Epistemic Networks, Collective Misperception, and the Manipulation of Social Knowledge

Mihnea C. Moldoveanu; Joel A. C. Baum

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Epistemic Networks, Collective Misperception, and the Manipulation of Social Knowledge

Mihnea C. Moldoveanu, Joel A. C. Baum

arXiv:2608.26075v1Aug 26, 2026cs.SI

TL;DR

The paper asks how collective beliefs arise from agents’ beliefs about one another, rather than from private opinions alone. It models these interactive states with a mutual-attribution tensor and signed network dynamics, showing that collective misperception separates into transient observation lag and persistent concealment. The framework also characterizes consensus, polarization, and instability through operator structure while marking the limits of its finite cognitive-consistency assumption.

Problem
First-order opinion models cannot represent the second-order structure of what agents take others to believe, while related misperception models often lack tractable higher-order dynamics.
Method
The paper represents interactive belief with a finite mutual-attribution tensor, separates private, expressed, and perceived belief, and models their evolution through signed influence and composition.
Results
Collective misperception decomposes into an observation-lag component that decays and a concealment component that persists while expression remains detached from belief.
Takeaways & Limitations
The framework makes collective epistemic pathologies analyzable as configurations and spectral regimes of a shared interactive-belief structure.
Takeaways & Limitations
Higher-order belief is generated from a finite array under cognitive consistency, setting aside the full modal structure of interactive knowledge.

Abstract

from arXiv · show

We investigate the structure of interactive beliefs in networks: the epistemic state in which agents hold, revise, and act on their models of the epistemic states of other agents. What a group believes depends on what each member agent takes the others to believe, and on what each takes the others to believe about still others. We posit that the proper unit of social-epistemic analysis is not the individual belief but the tensor of mutual attribution, the array that records what every agent takes every agent to believe. We separate three layers of this object: what is privately held, what is publicly expressed, and what is to be believed by others. Social belief evolves by contraction of the tensor against a signed matrix of epistemic influence. Collective misperception decomposes exactly into a component that observation dissolves and a component that observation cannot touch. Social conformity amplifies pluralistic ignorance without generating it. The stable forms of collective epistemic consensus, polarization, entrenched misperception, and instability are spectral regimes of a single operator. Higher-order belief reduces to walks in the epistemic network, under a stated assumption of cognitive consistency whose boundary we mark precisely.

1 Introduction: the epistemology of interactive belief

The paper treats collective belief as a structure of interactive attributions rather than an aggregate of private opinions. It develops a finite dynamical framework connecting higher-order belief, misperception, and social-epistemic pathologies.

The epistemology of interactive belief: Collective belief can diverge from every member’s private belief because it is produced by mutual attributions about what others believe.The seminar example illustrates a group acting on a false consensus that no individual privately holds.
The epistemology of interactive belief: The paper identifies the structure of mutual attribution—what agents believe, express, and take others to believe—as the unit of social-epistemic analysis.This structure is intended to explain collective epistemic states and their characteristic pathologies.
Interactive belief and its recursion: The paper connects higher-order belief to conventions, focal points, common knowledge, and epistemic game theory, where actions depend on beliefs about others’ beliefs.These traditions motivate the recursive structure the paper formalizes.
Interactive belief and its recursion: Interactive-epistemology traditions represent belief hierarchies as potentially infinite, whereas the paper uses finite models of mutual attribution.The finite representation is adopted under an explicit cognitive-consistency assumption.
Related literatures: Existing literatures separately address higher-order recursion, opinion dynamics, or misperception, while the paper combines these elements in one framework.The paper contrasts first-order opinion models with approaches that document private–public divergence but lack tractable higher-order dynamics.

2 The attribution tensor and the three layers of belief

The paper represents interactive belief with a tensor that records who attributes which belief to whom about each proposition. It separates private belief, public expression, and perceived belief so collective misperception remains visible and analyzable.

