Making sense of complex systems through resolution, relevance, and mapping entropy

TL;DR

This paper introduces a method using resolution, relevance, and mapping entropy to identify key degrees of freedom in complex systems, demonstrated on diverse datasets including spins, financial markets, and proteins.

Contribution

It develops new theoretical links among information measures and provides a general, software-implemented approach for analyzing complex systems' relevant variables.

Findings

Successfully identifies influential degrees of freedom in various systems.

Provides a quantitative framework for data-driven system simplification.

Demonstrates applicability across physical, biological, and social systems.

Abstract

Complex systems are characterised by a tight, nontrivial interplay of their constituents, which gives rise to a multi-scale spectrum of emergent properties. In this scenario, it is practically and conceptually difficult to identify those degrees of freedom that mostly determine the behaviour of the system and separate them from less prominent players. Here, we tackle this problem making use of three measures of statistical information: resolution, relevance, and mapping entropy. We address the links existing among them, taking the moves from the established relation between resolution and relevance and further developing novel connections between resolution and mapping entropy; by these means we can identify, in a quantitative manner, the number and selection of degrees of freedom of the system that preserve the largest information content about the generative process that underlies an empirical dataset. The method, which is implemented in a freely available software, is fully general, as it is shown through the application to three very diverse systems, namely a toy model of independent binary spins, a coarse-grained representation of the financial stock market, and a fully atomistic simulation of a protein.