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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>