Global and local observability of hypergraphs

TL;DR

We address observability for non-uniform directed hypergraphs with inputs and outputs by formulating a polynomial dynamical system on hypergraphs and framing state distinguishability in algebraic terms of $\mathbf{V}(\mathcal{J})$ vs $\mathbf{V}(\ell)$ for initial state $\sigma$. Global observability is characterized by finite-step criteria derived from polynomial ideals and tensor symmetrization, with Lie derivatives $L_f^r h$ corresponding to higher-order propagation paths along directed hyperedges. Local observability is provided by rank conditions on observability matrices, and a structural observability theory is developed using observational diameter and automorphism breaking to enable topology-driven observability guarantees, along with design principles for outputs and hyperedge weights. The framework is illustrated through numerical and higher-order population-model examples, showing both algebraic and structural insights for sensor placement, output design, and higher-order control on hypergraphs.

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.

Figures (5)

• Figure 1: Illustration of uniform and non-uniform hypergraphs. In (a), $e_1=\{x_1,x_2,x_3\}$, $e_2=\{x_3,x_4,x_5\}$, and $e_3=\{x_5,x_6,x_7\}$ are hyperedges. The cardinality of every hyperedge is equal to 3. In (b), $\bar{e}_1=\{x_1,x_2,x_3,x_4\}$, $\bar{e}_2=\{x_4,x_5,x_6\}$, and $\bar{e}_3=\{x_1,x_6,x_7\}$ are hyperedges. The cardinality of hyperedge $\bar{e}_1$ is 4 and that of $\bar{e}_2$ and $\bar{e}_3$ is 3. The non-uniform hypergraph can be used to represent a network system containing different (higher-)order interactions. Please see Section III-B for details.
• Figure 2: Outline of the proposed theorems.
• Figure 3: Illustration of higher-order network system modeling. In the left-side hypergraph, arrays in the same color are used to capture a group of interactions. For example, the blue arrays represent the joint influence from $x_1$, $x_2$ and $x_3$ acted on $x_1$ such that there is a directed hyperedge $(\{x_1,x_2,x_3\},\{x_1\})$; the orange arrays represent the joint influence from $x_1$ and $x_3$ acted on $x_2$ such that there is a hyperedge $(\{x_1,x_3\},\{x_2\})$, and so on. Furthermore, the weight of each hyperedge is designed as the corresponding tensor element. This represents the right-side higher-order network system.
• Figure 4: Example of a higher-order network with deficient observability matrix rank
• Figure 5: Examples with symmetric and asymmetric structures

Theorems & Definitions (9)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof