Fig. 2: Predicting the transition to synchronization.

a An all-to-all connected, symmetric, weighted graph of N = 10 nodes is considered. The graph is endowed with three symmetry orbits: the one composed by the red nodes {1, 2, 3}, the one made of the orange nodes {4, 5, 6}, and the one made of the four yellow nodes {7, 8, 9, 10}. In the sketch, the widths of the links are proportional to the corresponding weights, and the sizes of the nodes are proportional to the corresponding strengths. b The entries of S₁₀, which corresponds to λ₁₀ = 6. c S₈ (associated to λ₈ = 6), where the entries equal to 2 clearly define a cluster formed by nodes {7, 8, 9, 10}. The first predicted event in the transition then consists in the synchronization of such nodes at d₁ = ν^*/λ₈ = ν^*/6. d S₅ (related to λ₅ = 4), where additional entries become equal to 2, indicating a second foreseen event in which nodes {4, 5, 6} join the existent synchronization cluster at d₂ = ν^*/λ₅ = ν^*/4. e S₂ (corresponding to λ₂ = 1) where it is seen that nodes {1, 2, 3} also join the existing synchronized cluster at d₃ = ν^*/λ₂ = ν^* in a third predicted event where complete synchronization of the network takes place. f The expected events (and their exact sequence) occurring in the path to synchrony of the network’s architecture depicted in a. The bar at the bottom of the Figure gives the color code used in panels (b-e) for matrices' entries.