Many scientific areas, ranging from computer science to environmental science and finance, give rise to multivariate time series processes that exhibit long memory, that is, time-dependent data with slowly decaying autocorrelation. Efficient modeling in such settings is crucial for various analytical tasks, including inference and prediction; however, traditional approaches are challenging even in modest dimensions. Several settings are also characterized by a (possibly inferred) network encoding the presence or absence of component associations via its edge topology, yet current methods are often inadequate for successfully exploiting this structure. In this talk, we propose a novel approach to modeling multivariate long-memory time series by defining two network-based models, capturing both the temporal and the underlying network dynamics. Our proposed likelihood-based exact estimation method exhibits strong asymptotic properties and, thanks to the low-dimensional parameter space induced by the network, yields more stable estimates than traditional models, allowing us to tackle scenarios where current methods fail due to computational limitations.