Global and local observability of hypergraphs

TL;DR

This paper develops algebraic and matrix-based criteria for assessing both global and local observability in non-uniform hypergraphs with inputs and outputs, advancing the understanding of higher-order network dynamics.

Contribution

It introduces a novel framework combining polynomial ideals, rank conditions, and tensor similarity to analyze hypergraph observability comprehensively.

Findings

Provides algebraic conditions for global observability

Derives rank-based criteria for local observability

Proposes tensor similarity for comparing hypergraph observability

Abstract

This paper studies observability for non-uniform hypergraphs with inputs and outputs. To capture higher-order interactions, we define a canonical non-homogeneous dynamical system with nonlinear outputs on hypergraphs. We then construct algebraic necessary and sufficient conditions based on polynomial ideals and varieties for global observability at an initial state of hypergraphs. An example is given to illustrate the proposed criteria for observability. Further, necessary and sufficient conditions for local observability are derived based on rank conditions of observability matrices, which provide a framework to study local observability for non-uniform hypergraphs. Finally, the similarity of observability for hypergraphs is proposed using similarity of tensors, which reveals the relation of observability between two hypergraphs and helps to check the observability intuitively.