The causal interaction between the subnetworks of a complex network

TL;DR

This paper develops a formal framework for measuring causal interactions between large complex subsystems, such as brain regions, using information flow, with validated formulas and implications for various scientific fields.

Contribution

It extends previous formalism to bulk information flow between subsystems, providing analytical formulas and estimators validated through simulations.

Findings

Analytical formulas for bulk information flow derived in closed form.

Maximum likelihood estimators under Gaussian assumptions obtained.

Common proxies like averages and principal components often fail to accurately characterize subsystems.

Abstract

Information flow provides a natural measure for the causal interaction between dynamical events. This study extends our previous rigorous formalism of componentwise information flow to the bulk information flow between two complex subsystems of a large-dimensional parental system, in order to investigate problems such as the effective connectivity between two brain regions, each with millions of neurons involved. Analytical formulas have been obtained in a closed form. Under a Gaussian assumption, their maximum likelihood estimators have also been obtained. These formulas have been validated using different subsystems with preset relations, and they yield causalities just as expected. On the contrary, the commonly used proxies for the characterization of subsystems, such as averages and principal components, generally do not work correctly. This study can help diagnose the emergence of patterns in complex systems (e.g., the human brain), and is expected to have applications in many real world problems in different disciplines such as neuroscience, climate science, fluid dynamics, financial economics, to name a few.