Who is important in a network? Not always the one with the most connections — sometimes it’s the BRIDGE, the one every path has to cross. Betweenness centrality counts, for each node, the fraction of all shortest paths that pass through it. A single broker between two clusters can have huge betweenness with few links. Remove them and the network splits. Slide to reveal the bridge.
Betweenness centrality of a node v is the sum over all pairs (s,t) of the fraction of shortest s–t paths that pass through v: Cᵇ(v) = Σ σₛₜ(v)/σₛₜ. It measures a node’s role as a BRIDGE or broker — distinct from degree (raw connection count), a high-betweenness node can have few edges yet control flow between clusters, and removing it fragments the network. Brandes’ algorithm computes it in O(VE). It finds key infrastructure, influencers, and bottlenecks. A fail-loud self-check throws unless the bridging node scores highest on a two-cluster test graph. ◆ real network science, node-verified.
Computed on a small fixed graph via all-pairs shortest paths; the exact values scale with graph size, but the BRIDGE-scores-highest property is exact and is the point — centrality is about position, not just count.