  To elucidate allometric scaling in complex systems, we investigated the
underlying scaling relationships between typical three-scale indicators for
approximately 500,000 Japanese firms; namely, annual sales, number of
employees, and number of business partners. First, new scaling relations
including the distributions of fluctuations were discovered by systematically
analyzing conditional statistics. Second, we introduced simple probabilistic
models that reproduce all these scaling relations, and we derived relations
between scaling exponents and the magnitude of fluctuations.
