  Complex ecological networks are often characterized by intricate interactions
that extend beyond pairwise relationships. Understanding the stability of
higher-order ecological networks is salient for species coexistence,
biodiversity, and community persistence. In this article, we present complexity
analyses for determining the linear stability of higher-order ecological
networks through tensor decompositions. We are interested in the higher-order
generalized Lotka-Volterra model, which captures high-order interactions using
tensors of varying orders. To efficiently compute Jacobian matrices and thus
determine stability in large ecological networks, we exploit various tensor
decompositions, including higher-order singular value decomposition, Canonical
Polyadic decomposition, and tensor train decomposition, accompanied by in-depth
computational and memory complexity analyses. We demonstrate the effectiveness
of our framework with numerical examples.
