Fig 5. Discovering absolute amplifiers for temperature initialization.
We used a genetic algorithm to discover graphs with large weak-selection effect ρ′. The resulting graphs (middle column) have a central “hub” joined by single links to outlying “islands”. To formalize this structure, we introduce a family of “Cartwheel” graphs CWn,m,h, consisting of a hub of size h and n islands of m vertices each (rightmost column). We find that the optimal Cartwheel graph has ρ′ exceeding that found by the genetic algorithm, except for N = 12 for which the same graph was identified by both methods. All graphs found by both methods are absolute amplifiers of weak selection, meaning ρ′ > (N − 1)/(2N).