Phase 2 propagation study

Reproduce: python studies/phase2_propagation.py (add --quick for a fast run). Every number here comes from that script; raw output is in phase2-results.json, figures in figures/.

What this shows and does not show. This is measured on the ground-truth fake client, not a real model. It demonstrates that the M2 propagation mechanics behave as designed — bias weighted by blast radius, accumulated across execution — and quantifies the effect of where bias enters the graph. It is not a claim about real multi-agent systems.

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Setup

A hub topology: root → hub → {w1, w2, w3, w4}. The hub fans out to four workers, giving it the largest Katz blast radius; w1..w4 are terminal leaves. A seed node is given a fixed bias (beta = 3.0) that amplifies as it propagates downstream (capped); every node is scored through the real compute_local_bias → node_magnitude → Katz weight → NetworkAccumulator stack, one accumulator update per topological superstep. B_net is the fast-scale network signal.

1. Same bias, central vs. leaf seed

The headline result. Identical bias magnitude, seeded once at the central hub and once at the leaf w1:

Seed	Katz weight	Final B_net (fast)	Trajectory
hub (central)	0.316	0.160	0.0 → 0.10 → 0.16
w1 (leaf)	0.105	0.028	0.0 → 0.04 → 0.03

The same bias produces ~5.8× more network-level signal when it enters at a central node than at a leaf. A biased leaf reaches nothing downstream and carries little weight, so B_net barely moves and then decays; a biased hub contaminates all four workers and is weighted heavily, so B_net climbs and stays up. This is the entire justification for topological weighting: a per-node bias score is not actionable without knowing the node's blast radius.

2. B_net tracks blast radius

Seeding each of three positions of increasing downstream reach:

Seed	Katz weight	Downstream reach	Final B_net
w1 (leaf)	0.105	0 nodes	0.028
hub	0.316	4 nodes	0.220
root	0.263	5 nodes	0.335

B_net rises monotonically with the seed's downstream reach (0 → 4 → 5 nodes): a bias that can contaminate more of the graph produces a larger network signal, as intended.

An honest nuance worth recording. The ordering by B_net (root > hub) does not match the ordering by Katz weight (hub > root). Katz ranks hub above root because its α-discount (α=0.5) penalizes root's more-distant leaves; but the harness amplifies bias to all reachable nodes without that discount, so root — which reaches one more node than hub — accumulates more. This is a real interaction between the weighting attenuation and the propagation model, not a bug: if you want B_net ordering to track Katz weight exactly, the propagation decay and the Katz α have to be aligned. It is flagged here as a calibration consideration for M3's threshold work.

3. Two timescales

On the central-seed run, the fast and slow accumulators (α = 0.7 and 0.1):

Superstep	fast	slow
0	0.00	0.00
1	0.10	0.07
2	0.16	0.11

The fast scale leads — it reaches a higher level sooner as the bias spreads, which is what a spike detector should do; the slow scale lags, integrating the trend. Tripping on fast and drift-alerting on slow (an M3 concern) is what these two scales are for.

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Takeaways for M3

• Blast-radius weighting works: identical bias is ~6× more impactful from a central node, and B_net tracks downstream reach. A threshold on B_net is therefore meaningfully topology-aware.
• The Katz-α / propagation-decay interaction (§2) means B_net's absolute scale depends on both knobs; M3's threshold calibration should sweep them together rather than fixing one and tuning the other.
• Fast and slow scales separate cleanly, supporting a two-threshold (spike vs. drift) breaker.