Confirmation-Aware Social Dynamic Model

• The paper introduces a formal model that embeds confirmation bias into user feedback loops, social network interactions, and stochastic recommender dynamics.
• It demonstrates that alignment-sensitive mechanisms can lead to echo chambers, segregation, or polarization, depending on network topology and bias intensity.
• The study also evaluates mitigation strategies that trade off some recommendation accuracy for improved diversity and balanced opinion convergence.

Searching arXiv for the core paper and closely related confirmation-bias social dynamics work. Confirmation-aware social dynamic models are formal frameworks in which confirmation bias is encoded directly into interaction, learning, or recommendation dynamics, so that the influence of peers, sources, signals, or content depends on opinion alignment. In the narrow sense, the term names a recommender-system framework that incorporates user psychology and social relationships to simulate the actual user and recommender interaction process, and whose theoretical analysis proves that echo chambers and homogenization traps, defined respectively as reduced recommendation diversity and homogenized user representations, will inevitably occur (Tang et al., 15 Aug 2025). In the broader literature, closely related models implement confirmation awareness through asymmetric link decay, selective rewiring, biased DeGroot weighting, state-dependent source influence, and biased interpretation of ambiguous signals (Ngampruetikorn et al., 2015, Gallo et al., 2020, Fernandes, 2022).

1. Canonical recommender-system formulation

In the formulation explicitly called the Confirmation-Aware Social Dynamic Model, there are $n$ users, $m$ items, and $c$ categories. Each item $j$ is represented as a vector $\mathbf{v}_j \in \mathbb{R}^c$, and each user $i$ at time $t$ is represented by a category preference vector $\mathbf{u}_i(t)$. Initial user vectors are constructed from positive and negative item histories,

$$\mathbf{u}_i(0) = \frac{\sum_{j \in \mathcal{V}_i^+} \mathbf{v}_j - \sum_{j \in \mathcal{V}_i^-} \mathbf{v}_j}{\left\| \sum_{j \in \mathcal{V}_i^+} \mathbf{v}_j - \sum_{j \in \mathcal{V}_i^-} \mathbf{v}_j \right\|_2}.$$

Users are also embedded in a social network, with $\mathcal{N}_i$ denoting the social neighbors of user $m$0 (Tang et al., 15 Aug 2025).

Recommendation is social-aware and stochastic. The representation used by the recommender is

$m$1

where $m$2 controls the extent of social integration. The probability of recommending item $m$3 to user $m$4 at time $m$5 is

$m$6

with $m$7 controlling the stochasticity versus personalization of the recommender. Recommendation lists are generated by sampling $m$8 items without replacement from this distribution (Tang et al., 15 Aug 2025).

User feedback is explicitly psychological. Confirmation bias is parameterized by $m$9, and leniency bias by $c$0. Positive and negative feedback probabilities for an item $c$1 are

$c$2

$c$3

User preferences then evolve as

$c$4

where $c$5 takes value $c$6 with probability $c$7 and $c$8 with probability $c$9. An equivalent expected update is

$j$0

with $j$1 (Tang et al., 15 Aug 2025).

This formulation is notable because it makes the feedback loop between recommender stochasticity, social integration, user psychology, and representation drift mathematically explicit. The reported empirical program uses two real-world datasets and one synthetic dataset with five well-designed metrics, and studies root factors from three level perspectives: the stochasticity and social integration degree of recommender at the system level, the psychological mechanisms of users at the user level, and the dataset scale at the platform level; it also reports four practical mitigation strategies that alleviate echo chambers and user homogenization at the cost of some recommendation accuracy (Tang et al., 15 Aug 2025).

2. Principal mechanisms for encoding confirmation awareness

Across the literature, confirmation awareness is not a single operator but a family of state-dependent interaction mechanisms. The following formulations recur.

Setting	Confirmation-bias encoding	Reported consequence
Fluctuating opinion networks	Agreeing links decay at rate $j$2, disagreeing links at $j$3, with $j$4	Segregation and stabilization of consensus (Ngampruetikorn et al., 2015)
DeGroot learning with link cuts	If $j$5, then $j$6 and weight is added to $j$7	Slower learning and higher polarization in symmetric networks (Gallo et al., 2020)
Bounded-confidence variants	RBCM rewires discordant links; UCM adds repulsive updating; RUCM combines repulsion and rewiring	Consensus at lower $j$8 in RBCM; stable two-peak polarization in UCM and RUCM (Vicario et al., 2016)
Confirmation-biased DeGroot	Discount factor $j$9	Consensus on strongly-connected graphs, but persistent polarization in weakly-connected cases (Alvim et al., 2021)

In fluctuating-network models of binary opinions, confirmation bias is implemented through link dynamics rather than through static homophily alone. Agents hold $\mathbf{v}_j \in \mathbb{R}^c$0, unconnected pairs link at rate $\mathbf{v}_j \in \mathbb{R}^c$1, and agreeing and disagreeing links have different decay rates. The associated mean-field local opinion is

$\mathbf{v}_j \in \mathbb{R}^c$2

where $\mathbf{v}_j \in \mathbb{R}^c$3 is the global opinion (Ngampruetikorn et al., 2015).

