An arXiv preprint presents a mathematical model in which perception catches up at equilibrium while concealment linked to expressive bias remains. In the model, conformity amplifies the settled distortion, but this is a property of the equations, not a reported finding from a recruited participant sample.
The framework keeps private belief, expressed belief and perceived belief as separate layers of the epistemic state. It combines their updates into a single affine recursion governed by a stacked operator, T, so the three parts can be studied together.
Under the theorem’s stated assumptions, the closed system converges geometrically from every initial condition to a unique fixed point when the operator’s spectral radius, ρ(T), is below one. In plain terms, repeated updates settle to one mathematical endpoint; that endpoint can still contain concealment.
One framework, several possible endings
The theory puts consensus, polarization, entrenched distortion and perpetual instability into four spectral regimes of one common operator. They are mathematical regimes defined by operator properties, not four outcomes measured in a population.
Beliefs about other beliefs are represented as weighted directed walks through the network. Matrix powers add the weights of one-step, two-step and longer paths, making higher-order attribution finite and computable. The construction does not represent common knowledge as a full modal predicate.
The specified linear coupled operator permits the joint system to diverge even when every proposition is stable in isolation. The stated condition is ρ(W) < 1 while ρ(L)ρ(W) > 1. That result belongs to the coupled model, not to a general claim about networks.
Attacks leave different traces
The attack analysis sorts interventions by target: the proposition content, the expressed climate around it, or the observation medium. That taxonomy separates disturbances to the modeled state from disturbances to the channel carrying observations.
Under the stated stability conditions, a state perturbation has a transient signature. Its displacement decays geometrically, and the trajectory returns to the same fixed point as the unperturbed one.
Fixed corruption of the observation medium has a different signature. The system still converges, but equilibrium retains a nonzero perception-lag term and the fixed point is permanently shifted: M* = x* + A and P* = A, even in the stated case β = 0. In model terms, the channel change persists instead of washing out like a state perturbation.
For the stated symmetric influence family, expression-layer gain diverges as conformity α approaches one, while belief-layer gain is independent of conformity. The comparison therefore contains a conformity-dependent crossover in the modeled relative efficiency of the two attack families. The closed form applies to uniform injections and that specified network family; it is not a real-world cost estimate.
Another amplification measure comes from repeated attribution. For nonnegative primitive attribution weights, the resolvent-based echo multiplier becomes unbounded as the network’s spectral radius approaches one. The quantity tracks weighted walks, not common knowledge.
Stability can hide a short-term surge
A feed-forward hierarchy with fast perception exposes a different behavior. Its stacked operator is non-normal, its transient gain exceeds one, and its spectral radius can be arbitrarily small. The model therefore allows a short-term response to grow even under a very small long-run spectral radius.
A separate numerical calculation covered hierarchy sizes from three through 28, with perception rate 0.9 and spectral radius 0.10. Across that range, the gain fit Γ(N) ≈ 1 + 1.0√N, with R² = 0.9998; logarithmic and linear comparisons were lower, at 0.98 and 0.975. The fit is strong over the computed range, but it is not established as an asymptotic law.
In the instantaneous-perception limit, the calculation was extended to a hierarchy of 256 agents, and the gain saturated near 1.37 as the perception rate approached one. That saturation means the square-root pattern should not be read as an unlimited prediction; its asymptotic status remains unresolved.
The boundary of the result
The document is an arXiv preprint, version 1, dated 26 August 2026. It describes a finite set of agents and propositions, with no recruited participant sample.
That scope changes how the results should be read. The regimes, fixed points and attack responses are consequences of the specified mathematical operators; they do not establish that real people or institutions converge, polarize, conceal information or show the same vulnerability.
Two boundaries are especially important. The higher-order construction uses compositional attribution, so it makes recursive structure finite but does not capture the full modal structure of common knowledge. The convergence result is conditional on the theorem’s stated assumptions.
The supplied analysis leaves two questions open: how these model relationships compare with observations of real institutions and whether hierarchy-depth reactivity has an asymptotic form. For now, the work maps possible behavior inside a finite model: expressive concealment can remain at equilibrium, while hierarchical amplification can be transient.
Paper data and sources
Original title: Epistemic Networks, Collective Misperception, and the Manipulation of Social Knowledge
Authors: Mihnea C. Moldoveanu, Joel A. C. Baum
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text