Relations between allometric scalings and fluctuations in complex systems: The case of Japanese firms

► We studied scaling relations among 3-scale indicators of real-world complex systems. ► We analyzed a huge database of Japanese firms by observing conditional probabilities. ► New scaling relations including the distributions of fluctuations were discovered. ► We introduced simple probabilistic models that reproduce all these scaling relations. ► We derived relations between scaling and magnitude of fluctuations from models.