High Modularity Creates Scaling Laws

Scaling laws have been observed in many natural and engineered systems. Their existence can give useful information about the growth or decay of one quantitative feature in terms of another. For example, in the field of city analytics, it is has been fruitful to compare some urban attribute, such as energy usage or wealth creation, with population size. In this work, we use network science and dynamical systems perspectives to explain that the observed scaling laws, and power laws in particular, arise naturally when some feature of a complex system is measured in terms of the system size. Our analysis is based on two key assumptions that may be posed in graph theoretical terms. We assume (a) that the large interconnection network has a well-defined set of communities and (b) that the attribute under study satisfies a natural continuity-type property. We conclude that precise mechanistic laws are not required in order to explain power law effects in complex systems—very generic network-based rules can reproduce the behaviors observed in practice. We illustrate our results using Twitter interaction between accounts geolocated to the city of Bristol, UK.