Dissipation enables robust extensive scaling of multipartite correlations

We investigate the multipartite mutual information between $N$ discrete-state stochastic units interacting in a network that is invariant under unit permutations. We show that when the system relaxes to fixed point attractors, multipartite correlations in the stationary state either do not scale extensively with $N$, or the extensive scaling is not robust to arbitrarily small perturbations of the system dynamics. In particular, robust extensive scaling cannot occur in thermodynamic equilibrium. In contrast, mutual information scales extensively when the system relaxes to time-dependent attractors (e.g., limit cycles), which can occur only far from equilibrium. This demonstrates the essential role of dissipation in the generation and maintenance of multipartite correlations. We illustrate our theory with the nonequilibrium Potts model.