More connections should mean faster mixing. On a network, the averaging process — where adjacent nodes replace their values with their mean — converges to consensus. The more edges, the more averaging events per unit time, and the faster the system should reach equilibrium. This is the intuition: connectivity accelerates convergence.

On random regular graphs, connectivity accelerates convergence until degree 10. Then it slows down.

The phase transition is sharp and occurs at a specific, finite degree. For random d-regular graphs with d ≤ 10, the averaging process achieves L²-cutoff at the same time as the random walk — the mixing is limited by how quickly information spreads through the graph, and adding edges helps. For d > 10, a different mechanism dominates. The mixing time increases relative to the random walk, and the additional connectivity no longer helps — it changes the qualitative character of the convergence.

The mechanism involves a biased birth-and-death chain with a slow bond. The spectral analysis reveals that the bottleneck shifts: in sparse graphs, mixing is limited by the graph's expansion (how quickly information reaches distant nodes). In dense graphs, mixing is limited by a local equilibration process that becomes the rate-limiting step when there are too many neighbors to average with simultaneously. The global spread of information is fast, but the local reconciliation of values is slow.

The threshold d = 10 is not an asymptotic statement. It is an exact phase boundary — a finite number at which the dominant mixing mechanism changes. Below the threshold, more connections help. Above it, more connections change what "mixing" means, activating a slower process that the sparse regime never encountered.

The network doesn't slow down. It switches to a different dynamical problem.