Mean-field theory of vector spin models on networks with arbitrary degree distributions

Understanding the relationship between heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories spin models constitute fundamental tool to tackle this cornerstone statistical physics, with an impressive number applications condensed matter, biology, computer In work we derive mean-field equations for equilibrium behavior vector high-connectivity random arbitrary degree distribution randomly weighted links. We demonstrate that limit is not universal it depends full distribution. Such nonuniversal akin remarkable mechanism leads breakdown central theorem when applied effective local fields. Traditional fully-connected models, such as Curie-Weiss, Kuramoto, Sherrington-Kirkpatrick model, are only valid if highly concentrated around its mean degree. obtain series results highlight importance fluctuations phase diagram by focusing Kuramoto model synchronization spin-glasses. Numerical simulations corroborate our theoretical findings provide compelling evidence present theory describes intermediate regime connectivity, which average $c$ scales power $c \propto N^{b}$ ($b < 1$) total $N \gg 1$ spins. Our put forward novel class incorporate effects and, at same time, amenable exact analytic solutions.