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

Anqi Dong, Can Chen

TL;DR

This work tackles the scalability challenge of stability analysis in ecological networks with higher-order interactions by focusing on the HOGLV model. It develops a tensor-based framework that recasts the HOGLV dynamics through HOSVD, CPD, and TT decompositions to enable efficient computation of the Jacobian at equilibrium and its eigenvalues. The authors derive explicit memory and computational complexity bounds for full, HOSVD-, CPD-, and TT-based representations, highlighting that CPD often minimizes complexity but can be numerically unstable, while TT-based methods offer stable, scalable performance. Numerical experiments illustrate how higher-order terms influence stability and demonstrate the practical advantages of TT representations in large systems. The approach provides a foundation for scalable stability analysis in complex ecological networks and potentially other domains with high-order interactions.

Abstract

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.

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

TL;DR

This work tackles the scalability challenge of stability analysis in ecological networks with higher-order interactions by focusing on the HOGLV model. It develops a tensor-based framework that recasts the HOGLV dynamics through HOSVD, CPD, and TT decompositions to enable efficient computation of the Jacobian at equilibrium and its eigenvalues. The authors derive explicit memory and computational complexity bounds for full, HOSVD-, CPD-, and TT-based representations, highlighting that CPD often minimizes complexity but can be numerically unstable, while TT-based methods offer stable, scalable performance. Numerical experiments illustrate how higher-order terms influence stability and demonstrate the practical advantages of TT representations in large systems. The approach provides a foundation for scalable stability analysis in complex ecological networks and potentially other domains with high-order interactions.

Abstract

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.
Paper Structure (12 sections, 20 equations, 3 figures, 2 tables)

This paper contains 12 sections, 20 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Tensor Preliminaries
  3. The HOGLV Model
  4. HOSVD-based Representation
  5. CPD-based Representation
  6. TTD-based Representation
  7. Complexity Analyses of Linear Stability
  8. Numerical Examples
  9. Computing Jacobian Matrices
  10. Memory Comparison
  11. Computation Comparison
  12. Conclusion

Figures (3)

  • Figure 1: High-order network representation of an ecological system with seven species, where interactions can occur between pairs or groups of nodes, and can be positive or negative.
  • Figure 2: Two ecological networks, fully characterized by their pairwise and third-order interactions, with positive interactions colored in blue and negative interactions in red.
  • Figure 3: Comparison of computational time for computing the Jacobian matrix between the full and TTD-based representations.

Theorems & Definitions (4)

  • proof
  • proof
  • proof
  • proof