consensus is not reached. By treating dialogues as the fundamental units of social interaction, our probabilistic approach incorporates resistance and persuasiveness as independent agent attributes influencing opinion evolution. We present a rigorous formalization and simulation analysis that demonstrates how varying these attributes leads to diverse outcomes, such as opinion deadlocks, rapid convergence toward alternative opinions, and gradual shifts that reflect real-world opinion formation. Beyond its methodological contributions, the extended dialogue model addresses challenges central to contemporary psychological operations (psyops), where information is weaponized to manipulate narratives, polarize societies, and destabilize decision-making. The ongoing war in Ukraine illustrates these dynamics vividly and motivates the need for models that make the mechanics of influence observable at scale. Our framework provides a computational lens for analyzing, detecting, and anticipating manipulation strategies in networked communities, complementing empirical and policy work on countering disinformation.
Psychological operations (psyops) have become a defining feature of modern hybrid warfare, where
information is weaponized alongside kinetic force to shape perceptions, erode trust, and steer collective
behavior. The ongoing war in Ukraine illustrates these dynamics vividly: coordinated disinformation
campaigns, troll networks, and propaganda outlets target domestic and international audiences to
fracture consensus and destabilize decision-making [
In this environment, there is an urgent need for computational models that make the mechanics of
influence observable: how opinions evolve under strategic manipulation, which parameter regimes lead
to deadlock or rapid shifts, and when asymmetric attributes (e.g., high persuadability vs. high resistance)
create path-dependent outcomes. Opinion dynamics ofers the right formal lens. Classical models such
as DeGroot’s averaging process explain consensus formation under ideal conditions [
Such binarization is limiting in psyops contexts, where targets may reject both dominant and
counternarratives and pivot to alternative or adversarial viewpoints. It also obscures how persuasion and
resistance act as independent drivers of change — an asymmetry routinely exploited in coordinated
influence operations in and around Ukraine [
This paper answers that need. We extend a dialogue-based framework by (i) modeling persuasion and
resistance as independent agent attributes; (ii) introducing an explicit alternative-opinion outcome when
consensus is not reached; and (iii) defining a probabilistic transition process over pairwise dialogues
that yields rich behaviors (deadlock, rapid convergence to alternatives, gradual shifts). Through
simulations, we show how parameter regimes reproduce phenomena salient to psyops and information
warfare—rapid narrative pivots, echo-chamber stability, and asymmetric influence efects—thus ofering
a computational lens for analyzing, detecting, and anticipating manipulation strategies in networked
communities [
We next recall the baseline dialogue model that our extension builds upon and then formalize the proposed transition dynamics.
The model proposed in [
2. Adopt the opinion of their interlocutor (persuasion).
Opinion exchange occurs in a networked setting, where individuals interact according to predefined communication probabilities. Over time, these interactions shape the distribution of collective opinions within the community.
While this framework captures basic opinion exchange mechanisms, it assumes that an individual must either accept or reject an opinion of their interlocutor without any third option or consideration of an independent evaluation. As a result, the original model has two key limitations that restrict its applicability to real-world decision-making: • No opinion change in high-resistance cases: When both agents are highly resistant to persuasion, dialogues become inefective — no opinion shifts occur, leading to a static system even after many interactions. • Lack of an alternative option: The model assumes that agents must either keep their opinion or fully adopt their interlocutor’s opinion. However, individuals often reject both options and seek an independent alternative.
To address these limitations, we introduce an extended model that allows agents to select an alternative opinion by: • Incorporating a mechanism of choosing an alternative, allowing agents to reject both their current opinion and their interlocutor’s. • Introducing persuasion and resistance as independent attributes, enabling a more nuanced understanding of how opinions evolve. • Defining a probabilistic transition model that determines the likelihood of diferent opinion shifts based on these factors. • Simulating the extended model to analyze its behavior under diferent conditions and compare it to the original framework.
These enhancements specifically address critical gaps in the initial model [
Persuasion and resistance have long been fundamental concepts in social psychology, highlighted in
classical frameworks such as the Social Judgment Theory proposed by Sherif and Hovland [
Furthermore, real-world opinion dynamics often involve exploring alternatives rather than binary
acceptance or rejection, a behavior increasingly significant in contemporary sociopolitical environments.
Research by Mutz [
Thus, extending computational models with these social-scientific dimensions facilitates more realistic simulations and deeper insights into collective decision-making processes, enriching both the theoretical understanding and practical interventions in social networks and digital communication systems.