In DeGroot-based social learning, confirmation awareness is often represented by selective suppression of discordant influence. One model makes this suppression discrete and irreversible: agents cut neighbors whose initial beliefs are too far from their own, route the dropped weight to self-links, and then learn according to $\mathbf{v}_j \in \mathbb{R}^c$4 (Gallo et al., 2020). Another makes the suppression continuous: each influence term is discounted by the similarity factor $\mathbf{v}_j \in \mathbb{R}^c$5, yielding a nonlinear confirmation-biased extension of the classical DeGroot update (Alvim et al., 2021).

This range of mechanisms suggests that a confirmation-aware model is defined less by a particular state space than by the use of alignment-sensitive interaction kernels. In different papers those kernels operate on links, on weights, on source exposure, or on interpretation itself.

3. Consensus, segregation, and polarization

The macro-dynamics induced by confirmation awareness are model-dependent. In the fluctuating-network binary-opinion model, confirmation bias induces segregation of individuals with different opinions, and sufficient confirmation bias renders consensus globally stable for any local update rule within the spineless class. The global opinion obeys

$\mathbf{v}_j \in \mathbb{R}^c$6

and the update rules are classified into Type-I, Type-II, and the proportion rule. Type-I rules are concave for $\mathbf{v}_j \in \mathbb{R}^c$7 and drive consensus in the unbiased case; Type-II rules are convex for $\mathbf{v}_j \in \mathbb{R}^c$8 and support stable polarization in the unbiased case; the proportion rule is an interface with no drift in mean-field. Under confirmation bias, consensus time can be non-monotonic: for the proportion rule, small to moderate $\mathbf{v}_j \in \mathbb{R}^c$9 decreases consensus time, whereas large $i$0 increases it, and at strong bias $i$1 (Ngampruetikorn et al., 2015).

A different conclusion appears in the DeGroot model with confirmation-biased link cuts. In any symmetric network where agents put sufficient weight on themselves, $i$2, confirmation bias makes social learning slower, increases polarization at all intermediate steps, and redistributes asymptotic influence. Average convergence time is

$i$3

and slower mixing is characterized by a larger second-largest eigenvalue $i$4 of $i$5 (Gallo et al., 2020).

Continuous-opinion bounded-confidence variants generate yet another pattern. RBCM tends to promote consensus at smaller $i$6 thresholds than the BCM, while UCM and RUCM yield robust two-peak polarization and stable coexistence of two final opinions over a broad parameter range. In RUCM, the transition from multiple opinions to two groups and then to consensus is sharper and faster than in UCM, especially for larger $i$7 (Vicario et al., 2016).

In confirmation-biased DeGroot models based on the Esteban-Ray polarization measure, the existence of confirmation bias does not by itself imply permanent polarization. If the influence graph is strongly-connected, polarization eventually vanishes and agents’ beliefs converge to a common value; if the graph is a regular symmetric circulation, the unique belief value is determined; if polarization does not eventually vanish, then either there is a disconnected subgroup of agents, or some agent influences others more than she is influenced (Alvim et al., 2021). Mean-field analysis of a persuasive arguments model likewise identifies a transition from consensus to polarization induced by confirmation bias, with acceptance probability

$i$8

and reports polarized solutions emerging beyond a critical range of $i$9 (Banisch et al., 2024).

A common misconception is that confirmation bias has a single directional effect. The literature does not support that simplification. Depending on whether the bias acts through rewiring, averaging, repulsion, or source selection, it can stabilize consensus, slow convergence, amplify polarization, or make the long-run outcome depend primarily on connectivity conditions.

4. DeGroot, Bayesian, and source-driven extensions

Several confirmation-aware models generalize peer influence by adding external sources, ambiguous signals, or multidimensional state spaces. In cyber-social networks with individuals and news agencies, one extension of the DeGroot-Friedkin model makes agency influence state-dependent, so that individuals receive more weight from news agencies that are closer to their belief. The opinion update is a convex combination of innate opinion, neighbors’ opinions, and news agency content, and the paper characterizes conditions for convergence to a unique equilibrium as well as estimation and exact computation of steady-state values under non-linear and linear state-dependent weight functions (Mao et al., 2018).

A competitive-information variant formulates confirmation-aware information spread as a zero-sum game with two competing sources. The influence of source $t$0 and source $t$1 depends on current opinion through

$t$2

and the game admits a unique Nash equilibrium in pure strategies. The equilibrium depends on innate opinions, social topology, and confirmation-bias parameters, and confirmation bias moves the equilibrium towards the center only when the innate opinions are not neutral (Mao et al., 2019).

In a related inference problem, confirmation bias is represented as a strictly decreasing source-influence function $t$3, or in the piecewise linear case

$t$4

From observed opinion trajectories, the paper derives necessary and sufficient conditions for exact inference of network topology and confirmation-bias parameters, and presents an approximation algorithm for the case of unknown bias models (Mao et al., 2019).

