EXP-TOPOLOGY

informative

Topology Series — Chain vs Cycle vs Star Hierarchy Formation

2026-03-14 L3 level 6 runs

In plain language

Three AI agents communicate in a text space. Information flows in one direction: Alpha writes — Beta reads only Alpha — Gamma reads only Beta. No feedback: nobody reads Gamma. Will a hierarchy gradient emerge — Alpha = word source, Beta = relay, Gamma = receiver? Or will unidirectional flow break convergence entirely?

What we found

Topology constrains hierarchy: chain (directed acyclic), cycle (distributed), star (hub-mediated). All artifact on legacy platform — methodology lesson.

▶ Technical details

Research hypothesis

OPEN QUESTION (from alignment.yaml): "cyclicity confound: ANSWERED. cycle topology kills hierarchy. PROMOTE: topology × hierarchy mapping (star, chain next)."

CONTEXT — designed-topology series results so far: Complete (exp-asym, N=9): TR_range=0.444, VP=0.075, C=0.344 — STRONG hierarchy Star (exp-topo-star, N=1): TR_range=0.141, VP=0.243, C=0.778 — WEAK hierarchy, hub=SINK Cycle (exp-topo-cycle, N=1): TR_range=0.071, VP=0.167, C=0.715 — FLAT hierarchy

KEY FINDING FROM STAR: high in-degree = SINK, not source. Hub absorbs vocabulary from both spokes. This inverts naive "more connections = more influence" intuition. Mechanism: high in-degree → more vocabulary exposure → more imports.

CHAIN TOPOLOGY: A→B→C (directed, acyclic, NO feedback).

Alpha: reads ONLY seed text. In-degree=0. No agent input. Pure generator.
Beta: reads ONLY Alpha. In-degree=1 (from Alpha).
Gamma: reads ONLY Beta. In-degree=1 (from Beta). Nobody reads Gamma.

WHY CHAIN IS THE CRITICAL TEST: 1. Zero in-degree for Alpha → CANNOT absorb (in-degree=0 vs star hub's in-degree=2). If star hub was sink BECAUSE of high in-degree, then chain Alpha should be pure SOURCE. This tests the causal mechanism. 2. Acyclic → no feedback loops. Information flows ONE WAY: A→B→C. Unlike cycle (A→B→C→A) there is no return path. 3. Maximum asymmetry in a 3-agent graph. Every agent has a unique structural position (source/intermediate/sink). 4. Predicts GRADIENT: unlike star (binary: hub vs spoke) or cycle (uniform), chain should produce THREE distinct trophic levels.

BASELINE DATA: Complete (N=9): TR_range=0.444 ± 0.15, VP=0.075 ± 0.04, C=0.344 Star (N=1): TR_range=0.141, VP=0.243, C=0.778 Cycle (N=1): TR_range=0.071, VP=0.167, C=0.715 Full decay (N=1): TR_range=0.015, VP=0.206, C=0.763

THIRD in designed-topology series: cycle (done) → star (done) → chain (THIS) → complete (baseline).

Series (7 experiments)