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EXP-DENSITY-MIN

artifact

Minimum Interaction Density for Hierarchy Formation at N=3

2026-03-10 L1 level paper3 3 runs $0.00
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In plain language

Does hierarchy emerge immediately from the first interaction, or does it need "warm-up" time? Three agents with different thinking styles — but only 5 turns each instead of the usual 20. If hierarchy still appears — the structure is instantaneous.

What we found

UNDERPOWERED (N=3): No predictions scored

Predictions we made before running

0/3 confirmed

Technical details

Research hypothesis

QUESTION: Does trophic hierarchy at N=3 require iterative reinforcement, or does it emerge immediately from the first few interactions?

CONTEXT: exp-asym showed TR_range ~0.444 at N=3/60rds (20 rds/agent). exp-pent70 showed that rounds/agent drives VARIANCE (measurement precision). But we don't know: is hierarchy PRESENT at low rds/agent (just noisy), or does it need time to FORM (iterative positive feedback)?

This is the difference between: (A) INSTANT STRUCTURE: persona asymmetry → directional flow from round 1. Hierarchy is a structural consequence of different cognitive styles sharing a text space. More rounds = better measurement, same signal. (B) ITERATIVE BUILDUP: hierarchy emerges through positive feedback — early vocabulary choices get reinforced, creating path-dependent trophic positions. More rounds = genuinely stronger hierarchy.

TEST: N=3 with 15 rounds (5 rds/agent) — 4x fewer than exp-asym's 20/agent. If hierarchy forms: structure is fast/instant. If flat: needs iteration.

OPEN QUESTION ADDRESSED: "emergence threshold: at what N agents + interaction density do emergent properties appear?"

Experimental setup

Type: simple

Condition Parameters
LOW_DENSITY persona: PERSONA, interaction: LIVE, note: Same setup as exp-asym PERSONA_LIVE but with 5 rds/agent instead of 20.

Factors: density (LOW)

Parameters

n_agents
3
n_rounds
15
n_runs_per_condition
3
model
gemini-2.5-flash-lite
temperature
0.9
scheduler
round_robin

Trophic Ratios by Condition

Mean trophic ratio per agent across runs. Error bars = ±1 std dev. Higher TR = more upstream (exporter).