RED test commit: `fd661569` — CI will fail on this (stub returns empty
map; assertions fail by design). GREEN: `bf4b8592`.
## What
Implements **axis 2 of 4** for the repeater usefulness score per #672
([status
comment](https://github.com/Kpa-clawbot/CoreScope/issues/672#issuecomment-4484635378)).
The Bridge axis measures *structural importance*: how many shortest
paths between other nodes route through this one. A high-traffic
redundant node and a low-traffic critical bridge will no longer look
identical.
## Algorithm
**Brandes' weighted betweenness centrality** with Dijkstra for shortest
paths (`cmd/server/bridge_score.go`).
- Nodes: pubkeys in the `neighbor_edges` graph
- Edge weight: `Score(now) * Confidence()` — per the convention from
#1235 (count + recency decay scaled by observer-diversity confidence).
Geo-rejected edges already excluded at graph build time (#1230) so we
don't re-filter here.
- Dijkstra distance: `1 / max(epsilon, weight)` — high affinity = cheap
cost.
- Normalize: divide by max observed centrality so output is in `[0, 1]`.
Cost: `O(V · (E + V log V))`. Staging-scale (~600 nodes / ~2 000 edges)
≈ ~4.8M ops, completes in milliseconds.
## Where it lives
- `cmd/server/bridge_score.go` — pure algorithm, no locks
- `cmd/server/bridge_recomputer.go` — background recomputer (mirrors
#1240/#1262 pattern), 5-min default interval, initial sync prewarm,
snapshot stored in `s.bridgeScoreMap atomic.Pointer[map[string]float64]`
- `cmd/server/routes.go` — `handleNodes` adds `node["bridge_score"]` on
repeater/room rows; node-detail handler adds it on the single-node path
- `public/nodes.js` — separate **Bridge** row in the node detail panel,
alongside the existing **Usefulness** (Traffic) row. Distinct
colour-coded bar.
## What's NOT in this PR (still pending for #672)
- **Coverage axis** (axis 3) — unique observer-pair connectivity
- **Redundancy axis** (axis 4) — simulated node-removal impact
- **Composite** — once all 4 axes ship, swap the `usefulness_score`
formula from "traffic-only" to the weighted composite
`Refs #672` (not `Fixes` — issue stays open until all 4 axes + composite
ship).
## Tests
- `TestComputeBridgeScores_LineGraph` — 4-node line: middles non-zero,
leaves zero, max normalized to 1.0
- `TestComputeBridgeScores_TriangleNoBridge` — clique has zero bridges
- `TestComputeBridgeScores_Empty` — defensive nil-safety
- `TestComputeBridgeScores_WeightSensitive` — mutation guard: revert the
`1/w` inversion and this test fails
- `TestBridgeScore_HandleNodesSurface` — integration: `/api/nodes`
returns `bridge_score` on repeater rows; middle nodes > 0, ends == 0
---------
Co-authored-by: clawbot <bot@meshcore.local>
Kpa-clawbot