A Mereological Approach to Higher-Order Structure in Complex Systems: from Macro to Micro with M\"obius

TL;DR

This paper introduces a unified mathematical framework based on M"obius inversion to connect microscopic interactions with macroscopic observables in complex systems, revealing how different decompositions lead to various notions of interactions across disciplines.

Contribution

It presents a novel formalism that unifies diverse interaction concepts through system decomposition, enabling cross-domain insights and practical analysis tools.

Findings

Decomposition of Kullback-Leibler divergence identifies responsible variables.

Efficient derivation of renormalised couplings in a 1D Ising model.

Framework unifies interaction notions across genetics, physics, and AI.

Abstract

Relating macroscopic observables to microscopic interactions is a central challenge in the study of complex systems. While current approaches often focus on pairwise interactions, a complete understanding requires going beyond these to capture the full range of possible interactions. We present a unified mathematical formalism, based on the M\"obius inversion theorem, that reveals how different decompositions of a system into parts lead to different, but equally valid, microscopic theories. By providing an exact bridge between microscopic and macroscopic descriptions, this framework demonstrates that many existing notions of interaction, from epistasis in genetics and many-body couplings in physics, to synergy in game theory and artificial intelligence, naturally and uniquely arise from particular choices of system decomposition, or mereology. By revealing the common mathematical structure underlying seemingly disparate phenomena, our work highlights how the choice of decomposition fundamentally determines the nature of the resulting interactions. We discuss how this unifying perspective can facilitate the transfer of insights across domains, guide the selection of appropriate system decompositions, and enable the search for new notions of interaction. To illustrate the latter in practice, we decompose the Kullback-Leibler divergence, and show that our method correctly identifies which variables are responsible for the divergence. In addition, we use Rota's Galois connection theorem to describe coarse-grainings of mereologies, and efficiently derive the renormalised couplings of a 1D Ising model. Our results suggest that the M\"obius inversion theorem provides a powerful and practical lens for understanding the emergence of complex behaviour from the interplay of microscopic parts, with applications across a wide range of disciplines.