Belief Dynamics

Opinions move only when they already almost agree. On a complete graph of pairwise encounters, a wide confidence bound can pull the population toward one value; a narrow bound leaves several camps that no longer hear each other. Dispersion and cluster count are the regime readout—not a forecast of politics.

The Deffuant–Weisbuch bounded-confidence model represents each agent’s opinion as a continuous number on [0,1]. At each step two agents are chosen. If their absolute difference is smaller than a confidence threshold ε, they compromise by a factor μ:

x ← x + μ (x′ − x), x′ ← x′ + μ (x − x′),

with μ typically in (0, ½]. If the difference exceeds ε, nothing happens. Deffuant, Neau, Amblard and Weisbuch showed that high ε yields convergence toward a shared opinion, while low ε produces several opinion clusters: members of a cluster agree among themselves and have drifted out of hearing range of other clusters.

The rough cluster-count lore—on the order of 1/(2ε) peaks for opinions on the unit interval, with edge corrections—is a finite-population, detection-dependent guide from their simulations, not an exact theorem for every run. Population size, μ, initial conditions and how clusters are counted all move the observed number. This sphere’s telemetry therefore reports an operational cluster count (gap and minimum-share thresholds) and a dispersion equal to opinion variance. Dispersion is not “polarization”; separated camps may be mild or extreme.

Homogeneous ε is the canonical paper’s working assumption. Heterogeneous openness—different agents with different ε—and fixed agents who never update (zealots / anchors) are explicit extensions. Heterogeneous thresholds can create bridge agents who still hear across camps that narrower agents ignore. Anchors pin local opinion and can drag neighbours who remain within ε.

AL.Floys places agents on a fixed complete graph: every pair is eligible; the interaction network does not rewire. That distinguishes Belief Dynamics from Adaptive Networks on ecospheres, where topology co-evolves and acceptance is a soft linear function of state gap. Here acceptance is the hard DW threshold (or one-sided acceptance when heterogeneous ε makes only one agent willing). Screen position is not social space: the renderer maps opinion to the horizontal axis and confidence to the vertical axis (y ∝ 1 − ε). Points that look “near” vertically may still refuse to interact if their opinions differ by more than ε.

Most early opinion models used binary states and imitation. Deffuant et al. asked what changes when opinion is continuous and influence requires prior proximity—an “already close enough to discuss” rule. Their 2000 Advances in Complex Systems paper studied both well-mixed encounters and lattice neighbourhoods, documenting consensus versus fragmentation as ε varies. Later work added extremists, media, networks and heterogeneous confidence; those extensions matter for reading this sphere’s anchor and bridge presets, but the core mechanism remains pairwise bounded compromise.

Aspect	Current implementation
Graph	Fixed complete graph; random pair sampling.
State	Opinion in [0,1]; optional per-agent ε.
Update	Symmetric DW compromise when both (or the willing side under hetero ε) accept.
Extensions	Heterogeneous ε, zealot fraction, PERSUADE brush.
Clusters	Gap Δ = 0.04, minimum share 0.03; not a force-layout community.
Dispersion	Opinion variance.
Visual	x = opinion, y = 1 − ε; optional ε halos; zealots larger.

Under Wide Hearing (ε = 0.4) the cloud should tighten and cluster count fall toward one. Under Many Camps (ε = 0.12) several vertical bands persist; accepted-interaction ratio stays lower because many random pairs are out of range. Anchored Poles seeds immobile extremes—watch whether free agents freeze into camps pulled by those poles rather than a single mean. Bridge Agents turns on heterogeneous ε so wide-hearing individuals can still couple camps that narrow agents cannot. Movement telemetry near zero means the configuration has effectively settled, not that the model “decided the truth.”

Start from a preset chosen for the regime you want, then vary ε or Anchors and restart. Interactions/s changes tempo only. Agents changes finite-size grain. On the ecospheres simple dock, use PERSUADE to pull nearby non-zealot opinions toward the pointer’s opinion coordinate; do not rely on RELEASE (the force tool is listed but not implemented as a pointer path) or on Toggle anchor (field actions are hidden on the simple surface). Prefer the Anchored Poles / One-Sided Anchor presets and the Anchors slider when you want fixed agents.

• Confidence ε. The hearing radius. This is the primary bifurcation parameter in the canonical model.
• Compromise μ. How far each accepted pair moves. Mainly affects speed and residual width, not the qualitative cluster count story.
• Anchors (zealotFraction). Implementation extension: immobile agents that still influence others who hear them.
• Heterogeneous ε / ε spread. Opens bridge-agent regimes; not in the homogeneous 2000 simulations as a varied law.
• Interactions/s. Presentation rate of pairwise encounters.

Wide Hearing

ε = 0.4, no anchors. Showcase consensus drift.

Many Camps

ε = 0.12. Persistent fragmentation under narrow confidence; selected from regime-sweep evidence.

Anchored Poles

ε = 0.25 with 6% zealots. Fixed extremes shape the field.

Bridge Agents

Heterogeneous ε with spread 0.18. Wide-hearing agents can couple camps.

One-Sided Anchor

Fewer zealots, slightly narrower ε—asymmetric pull.

Many Camps Mobile

Same narrow-ε idea at N = 550 for mobile budgets.

Recommended order: Wide Hearing → Many Camps → Anchored Poles → Bridge Agents.

• PERSUADE moves non-zealot agents near the pointer toward the opinion implied by pointer x.
• RELEASE is not a reliable visitor experiment on this dock—prefer presets and the Anchors control.
• Cluster and dispersion chips read model state, not a layout clustering algorithm.

This is not a predictor of elections, polarisation in the political sense, media effects, or individual psychology. Opinions are abstract scalars. “Belief” in the public title is visitor language for continuous opinion under bounded confidence; the equations do not encode truth, identity or emotion.

Separated clusters are fragmentation under a hearing threshold. Calling every multi-cluster state “polarization” overclaims. Anchors are an authored extension, not Deffuant et al.’s homogeneous baseline. Adaptive Networks on this atlas is a different model—do not narrate Belief Dynamics as if edges were rewiring.

Belief Dynamics isolates social influence without adaptive topology. Beside Associative Memory’s attractor repair and Naming Game’s convention formation, it shows another Mind mechanism: local rules that create global agreement or lasting camps. It also clarifies, by contrast, what Adaptive Networks adds when the graph itself can change.

Foundational and primary sources

1. Guillaume Deffuant, David Neau, Frédéric Amblard and Gérard Weisbuch, “Mixing beliefs among interacting agents,” Advances in Complex Systems 3, 87–98 (2000). DOI: 10.1142/S0219525900000078.

Implementation evidence

2. AL.Floys belief-dynamics regime sweep under docs/evidence/mind-expansion/belief-dynamics/ (pack @uridolan/al-floys-engines@0.10.1): ε∈{0.1,0.15,0.2,0.3} cluster medians 4→1 at N = 300 in the harness.