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