To address the limitations of the original model, we introduce an extended opinion dynamics framework that allows agents to not only retain or adopt an interlocutor’s opinion but also select an alternative option when consensus is not reached. This modification makes the model more flexible and realistic, reflecting real-world decision-making processes.
This section formalizes the dialogue process, defines the state space, and introduces a probabilistic transition function that governs opinion evolution.
We begin by introducing assumptions to formalize opinion exchange:
• In real-world discussions, when people reject both available options, they often consider multiple alternatives without an immediate preference for one. • Instead of modeling specific alternatives, we aggregate all alternatives into one efective option. • This simplification reduces computational complexity while preserving the model’s core dynamics.
• The initial state distribution of opinions is assumed to be statistically independent between agents. • This simplifies analytical computations while allowing the model to focus on how interactions influence opinion shifts.
Let denote the set of possible opinions, where | | > 2. The special case when | | = 2 was already
covered in detail by the original model introduced in [
A preference density function () of an agent over a set of possible opinions is a probability distribution defined on , assigning each opinion ∈ a probability () representing how strongly the agent initially favors that opinion. Formally, it satisfies: ∑︁ () = 1, () ≥ 0, ∀ ∈ ∈ (), () ∀ ∈
For two interacting agents, Alice () and Bob (), their individual preferences are denoted as: The state of the dialogue between two agents, Alice and Bob, is formally represented by an ordered pair of opinions:
= (, ), , ∈ where denotes Alice’s opinion and denotes Bob’s opinion immediately before their interaction. Consequently, the complete dialogue state space is the Cartesian product: ∈ ×
Given Assumption 4, which asserts that the initial opinions of agents are independent, we define the joint preference density of the dialogue state as the product of the individual preference densities: (, ) = () · () Here, () and () represent the probabilities that Alice and Bob independently hold opinions and , respectively, before their interaction.
(, ) is also a preference density function defined on that satisfies conditions ∑︁ (,)∈
(, ) = 1, (, ) ≥ 0, ∀( , ) ∈
Since agents can choose an alternative when they disagree, we formally define the space of other options.
If ̸= then • for | | = 3 the alternative is uniquely given by the single element (,) in ∖ {, }; • for | | > 3, agents, however, have more than one alternative ∈ (, ) = ∖ {, } to choose from. To maintain computational simplicity while preserving model robustness, under the Assumption 3, we assume: – The alternative is selected uniformly at random from the set ∖ {, }.
– It means the efective alternative is represented by a single aggregated option, . Importantly, this aggregated alternative should not be interpreted as a singular or uniform stance adopted equally by all agents. Instead, it reflects a generalized category of alternatives, potentially unique to each individual, signifying dissatisfaction with currently available options and openness toward exploring other viewpoints. In practical terms, when agents transition into this residual category, they individually enter exploratory or uncertain states, each potentially distinct from others’ interpretations and motivations. This modeling approach aligns with common probabilistic and decision-theoretic methodologies, where varied but infrequently occurring alternatives are grouped into a general “other” category for analytical convenience. The model, therefore, captures the essential dynamics of opinion shifts without the intractable complexity of enumerating all possible alternatives explicitly. Future research could further disaggregate this generalized alternative category to explore detailed, actor-specific opinion dynamics, enriching the model’s descriptive power and practical applicability.
Thus, the opinion set is reduced to: = {, , }, = × , || = 9.
It allows the transition function to be tractable and interpretable while capturing the impact of alternative selection.
Given a state (, ), the transition function : → () defines the set of possible opinion shifts after a dialogue: (, ) = {︃{(, )} if = = if ̸= (1) It means that: • If = , the state is absorbing (no change occurs). • If ̸= , the dialogue may lead to any of the nine possible states where agents can retain, adopt, or switch to an alternative.
Each possible outcome can be represented as:
Outcome 1. ((, )) → (, ) 2. ((, )) → (, ) 3. ((, )) → (, ) 4. ((, )) → (, ) 5. ((, )) → (, ) 6. ((, )) → (, ) 7. ((, )) → (, ) 8. ((, )) → (, ) 9. ((, )) → (, )
Interpretation Both keep their opinions Bob switches to Alice’s opinion Alice switches to Bob’s opinion Both switch (swap opinions) Alice keeps, Bob chooses alternative Alice chooses alternative, Bob keeps Both choose alternative Alice switches to Bob’s, Bob chooses alternative
Alice chooses alternative, Bob switches to Alice’s This formalism provides a probabilistic representation of opinion shifts in a network.
