  We utilise the graphon--a continuous mathematical object which represents the
limit of convergent sequences of dense graphs--to formulate a general,
continuous description of quantum spin systems in thermal equilibrium when the
average co-ordination number grows extensively in the system size.
Specifically, we derive a closed set of coupled non-linear Fredholm integral
equations which govern the properties of the system. The graphon forms the
kernel of these equations and their solution yields exact expressions for the
macroscopic observables in the system in the thermodynamic limit. We analyse
these equations for both quantum and classical spin systems, recovering known
results and providing novel analytical solutions for a range of more complex
cases. We supplement this with controlled, finite-size numerical calculations
using Monte-Carlo and Tensor Network methods, showing their convergence towards
our analytical results with increasing system size.
