Structural Effects in Network Dynamical Systems: From Reconstruction to Pattern Formation in Hypergraphs

Abstract:

This dissertation studies how interaction structure influences the behavior of network dynamical systems and, more fundamentally, which aspects of that structure are dynamically observable. While complex systems are often modeled through underlying interaction networks or hypergraphs, the relationship between structure and dynamics is not direct: different analytical frameworks reveal different structural projections.

First, we study the inverse problem of reconstructing higher-order interaction structure from pairwise observations. We show that such reconstruction is fundamentally non-unique, establishing intrinsic limitations on structural inference from graph data.

Next, we analyze network dynamical systems on graphs and show that, in the linear regime, structural effects are mediated through coupling operators and their associated spectral and degree-based representations. We further identify intrinsic obstructions to coupling-induced stabilization.

Finally, extending these ideas to reaction–diffusion systems on directed hypergraphs, we develop a weakly nonlinear reduction framework for pattern formation near bifurcation. We show that the resulting nonlinear dynamics depend not on the full higher-order interaction structure, but on specific projected quantities, termed packing contributions, which govern pattern selection and saturation. This leads to a characterization of the notion of dynamical graph surrogacy, under which higher-order interactions become dynamically indistinguishable from pairwise ones.

Taken together, these results show that structural effects are fundamentally analysis-dependent and provide a unified perspective on the limits of structural inference and the role of higher-order interactions in complex dynamical systems.