Why a tensor: A first-order N × K belief array cannot record what agent i takes agent j to believe, so interactive belief requires a third index.Fixing one proposition yields an N × N matrix of mutual attributions.
The attribution tensor: The attribution tensor B ∈ R^N×N×K records agent i’s estimate of agent j’s degree of belief in proposition p.Its indices identify the observer, target, and proposition.
Three layers of belief: The tensor’s diagonal recovers private beliefs, while off-diagonal entries represent agents’ models of other agents.For a fixed proposition, Bii is agent i’s private belief and Bij captures i’s attribution about j.
Scope of the tensor representation: The tensor is mainly a stack of proposition-separable attribution matrices until inferential coupling is introduced in Part III.The paper explicitly limits its claim to novel tensorial bookkeeping plus operator theory, leaving richer tensor factorizations open.
Three layers of belief: The framework distinguishes private belief bi, expressed belief xi, and perceived belief Bij as separate epistemic layers.The concealment gap is the difference between what an agent expresses and privately holds.
Three layers of belief: Keeping the layers distinct makes collective misperception a structured object rather than an undifferentiated failure.Epistemic error is Eij := Bij − bj, and the paper separates its possible sources for analysis.

3 The dynamics of interactive belief

Agents update private beliefs from the beliefs they attribute to others, with signed epistemic influence allowing cooperation, antagonism, and polarization. The resulting forcing term decomposes collective misperception into transient perceptual lag and persistent concealment.

Epistemic influence and updating: Agents revise private beliefs using an influence-weighted aggregate of the beliefs they attribute to others, not necessarily others’ actual beliefs.Positive weights encode movement toward attributed positions, while negative weights encode antagonistic movement away from them.
Epistemic influence and updating: Signed epistemic influence extends classical stochastic opinion dynamics by allowing antagonism and polarization as part of the same framework.The veridical-perception case recovers DeGroot dynamics; the distinctive behavior comes from departing from veridical perception and allowing signed weights.
Truth–misperception decomposition: Theorem 3.1 writes the update as bt+1 = W bt + ht, with ht = (W ⊙Et)1, separating transmitted private belief from misperception-driven forcing.The forcing vanishes identically when perception is veridical.
Truth–misperception decomposition: The forcing is endogenous: collective misperception generates a structured driving term through the Hadamard contraction of influence and epistemic error.Manipulation can therefore target the layers and channels that write into this forcing term.
Closing the system: Perception tracks public expression rather than private belief, so expression and perception dynamics are required to close the system.The closed state is the triple (bt, xt, Bt), whose forcing is generated by laggy perception and strategic expression.
The two channels of collective misperception: The perception-lag channel decays when expression becomes stationary, whereas the concealment channel persists because observation tracks the distorted expression.Thus better observation closes lag but cannot remove concealment while agents continue not to say what they believe.
The two channels of collective misperception: Pluralistic ignorance is the residual forcing after perception lag closes, and conformity amplifies this concealment residue without generating it.A systematic expressive bias supplies the seed; conformity increases its magnitude.

4 Higher-order belief as composition, and its philosophical boundary

The paper replaces infinite higher-order belief hierarchies with finite compositional walks, while marking the precise boundary where modal content such as common belief is lost. Its first-order dynamical results remain invariant to this higher-order representation, whereas walk-based applications depend on compositional consistency.

Philosophical boundary: The compositional representation replaces an infinite hierarchy with a finite, tractable structure, but it does not represent common belief as a modal limit.The paper treats this as a deliberate trade-off for modeling finite agents who observe, express, conform, and update in real time.
Higher-order belief as walks: Higher-order belief is represented by powers of the first-order attribution slice, so an L-level recursion corresponds to directed walks of length L.The path-aggregation theorem identifies matrix powers with weighted walks in the epistemic network.
Higher-order belief as walks: Closed walks are collected by diagonal entries and summed by the trace; their contributions decay when ρ(B) < 1 and may become unbounded when ρ(B) ≥1.The spectral radius determines whether recursive attribution remains bounded as belief depth increases.
Philosophical boundary: The first-order dynamical theorems—including consensus, polarization, distortion, oscillation, closure, and spectrum coupling—do not depend on composed higher-order attributions.The stacked operator, its spectrum, and its fixed point use only first-order entries B_ij.
Philosophical boundary: The walk calculus and its adversarial descendants are the results for which the cognitive-consistency assumption is genuinely load-bearing.The paper distinguishes these composition-dependent results from the first-order results that remain valid when higher-order attributions are free.
Philosophical boundary: Walk-based applications use quantitative attribution weights rather than modal predicates, so echo multipliers, closed-walk mass, audits, and compression remain valid without asserting common belief.These quantities are defined through walk sums or resolvent entries, and the paper states that the modal structure they omit is unused.

