concept

Percolation threshold

The critical fraction of removed nodes (or links) at which a network abruptly breaks from one connected whole into a dust of isolated fragments. Written p_c. Below it the network conducts — a signal can cross from one side to the other; above it the network insulates — no path links the two sides. The counter-intuitive part is that this transition is sharp, not gradual: removing one more small batch of nodes, no different from the batches before, can tip the whole system over the edge. For a scale-free network under random loss, p_c is pushed close to 1 — you must remove very nearly everything before it fails.