On Complexity of Stability Analysis in Higher-order Ecological Networks through Tensor Decompositions

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.