Count time series arise naturally in many scientific applications. Examples include species abundances collected over time, and gene expression counts over pseudotime. Network inference from such data can yield important insight into underlying biological mechanisms, but is challenging because count observations are discrete, overdispersed, and temporally dependent. We propose a multivariate Poisson log-normal model in which observed counts arise from a Poisson distribution with rates determined by a latent vector autoregressive (VAR) process. The latent VAR transition matrix induces directed temporal dependencies among components and serves as the target network parameter. We establish identifiability of the model under suitable conditions and develop two procedures for sparse estimation of the VAR transition matrix. Our first estimator is a sparse method-of-moments estimator, for which we establish consistent network recovery and parameter estimation in high-dimensional settings. Our second estimator is an L1-penalized variational inference estimator that is motivated by recent approaches for count data. We evaluate the proposed estimators through simulations under a range of sampling regimes. We further demonstrate the method on a macroinvertebrate abundance time series collected from European freshwater ecosystems, obtaining an interpretable ecological network.

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