Degree Normalization Reshapes Collective Synchronization in Adaptive Kuramoto Networks

Collective dynamics in adaptive oscillator networks are shaped by both network structure and coupling-adaptation mechanisms, and separating the contributions of the two to synchronization is often difficult. We study a Kuramoto network with degree normalization and adaptive Hebbian weights. Introducing the degree-normalization exponent α, we examine how it affects the final synchronization order parameter Rfinal, and we separate the two effects by turning the mechanism off and running a paired control. We find that the increase of Rfinal with α is driven primarily by degree normalization rather than by the adaptive weight dynamics, and that it survives a control in which wij ≡ 1 is held fixed. The structural coupling budget takes the form Kieff = K (ki/N)1−α , whose dependence on node degree weakens as α grows; the coupling heterogeneity of the fixedweight control follows from the analytic degree distribution, and the numerical results match the analytic ones to machine precision. The effect of the adaptive weights on Rfinal, by contrast, carries a network-dependent sign: for two of the three network types it is negative at every sampled α , while for the third it crosses zero near α ≈ 0. 55 and turns positive. Because the three network types differ simultaneously in degree distribution, spatial embedding and average path length, this sign change cannot yet be attributed to degree heterogeneity alone. Mechanism-off controls, paired comparisons and analytic verification together resolve the relative roles of degree normalization and adaptive weights in the models studied here.