  Theoretical research into many-body quantum systems has mostly focused on
regular structures which have a small, simple unit cell and where a vanishingly
small number of pairs of the constituents directly interact. Motivated by
advances in control over the pairwise interactions in many-body simulators, we
determine the fate of spin systems on more general, arbitrary graphs. Placing
the minimum possible constraints on the underlying graph, we prove how, with
certainty in the thermodynamic limit, such systems behave like a single
collective spin. We thus understand the emergence of complex many-body physics
as dependent on `exceptional', geometrically constrained structures such as the
low-dimensional, regular ones found in nature. Within the space of dense graphs
we identify hitherto unknown exceptions via their inhomogeneity and observe how
complexity is heralded in these systems by entanglement and highly non-uniform
correlation functions. Our work paves the way for the discovery and
exploitation of a whole class of geometries which can host uniquely complex
phases of matter.
