The shortcut a network can lose.

A node can sit on every cheapest route without being the only way through. The difference matters when a network measurement is used to make a claim about resilience or control.

A busy junction may owe its importance to a shortcut. Close it and the traffic takes a longer route. Another junction may be the only crossing between two districts. Close that one and no diversion exists. A ranking that puts both junctions first has not told us which problem we face.

My paper with Marco Alberto Javarone, From Bitcoin to Bitcoin Cash: a network analysis, examined the structure of historical peer-to-peer network samples. Its questions about connections and short paths motivate a further distinction: how much does a node shorten a route, and which connections disappear when it is removed? The example below is a deliberately constructed graph, not a new measurement of Bitcoin, Bitcoin Cash or BSV.

A shortcut used by every shortest route

Take four outer nodes, A, B, C and D, arranged in a ring, and a centre H connected to all four. Give each spoke a cost of 1 and each ring edge a cost of 3. These are fixed illustrative routing costs, not measured network latencies.

There are six unordered pairs of outer nodes. Each pair has a unique shortest route through H, costing 2. Going directly around one ring edge costs 3; reaching the opposite outer node around the ring costs 6. H therefore lies on every shortest path between the other nodes.

Under the endpoint-excluding definition of shortest-path betweenness, H receives all six pair contributions. Normalising by the six available pairs gives it a score of 1, the maximum for this undirected five-node graph.

Remove the centre

Now remove H and its four spokes. The outer ring survives. All six pairs can still communicate. Four neighbouring pairs have distance 3; two opposite pairs have distance 6. Their mean shortest-route cost rises from 2 to 4. The loss is a doubling of this illustrative cost, not a loss of connectivity among the surviving nodes.

For comparison, erase the ring edges from the original graph, leaving only the four spokes. H still has normalised betweenness 1. Remove H from that star and all four remaining nodes are isolated: none of their six pairs remains reachable. The same maximum score accompanies two different failure outcomes.

A node whose removal increases the number of connected components is an articulation point. H has that property in the star, but not in the ring-and-spokes graph.

From a graph to an operating claim

The calculation assumes that routes minimise assigned additive costs. A real service may broadcast, race several paths, select peers by policy or change connections after a failure. Congestion may change an edge's cost. A nominally available diversion may lack capacity for the displaced work. Those facts require evidence beyond a centrality score.

Nor does the graph identify the people controlling its nodes. Four visible alternatives could depend on one operator or upstream service. An operational study should record that dependence separately, rather than treating every address as an independent decision-maker.

A useful extension of topology research would report three things together: the routing rule behind a centrality measure, the change in reachability and cost after specified removals, and evidence about the capacity and independent control of the remaining routes. The historical study supplies a reason to investigate structure. This example supplies a reproducible reason not to ask one structural statistic to answer every question.