5 Regime I: Consensus

With cooperative influence, veridical perception removes misperception and private beliefs converge to a network-weighted consensus. The outcome reflects influence centrality rather than a neutral average of initial opinions.

Consensus conditions: Under row-stochastic, primitive influence and zero misperception forcing, private beliefs converge to a common value.The dynamics reduce to bt+1 = Wbt.
Consensus value: The consensus value is the π-weighted average of initial beliefs, where π is the normalized left Perron eigenvector of W.π measures each agent’s eigenvalue centrality in the influence network.
Network weighting: Network position determines epistemic weight: agents attended to by influential agents receive greater weight in the final consensus.Consensus therefore reflects structured aggregation rather than simple averaging.
Worked example: The central placement of disagreeing agents pulls the consensus below the naive mean, while peripheral agreement receives less weight.Agents 2 and 4 each carry influence weight 0.2222, whereas agents 1 and 5 each carry 0.1746.
Worked example: 0.0444: all five beliefs converge to this mild net agreement within roughly twenty rounds in the worked example.The result is below the unweighted initial mean of 0.08 because disagreeing agents occupy central positions.

6 Regime II: Polarization

Signed influence extends the consensus dynamics to a polarization regime in which structurally balanced antagonism produces two internally coherent, mutually opposed camps. The bipolar outcome is encoded in the sign structure rather than generated by linear dynamics from an unpolarized state.

Structural balance: Under structural balance, W = DW+D with primitive row-stochastic W+, the signed dynamics are a sign-gauge transformation of cooperative consensus.D assigns agents to camps and flips exactly the cross-camp influence signs.
Polarized attractor: The limiting beliefs form equal-and-opposite blocs, ±(π⊤Db0), with camp membership determining the sign.The same cooperative dynamics operating underneath determine the common magnitude.
Polarization mechanism: Signed influence can produce two internally coherent camps that repel one another instead of reaching agreement.Negative weights encode antagonistic relations such as moving away from another agent’s position.
Scope: The linear result characterizes consequences of existing antagonistic structure but does not explain how an initially unpolarized population becomes polarized.That emergence requires a change in sign structure or a coupling-induced instability outside the basic linear result.
Worked example: +0.1 and −0.1: the balanced six-agent example converges to these exact camp values, with no agent remaining near zero.Cooperative within-camp influence erases initial disagreement, while antagonistic cross-camp influence drives separation.

7 Regime III: Entrenched distortion and the mechanism of pluralistic ignorance

Entrenched distortion arises when expression departs from private belief and conformity feeds that biased climate back into belief. Perception eventually tracks expression, but cannot remove the persistent concealment fixed point.

Fixed-point structure: The closed system maps expressive bias to steady-state concealment through an invertible linear image, producing a fixed point displaced from sincere dynamics.The displacement is cumulative distortion imposed by network misperception.
Two-channel decomposition: At the fixed point, perception lag vanishes while concealment persists, so the remaining collective distortion is the expression–belief gap.Perception tracks expression rather than private belief.
Source and amplification: Persistent pluralistic ignorance requires expressive bias; conformity amplifies distortion but cannot generate it from sincere expression.When β = 0, even a conformist population converges without collective self-deception.
Worked example: ∥c∗∥≈0.739: the worked example yields mild, clustered private beliefs but widely dispersed expressions that differ substantially from those beliefs.Agent 1 expresses +0.60 while agent 5 expresses −0.16, and agents conform to the biased perceived climate.
Parameter dependence: Concealment equals the expressive bias at α = 0, grows with conformity, and diverges as α →1.At the limit, agents abandon private conviction entirely and the population loses anchoring to actual beliefs.
Parameter dependence: The steady-state distortion is independent of perception rate γ, although γ controls how quickly the system reaches it.Pluralistic ignorance is therefore characterized as a pathology of the expression environment rather than perception.

