A categorical framework for quantifying emergent effects in network topology

Emergent effect is crucial to the understanding of properties of complex systems that do not appear in their basic units, but there has been a lack of theories to measure and understand its mechanisms. In this paper, we established a framework based on homological algebra that formulates emergence as the mathematical structure of derived functors, and then applied it to network models to develop a computational measure of emergence. This framework ties the emergence of a system to its network topology and local structures, paving the way to predict and understand the cause of emergent effects. We show in our numerical result that our measure of emergence correlates with the existing information-theoretic measure of emergence as information loss.