  Emergent effect is crucial to the understanding of the 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 encodes emergence as
the mathematical structure of cohomologies 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 experiment that our measure of emergence correlates with the existing
information-theoretic measure of emergence.