8 Regime IV: Perpetual instability

Perpetual instability occurs when the influence operator has a unit-modulus eigenvalue other than 1, producing belief cycles that neither decay nor grow. Pure mutual antagonism yields a period-two exchange with no consensus or stable resting state.

The oscillation regime is the failure to settle: beliefs cycle indefinitely without converging.
Two purely antagonistic agents swap beliefs every round, creating a period-two cycle from (1, −0.3) to (−0.3, 1).
The eigenvalue λ = −1 on the unit circle obstructs convergence in the two-agent antagonistic example.
This instability differs from consensus, stable polarization, and fixed distortion because the population remains in perpetual churn.
A unit-modulus eigenvalue off the real axis produces persistent quasi-periodic churn, placing the dynamics at the boundary of stability.Such modes neither decay nor grow; they rotate and prevent convergence.

9 The four regimes as limits of one process

The paper unifies consensus, polarization, entrenched distortion, and oscillation as parameter-dependent regimes of one closed belief, expression, and perception process. A stacked operator governs the coupled dynamics, while the closure theorem separates transient perception effects from persistent expressive bias.

Four collective epistemic regimes are limits of one generative process, reached by varying conformity, perception, and influence-sign parameters.
At the fixed point, perception becomes exact and the perception channel vanishes, leaving concealment as the asymptotic forcing.
Steady-state concealment is a nonsingular linear image of expressive bias: it vanishes exactly when β = 0 and is amplified without bound as conformity approaches 1.
The three-layer dynamics are stacked into one operator T whose spectral properties determine the joint evolution of private, perceived, and expressed belief.
Under ¯w < 1, ¯α < 1, and positive minimum perception rate, the closed system converges geometrically to a unique globally attracting fixed point.
Changing conformity, trust-sign structure, and influence coherence moves populations continuously among consensus, polarization, distortion, and instability.

10 Activating the third mode: inferential coupling

Inferential coupling activates the proposition dimension, making belief dynamics depend on a Kronecker-product operator rather than independent proposition slices. Strong enough cross-proposition reinforcement can destabilize individually stable issues.

Inferential coupling contracts the belief tensor across both agent and proposition modes, yielding the multilinear update bt+1 = WbtL⊤.
The coupled spectrum consists of products µλ of eigenvalues from L and W, so stability depends on L ⊗ W.
If ρ(W) < 1 but ρ(L)ρ(W) > 1, coupled beliefs diverge even though every proposition slice is stable in isolation.
Cross-proposition reinforcement creates instability through loops in which belief in one issue raises belief in another and feeds back.
The coupled closure theorem holds when ¯ℓ¯w < 1, ¯α < 1, and γmin > 0, with channel-separation conclusions unchanged.
In the worked example, δ = 0.15 gives ρ(L ⊗W) = 0.92 and decay, whereas δ = 0.35 gives ρ = 1.08 and divergence.

11 The attack surface: three layers, three families

The paper classifies social-knowledge attacks by the epistemic layer they target: content, perceived climate, or observation medium. One-shot state attacks decay under stability, whereas persistent or channel-based attacks can produce durable displacement and shift in relative cost with conformity.

Three attack families: Inferential obstructions target content by altering beliefs or the inference operator, while dramaturgical exploitations target perceived climate through perception and expression.
Three attack families: Context manipulations target the observation medium by corrupting how one layer observes another without directly changing beliefs or expressions.
Robustness and persistence: Any one-shot perturbation of the state decays geometrically and converges to the same fixed point as the unattacked trajectory.
Robustness and persistence: The transience guarantee assumes stable dynamics and an honest observation medium; persistent injection and corrupted media fall outside those conditions.
Cost and vulnerability: For symmetric influence, dramaturgical gain grows as 1/(1 − α), whereas inferential gain is α-independent.
Cost and vulnerability: Above α∗, climate and medium attacks dominate content attacks, and their advantage grows without bound as conformity rises.
Cost and vulnerability: The efficient attack surface shifts from content in low-conformity populations to climate and medium in high-conformity populations.
Cost and vulnerability: In the worked instance, the crossover is α∗ = 0.64, and at α = 0.9 dramaturgical advantage is 3.6-fold.

