Proc-to-Spec: A Functorial Map of Network Processes

TMLR Paper5877 Authors

Abstract: The analysis of dynamic networks is central to understanding complex environmental systems in nature, yet traditional methods often focus on describing changing states rather than formalising the underlying processes of change. In this work, we introduce a category-theoretical framework, Proc-to-Spec, that provides a principled, functorial method for analysing the transformations that govern network evolution. We model resource-constrained systems, such as those commonly found in biology and ecology, within a source category Proc, where morphisms represent dissipative physical processes. We then construct a spectral functor, $\chi: Proc \to Spec$, that maps each process to a unique linear transformation between the eigenspaces of the network's symmetrised Laplacian. This framework allows us to establish a set of rigorous theorems. We prove that physical conservation laws in Proc correspond directly to spectral invariants in Spec, such as the conservation of the Laplacian's trace. We derive a spectral sensitivity theorem that formally links resource dissipation to network fragmentation via the Fiedler value. We also establish a stability-spectrum equivalence theorem, proving that a system's physical dynamics converge to a stable state if and only if its spectral geometry converges. We validate our theory with numerical experiments and demonstrate its utility as a tool for scientific discovery in a case study of the Serengeti food web in northern Tanzania. Using a large collection of 1.2 million classified image sets of animal activity from 225 camera traps spread across 1,125 km$^2$ of the Serengeti National Park from 2010 to 2013, we show that our framework can detect the subtle, cyclical signature of seasonal change and identify the unique geometric fingerprint of the 2011 East Africa drought. Our work provides a different way of thinking about dynamic systems, shifting the focus from describing states to understanding the fundamental geometry of change. Code to reproduce all results in the paper is released at https://anonymous.4open.science/r/tmlr_pts

Submission Length: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Mauricio_A_Álvarez1

Submission Number: 5877