Fig. 1: In this work we consider a many-body spin s Hamiltonian—see Eq. (1)—defined over a simple graph \({{{{{{{\mathcal{G}}}}}}}}(V,E)\) where the vertices represent the spins and the edges the pairs of spins upon which the two-body terms act.

We prove for a graph chosen uniformly at random from all simple graphs that, as the graph size L increases, the equilibrium properties of the system become increasingly like that of a single collective spin and any many-body effects vanish as L → ∞. In order for this not to be true, the graph must possess a non-trivial cut and we prove that even for such a graph, chosen at random amongst all those with a non-trivial cut, the system can effectively be reduced to that of two interacting collective spins. The emergence of complex, non-collective physics is thus dependent on more structured, ‘exceptional' graphs which exist in a vanishingly small subspace of the space of all simple graphs. These include the well-known sparse, regular structures that arise in nature and a new class of graphs we identify here: irregular dense structures.