VAR(1) long memory from bottlenecks and long cycles in network dynamics

This paper develops a spectral–topological foundation for long memory in network autoregressions. We consider VAR(1) dynamics driven by the node Laplacian of a sequence of weighted graphs with weakly dependent innovations. Long memory emerges when the near-zero spectrum of the Laplacian is thickened by either of two structural mechanisms: (i) bottlenecks, which trap flows across network cuts; and (ii) long cycles with regularly varying distributions, which yield near-harmonic modes. These imply long-memory bounds for the autocovariances of linear observables and, along with harmonic components, they allow coexistence of random-walk and stationary long-memory behaviors. The framework could be useful to the design of new applications of network models in economics.