14 Context manipulation: when robustness is revoked

Context manipulation corrupts the observation medium rather than injecting false content, producing permanent belief displacement that disclosure cannot remove. The section distinguishes this architecture-level attack from episodic state attacks and expressive-climate manipulation, which require different remedies.

Context manipulation: Under context manipulation, the system still converges, but P∗ = A and beliefs remain permanently displaced even with fully sincere agents.The displacement is a nonsingular linear image of the corruption.
Attack classes: Episodic state attacks decay after injections stop and can be addressed by exposure, whereas structural sedimentation persists independently of discovery.The two attack classes therefore do not admit the same cure.
Context manipulation: Context manipulation corrupts the medium through which agents observe one another, creating a boundary condition rather than a state attack.Theorem 14.1 treats the resulting distortion as architecture-level.
Remedy: Disclosure alone leaves the displacement unchanged while the corrupted medium remains, so architectural repair is required.Restoring an honest medium makes the perception term vanish and allows residual displacement to decay.
Expression environment: Expressive attacks create ambient pluralistic ignorance through an induced bias β̸ = 0, while conformity α amplifies the resulting concealment equilibrium.The equilibrium is c∗ = Lβ and depends on the standing expression environment.
Expression environment: Making endorsement more visible does not change c∗ when dissent costs remain, and can deepen distortion by increasing effective conformity.Interventions that lower the cost of sincere dissent are the ones that move the equilibrium.

16 Backchannels, trust conduits, and the closed epistemic walk

Higher-order attribution is analyzed as a network of walks, where open chains terminate in private belief and closed chains generate apparent corroboration through recirculation. The echo multiplier is governed by the resolvent and becomes unbounded as attribution spectral radius approaches one.

Open versus closed walks: Closed walks manufacture apparent evidential weight through network closure rather than supplying an independent source.The backchannel and trust conduit are formally the same closed-walk object.
Walk calculus: Higher-order attribution reduces to weighted walks in the attribution network, with each chain weighted by the product of its edge weights.When ρ(B) < 1, deep backchannel gossip decays geometrically with depth.
Echo multiplier: As ρ(B) approaches 1, the resolvent develops a pole and a planted attribution generates unbounded apparent corroboration.The echo multiplier is a resolvent entry, while sensitivity of total closed-walk mass is given by the squared resolvent.
Defense: Operating with spectral slack and discounting corroboration from closed chains are the proposed defensive rules against recirculated attribution.The danger is closure, not merely attribution-chain depth.
Open versus closed walks: Open attribution chains terminate in a private belief and constitute genuine evidence, whereas closed chains return to their origin without an independent believer.The audit rule is to trace each attribution to its terminal.
Cognitive consistency: Compositional cognition stores N^2 first-order attributions rather than N^L free depth-L parameters, making inconsistent higher-order claims detectable.A claimed second-order value can be checked against the agent’s own composed estimate.

17 Defense by design

The paper matches defenses to the layer under attack: exposure for inferential obstruction, lower conformity for expressive climates, architectural repair for corrupted media, and spectral slack or reactivity monitoring for structural hazards. It also shows that asymptotic spectral safety does not guarantee transient safety in hierarchical networks.

