A mathematical approach to resilience

TL;DR

This paper introduces the mathematics of resilience, integrating concepts from category theory, dynamical systems, and biology to formalize robustness and resilience in complex systems.

Contribution

It develops a formal framework connecting biological degeneracy, dynamical systems, and statistical testing through the Multiplicity Principle.

Findings

Two families of statistical tests satisfy the Multiplicity Principle

Formal connection between biological degeneracy and resilience

Integration of category theory with dynamical systems and signal processing

Abstract

In this paper, we evolve from sparsity, a key concept in robust statistics, to concepts and theoretical results of what we call the mathematics of resilience, at the interface between category theory, the theory of dynamical systems, statistical signal processing and biology. We first summarize a recent result on dynamical systems [Beurier, Pastor, Spivak 2019], before presenting the de-generacy paradigm, issued from biology [Edelman, Gally 1973] and mathematically formalized by [Ehresmann Vanbremeersch 2007] [Ehresmann Vanbremeersch 2019] as the Multiplicity Principle (MP). We then make the connection with statistical signal processing by showing that two distinct and structurally different families of tests satisfy the MP.