A Mereological Approach to Higher-Order Structure in Complex Systems: from Macro to Micro with Möbius (2404.14423v5)

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.

• The paper introduces a novel framework leveraging the Möbius inversion theorem to quantify and decompose higher-order interactions in complex systems.
• It demonstrates how decomposing systems into constituent parts reveals non-additive effects and synergy across fields such as physics, biology, and AI.
• The approach unifies existing interaction models by inferring microscale dynamics from observable macroscopic phenomena, offering actionable analysis tools.

A Mereological Approach to Higher-Order Structure in Complex Systems: From Macro to Micro with Möbius

The paper presents a novel mathematical framework for understanding higher-order interactions in complex systems through a mereological lens. It posits that the decomposition of a system into its constituent parts is crucial for defining and identifying the nature of interactions that might appear trivial when viewed in isolation but become significant in combination. By using the Möbius inversion theorem, the framework unifies various existing concepts of higher-order interactions, demonstrating their common structure across diverse domains such as genetics, physics, game theory, and artificial intelligence.

Central Thesis

The thesis of the paper revolves around the formulation of a generalized method to quantify and understand interactions within complex systems. The authors argue that while pairwise interactions have been extensively studied and understood, higher-order interactions require a more nuanced mathematical formalism. This is because such interactions can capture synergy and non-additive effects that are not evident when examining components individually.

Methodology

The core approach of the paper is the application of the Möbius inversion theorem to mereological decompositions. Given a system, it can be decomposed into various parts. The authors define a mereology as a co-rooted poset with the system as its largest element. The Möbius inversion is used to reverse-engineer microscopic interactions from observed macroscopic phenomena—allowing researchers to infer the underlying structure of complex behaviors from observable data.

Applications and Implications

The paper provides a broad range of applications to illustrate the utility and generality of the framework:

1. Information Theory: By decomposing the entropy of a system through a mereological approach, the framework can rediscover traditional mutual information and uncover redundancy and synergy through the partial information decomposition (PID).
2. Biology: The formulation identifies genetic epistasis as part of a larger structure of interactions within the genetic makeup. The decomposition allows for the quantification of these genetic interactions in relation to phenotypic expressions.
3. Physics: The framework aids in analyzing equilibrium dynamics and statistical mechanics, providing an avenue to infer microscopic coupling parameters in models like Ising models through macroscopic properties such as phase behaviors and correlation functions.
4. Game Theory and AI: It is used to calculate Shapley values in cooperative games, illustrating how contribution and value can be decomposed into baseline and interactive components. Similarly, it supports feature importance in machine learning, allowing the decomposition of model predictions into the contributions of individual features.
5. Chemistry: In chemical systems, the mereological approach allows for the partitioning of molecular systems into effective interaction subunits, which can better predict molecular properties.

Future Directions

The framework offers a compelling viewpoint that could reshape how higher-order phenomena are perceived in scientific research. Future work might delve into exploring more complex mereological structures beyond typical finite posets, potentially extending the applicability of the Möbius approach to other scientific domains or systems that are traditionally seen as indivisible. Moreover, exploring order dualities could provide more profound insights into the inherent symmetrical properties of higher-order interactions.

Conclusion

This framework underscores the indispensability of decomposition choices, often overlooked in traditional analysis, in defining and estimating higher-order interactions. By leveraging the Möbius inversion theorem, the research sets a precedent for a systematic exploration of complex systems through a unified lens, opening pathways to uncover hidden interactions that dictate the emergent properties of systems across multiple disciplines.