Layer-specific defenses: Exposure is sufficient for inferential obstruction because one-shot state attacks decay after the falsehood is revealed.This remedy applies to content-layer attacks, not to climate- or medium-layer attacks.
Layer-specific defenses: Lowering conformity and protecting sincere expression defend against dramaturgical exploitation of the expression environment.The secret ballot approximates α ≈ 0 by reading private belief rather than public discourse.
Layer-specific defenses: Context manipulation requires verifiable, unranked access to actual expressions because disclosure is inert against medium-level distortion.The distortion is located in the observation architecture.
Layer-specific defenses: Spectral slack provides stability reserves against structural attacks, while the paper recommends flattening deep hierarchies or monitoring them at the reactivity timescale.The relevant reserves are 1 − w̄ and 1 − ρ(L)ρ(W).
Transient safety: In a depth-16 hierarchy with ρ(T) = 0.10, the pseudospectrum bulges toward the unit circle, with a pseudospectral-radius excursion of 0.85.The contrast marks non-normality as the source of transient amplification.
Transient safety: Hierarchical operators can have Γ > 1 while ρ(T) is arbitrarily small, so eigenvalue-based safety does not bound transient response.This is the proved reactivity result of Proposition 18.1.
Transient safety: The depth-scaling of transient gain is numerical rather than asymptotic: it is observed through N ≤ 28 and saturates at Γ ≈ 1.37 when γ → 1.The paper separates this empirical scaling from the theorem that reactivity is possible at arbitrarily small spectral radius.
Adversarial reactivity: An optimal injection reaches 4.7 times the response of an equal-magnitude random injection, and repetition at k∗ can sustain distortion.The repeated attack exploits the reactivity horizon despite eventual asymptotic decay.

19 What the theory says about social knowledge

The paper makes mutual attribution, rather than individual belief, the unit of collective epistemology and uses it to unify belief dynamics, misperception, manipulation, and robustness. Its conclusions are exact within a fixed linear scaffold, while nonlinear dynamics, endogenous scaffold formation, and strategic attackers remain outside scope.

Core object: The attribution tensor records who takes whom to believe what, making interactive structure primary over the distribution of private convictions.Private belief is a diagonal special case of attribution.
Collective misperception: Collective misperception decomposes into perception lag, which observation dissolves, and concealment, which observation cannot touch.Pluralistic ignorance is the residue of concealment after perception lag closes.
Collective misperception: Conformity amplifies pluralistic ignorance but cannot generate it without an expression bias that gives agents a reason to say other than they believe.Reducing the cost of sincere dissent, rather than increasing visibility, changes the equilibrium.
Spectral regimes: Consensus, polarization, entrenched distortion, and instability are spectral regimes of one parameter space shaped by conformity, trust signs, and influence coherence.Small parameter changes can move a population across qualitative boundaries.
Layered manipulation: Manipulation targets content, expression climate, or observation medium, and the corresponding remedies are not interchangeable.Exposure can be insufficient or counterproductive when the attack is architectural or expressive.
Robustness: Hierarchical epistemic structures store transient attack capacity invisible to eigenvalue analysis, with computed gain increasing over the accessible depth range.The robust conclusion is reactivity; the precise growth law remains an observation rather than a theorem.
Scope: The linear model’s superposition makes one-shot attacks globally decay, whereas nonlinear thresholds or cascades could permit transient tipping into another basin.The reactivity result identifies high-gain directions where such tipping may be easiest to induce.
Scope: The analysis fixes the epistemic-institutional scaffold and leaves its endogenous formation, including how Q, W, and L are produced, for complementary inquiry.A future bridge would study co-evolution between belief dynamics and epistemic structure.

21 Conclusion

The paper identifies mutual attribution as the proper unit for social-epistemic analysis and represents collective belief as a tractable dynamical system. Its regimes, manipulation costs, and transient fragility become explicit and calculable within this framework.

21 Conclusion: Collective belief is modeled as a tensor of mutual attribution rather than as individual beliefs or their aggregation.The tensor records what each agent takes each agent to believe, express, and perceive.
21 Conclusion: The tensor representation turns collective belief into a dynamical system whose stable forms are spectral regimes of one operator.The framework includes consensus, polarization, stationary distortion, and oscillatory instability regimes.
21 Conclusion: Collective self-deception is the residue of a concealment channel that conformity amplifies but cannot create.The framework distinguishes the dynamics of collective misperception from the mechanisms that intensify it.
21 Conclusion: Hierarchical structure creates transient, non-normal amplification that eigenvalue analysis alone cannot detect.This transient fragility is presented as a theorem within a system small enough to compute by hand.
21 Conclusion: The four regime results are restated from classical literature to make the paper self-contained, while the closure theorem operates on fully specified objects.The consensus proof uses Perron–Frobenius structure, and the oscillation proof shows generic nonconvergence for unit-modulus eigenvalues.