To model how agents update their opinions, we introduce two key attributes:
• Resistance ∈ [
In sociopsychological terms, “resistance” aligns closely with constructs such as attitude strength (the firmness or certainty of one’s beliefs) and ego involvement (the extent to which an opinion is tied to personal identity). Individuals with high resistance may be less receptive to contrary arguments, reflecting firm conviction or attachment to their existing viewpoint. On the other hand, “persuasiveness” resonates with frameworks like communicator credibility in social psychology, which highlights how a speaker’s perceived expertise and trustworthiness shape their influence over others. Agents endowed with high persuasiveness may shift others’ opinions more readily, mirroring how credible or charismatic communicators often succeed in swaying audiences.
For Alice () and Bob (), these parameters are denoted as:
, , , Each agent has three possible actions after a dialogue: 1. Retain their opinion. 2. Adopt the interlocutor’s opinion. 3. Choose an alternative opinion.
The probabilities of these actions are determined by resistance and persuasiveness. For Alice, they are defined as: 1 = (1 −
) 2 = (1 − )
(Probability of retaining opinion) 3 = (Probability of choosing an alternative)
where is a normalization constant ensuring that the probabilities sum to 1:
(Probability of adopting Bob’s opinion) Similarly, for Bob: with: = (1 −
Interpretation = (1 −
) + (1 − ) + 1 = (1 − ) , 2 = (1 − ) , 3 =
) + (1 − ) + • If resistance is high ( ≈ 1), Alice is more likely to retain her opinion. • If Bob’s persuasiveness is high ( ≈ 1), Alice is likelier to adopt Bob’s opinion. • If both resistance and persuasiveness are high, Alice may reject both opinions and select an alternative.
These probabilities define how agents transition between states in the dialogue process.
Since each agent acts independently, the probability of transitioning from state (, ) to a new state (′, ′) is the product of individual transition probabilities: where:
((, ), (′, ′)) = (′|) · ( ′|) (′|) = 1[′ = ] + 2[′ = ] + 3[′ = ] (′|) = 1[′ = ] + 2[′ = ] + 3[′ = ] These probabilities define how the system evolves, forming a Markov process. For each ̸= , the nine possible outcomes and their associated probabilities are:
Outcome 1. (, ) 2. (, ) 3. (, ) 4. (, ) 5. (, ) 6. (, ) 7. (, ) 8. (, ) 9. (, )
Interpretation (︀ (, ), (·, ·) )︀ Both keep their opinions : 1 1 Bob switches to Alice’s opinion : 1 2 Alice switches to Bob’s opinion : 2 1 Both switch (swap opinions) : 2 2 Alice keeps, Bob chooses alternative : 1 3 Alice chooses alternative, Bob keeps : 3 1 Both choose alternative : 3 3 Alice chooses Bob’s, Bob – alternative : 2 3
Alice chooses alternative, Bob – Alice’s : 3 2
2.6.1. Matrix Representation
For convenience, we enumerate the state space , reducing the joint preference density into a row
vector, and express transitions as a 9 × 9 matrix : × → [
, = (, ) represents the probability of moving from state to state . This matrix satisfies the following key properties: 1. Non-negativity: (, ) ≥ 0, ∀ , ∈ 2. Valid Transition Probabilities: (, ) = 0,
transition, its probability is zero. 3. Normalization (Stochastic Matrix Property): ∑︀∈() (, ) = 1, ∀ Each row of T sums to 1, ensuring that the agent must transition to some valid state for any given state. ∀ ∈/ () If the model dynamics do not allow a
To illustrate the behavior of the extended model, we conduct simulations where two agents, Alice and Bob, engage in a series of dialogues. These simulations examine how diferent values of resistance ( ) and persuasiveness ( ) influence opinion evolution. We aim to analyze whether agents reach consensus, polarization, or alternatives under diferent conditions.
We begin by demonstrating the limitations of the original model, followed by the impact of introducing persuasiveness and alternative options. Finally, we discuss practical applications of these dynamics in real-world decision-making and computational modeling.
The following simulation results and illustrations were generated using publicly accessible software,
available online for review and replication at [
• Fig. 1 illustrates the limitation of the original dialogue model [
As discussed, the original model in [
In Fig.1, Alice and Bob initially have a strong preference for their own options and high resistance, ignoring the alternative completely. After repeated dialogues, her opinion remains unchanged, as resistance prevents any shifts.
(a) Alice
The extended model introduces persuasiveness ( ), representing an agent’s ability to influence their interlocutor. It allows persuasion to counteract resistance, leading to potential opinion shifts even when agents start from opposing views.
If Alice and Bob have zero persuasiveness ( = = 0), the model behaves identically to the
original model in [
However, when both Alice and Bob are fully resistant ( = = 1) but also entirely persuasive ( = = 1), the system quickly reaches consensus on an alternative option (Fig.2).
It is important to emphasize that, although Alice and Bob converge rapidly to the alternative opinion () under conditions of maximum resistance and persuasiveness, this state does not imply they necessarily adopt the same viewpoint. Instead, as explained in Section 2.3, represents a generalized category of diverse opinions, individually interpreted by each agent. Hence, real-world scenarios might require further diferentiation among alternative opinions (e.g., , ) or the inclusion of stochastic elements to prevent unrealistic deadlocks and more accurately capture nuanced opinion dynamics.
• Initially, Alice and Bob hold distinct opinions. • Since they are both highly persuasive yet unwilling to adopt the other’s viewpoint, they converge on an alternative option () instead. • This shift happens immediately after the first dialogue iteration, unlike the no-change scenario observed when = = 0.
If Bob, however, is resistant but not persuasive ( = 1, = 0), he rejects Alice’s option but still moves toward the alternative, as shown in Fig.3. This case demonstrates that a persuasive agent (Alice) can influence the dynamics while a non-persuasive agent (Bob) remains inert.
(a) Alice (b) Bob
Tuning up Bob’s persuasion power does not afect his own choices but influences Alice, who, after the first iteration, opens up for an alternative option (Fig.4)
In most real-world scenarios, agents are neither fully resistant nor fully persuadable. To explore this, we simulate cases where Alice and Bob have moderate levels of resistance and persuasion ( = = 0.8, = = 0.8).
As Fig.5 illustrates: • Gradual shifts occur over multiple dialogue rounds. • Both agents move toward a mixed state, partially adopting each other’s perspectives while considering alternative options. • Unlike the extreme cases in previous sections, where no change or instant consensus occurred, this scenario represents a more realistic, gradual evolution of opinions.
(a) Alice (b) Bob
In this paper, we enhanced the original dialogue-based model of opinion dynamics by allowing agents to select alternative options beyond mere acceptance or rejection of opposing viewpoints. Our simulations illustrate how varying resistance and persuasiveness parameters significantly shape dialogue outcomes, yielding three behaviors: • Opinion Deadlock (Fig. 1): High resistance and low persuasiveness lead to persistent disagreement without convergence. • Rapid Consensus on an Alternative (Fig. 2, Fig. 3): High resistance combined with strong persuasive abilities quickly directs agents toward alternative opinions, avoiding mutual acceptance. • Gradual Opinion Shifts (Fig. 5): Moderate persuasion and resistance levels result in incremental opinion changes, closely resembling real-world gradual consensus-building.
These results have practical implications for studying and countering psyops and contemporary
information warfare. By simulating adversarial influence strategies and community-level responses,
the model provides a computational tool for (i) identifying parameter regimes that make populations
resilient or vulnerable to manipulation, (ii) testing detection heuristics for coordinated influence, and
(iii) stress-testing mitigation strategies before deployment [
The novelty of the proposed framework lies in explicitly incorporating an alternative-opinion outcome
and separating persuasion from resistance as independent drivers. Unlike traditional binary approaches,
this structure captures dynamics observed in influence campaigns—rapid narrative pivots, echo-chamber
stability, and asymmetric efects across subpopulations: phenomena documented in the context of the
war in Ukraine [
These limitations suggest that while the model ofers valuable insights into opinion evolution mechanisms, caution is warranted when directly extrapolating the findings to real-world scenarios without additional context-specific calibration. 4.0.2. Future Research Directions • Extending the model for larger, more complex network interactions. • Investigating the dynamic evolution of agent preferences based on interaction histories and network changes. • Exploring the disaggregation of the aggregated alternative category into multiple distinct alternatives to capture richer, individual-specific opinion dynamics. • Assigning empirically grounded values to resistance and persuasion parameters by – conducting empirical studies and controlled experiments to measure resistance and persuasiveness within defined populations or contexts; – analyzing historical datasets or opinion surveys to infer realistic parameter ranges and distributions, enhancing the model’s calibration; – incorporating machine learning techniques to estimate these parameters dynamically from observed interactions in online communities or social media platforms’ • Introducing individualized or context-dependent persuasiveness and resistance parameters. • Empirically validating predictions through observational studies on social media interactions and controlled experiments with structured dialogues. • Applying and calibrating the model to real-world datasets related to psyops (e.g., platform takedown datasets, messaging telemetry, and annotated narrative corpora) to evaluate detection/mitigation strategies in situ.
During the preparation of this work, the authors used ChatGPT and Grammarly to: Grammar and spelling check, Paraphrase and reword. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the publication’s content.