Social influence rules produce either agreement or entrenched camps, depending on a single threshold of confidence.
Each person in a population holds an opinion on a scale from 0 to 1. Pairs meet and nudge each other toward compromise — but only if they already mostly agree. If the gap is too wide they ignore each other.
This is the Deffuant model. Its single parameter, the confidence threshold, decides whether the society ends in agreement or deadlock.
Each dot is an agent. At each step, two agents meet; if they agree within the threshold, they move halfway toward each other.
Figure 1. Agents arranged vertically by index; horizontal position is their opinion (0 on the left, 1 on the right). Dot color reinforces this: blue = 0, red = 1. Two agents within the threshold average toward each other; pairs further apart ignore each other. Hover any agent to see its confidence window. A high threshold leads to consensus; a low one leads to polarization.
The threshold 0.25 is critical. Near it, small differences in starting opinions tip the population toward consensus or fragmentation.
As the threshold drops, the number of surviving clusters jumps from one to many — a phase transition.
Figure 2. The number of surviving clusters jumps as the threshold crosses the critical region around 0.25. Hover to see the simulated values.
Restricting interactions to a network of neighbors creates echo chambers. A few random long-range connections break them open.
Figure 3. Opinion dynamics on a small-world network. Green edges connect agents within the confidence threshold (active); grey edges exceed the threshold (ignored). Node color encodes opinion from blue (0) to red (1). Rewiring shortcuts break down local echo chambers and encourage global consensus.