Bayesian and signal-interpretation models push confirmation awareness further inside the cognitive update. In one well-mixed population model, agents probabilistically misperceive disagreeing incoming signals and then update by Bayesian inference; consensus is obtained only when confirmation bias is weak, while stronger bias yields disagreement configurations, with the critical value $t$5 for $t$6 (Nishi et al., 2013). In a networked learning model with ambiguous public signals, each agent interprets ambiguity according to a confirmation-bias parameter $t$7, and the long-run consensus remains biased; only two types of opinions can be formed, and both are biased (Fernandes, 2022).

Recent work also extends confirmation-aware dynamics to multidimensional opinions on multilayer systems. In the multi-layer framework of vector-valued opinions, source influence is modeled by nonnegative state-dependent functions of agent–source opinion mismatch, sufficient conditions are given for contractive convergence to a unique steady state independent of the initial condition, and affine confirmation-bias functions permit steady-state computation through a finite sign-consistency search (Abedinzadeh et al., 22 Mar 2026). This suggests that confirmation awareness can be incorporated without hard thresholds, and that source-design conclusions can change qualitatively when confirmation bias is ignored (Abedinzadeh et al., 22 Mar 2026).

5. Elections, epidemics, and online communities

Confirmation-aware social dynamics has been applied to political learning, epidemic response, and online-platform analysis. In political networks, confirmation bias can change both the speed and the path of collective learning. When agents cut links to ideologically distant neighbors and redirect weight to themselves, confirmation bias increases polarization in society, identifies a subset of agents that become more or less influential, and increases the likelihood of shock elections. Under a mean-field assumption and regular network, shock elections never happen without confirmation bias, but become possible with it; the same model also implies that fringe media move to a more extreme ideology as confirmation bias increases (Gallo et al., 2020).

On multiplex networks coupling communication and contact layers, mutual confirmation has been introduced into information-disease interacting dynamics. Individuals increase information transmission rate and willingness to adopt protective measures once they confirm the authenticity of news and severity of disease from neighbors’ status in multiple layers. Using a microscopic Markov chain approach, the model yields an epidemic threshold

$t$8

and shows that confirming aware neighbors on the communication layer increases the epidemic threshold more than confirmation on the contact layer, whereas confirmation from the contact layer yields a lower final infection density and a higher awareness density than confirmation from the communication layer. The reported implication is that explicit exposure of infection and awareness status to neighbors, especially those with real contacts, is helpful in suppressing epidemic spreading (Chen et al., 2021).

Online-community modeling has also adopted explicitly measured confirmation-bias variables. A gravity-well echo-chamber model replaces the originally constant user term with a dynamic user-specific confirmation bias variable computed from a user’s posting history and responses to posts of a wide range of viewpoints. The modified attraction is

$t$9

where $\mathbf{u}_i(t)$0 is derived from LLM-based support and alignment scores. The approach was validated on nineteen Reddit communities and is reported to improve detection of echo chambers while revealing community-level markers of information health (Jackson et al., 4 Sep 2025).

These applications show that confirmation awareness is not limited to abstract opinion exchange. It has been used to model electoral reversals, media positioning, public-health behavior, and echo-chamber identification. A plausible implication is that the same mathematical motif—alignment-sensitive weighting—can operate across substantively different domains.

6. Design implications, mitigation, and open theoretical themes

A recurrent design question is whether confirmation-aware systems can be made robust. In a social-planner model of confirmation-biased DeGroot learning, when the planner cannot observe agents’ beliefs, the optimal network is symmetric, vertex-transitive, and has no self-loops (Gallo et al., 2020). In agent-based opinion dynamics with confirmation bias and peer pressure, the number of opinion fragments first increases and then decreases to one as population identity scope becomes larger in a homogeneous population, and in heterogeneous settings even a small fraction of impressionable individuals who are sensitive to peer pressure could help eliminate public polarization when population identity scope is relatively large. The same model highlights the emergence of “impressionable moderates,” who are easily influenced, hold wavering opinions, and are of vital significance in competitive campaigns (Liu et al., 2020).

In recommender systems, mitigation appears as a controlled tradeoff rather than a full removal of the phenomenon. The confirmation-aware recommender model reports four practical mitigation strategies that alleviate echo chambers and user homogenization, but only at the cost of some recommendation accuracy (Tang et al., 15 Aug 2025). In fluctuating social networks, the non-monotonic dependence of consensus time on confirmation bias suggests an avenue for large-scale opinion engineering, since moderate bias can accelerate convergence while strong bias can slow or freeze it (Ngampruetikorn et al., 2015).

Theoretically, confirmation-aware models remain heterogeneous in their assumptions about topology, reversibility, and cognition. Some assume fast stochastic rewiring and spineless agents; some assume irreversible link cuts and increased self-weight; some treat discordance as repulsive rather than merely discounted; some embed the bias in source selection rather than peer interaction. This suggests that “confirmation-aware social dynamic model” is best understood as a model class centered on state-dependent acceptance, exposure, and influence, rather than as a single universally agreed formalism. Within that class, the principal comparative lesson is precise: confirmation bias changes not only equilibrium beliefs, but also who interacts with whom, which channels remain open, how quickly learning mixes, and whether consensus, segregation, or polarization becomes structurally favored.