Structural Visibility in Dynamical Systems on Hypergraphs: A Pattern Formation Perspective

Hypergraphs encode rich multiway interactions, but not all structural information is equally accessible through the dynamics.

By analyzing pattern-forming instabilities in reaction-diffusion systems on directed hypergraphs, this work develops a theory of structural visibility that characterizes which features of higher-order structure survive successive levels of dynamical reduction.

It is established that higher-order structure is not automatically dynamically relevant.

Linearization destroys most higher-order information.

Meanwhile, nonlinear reduction recovers only specific higher-order marginals of the adjacency tensor, and projection along critical directions further filters what is dynamically visible.

First, we show that the linearized dynamics depends on the hypergraph only through its first-tail-moment statistics, termed exposure.

Consequently, exposure-equivalent hypergraphs are linearly indistinguishable in the sense that they exhibit identical dispersion relations and instability thresholds.

Next, we define a hierarchy of hyperedge tail-moments that captures progressively detailed co-occurence, and we prove a structural decomposition theorem describing how contractions of these tensors, termed packing effects, influence the reduced amplitude dynamics.

This leads to a visibility hierarchy in which successive asymptotic orders reveal increasingly richer structural information.

More specifically, exposure governs linear onset while packing effects control post-onset dynamics.

Finally, we establish results on nonlinear distinguishability, characterizing when linearly indistinguishable higher-order systems may exhibit different post-onset behaviors.

In addition, we formalize when higher-order systems become dynamically indistinguishable from pairwise systems, leading to the notion of dynamical graph surrogacy.

Numerical simulations support the theoretical predictions.