Size and History Combine in Allometry Relation of Technology Systems

Complex systems often exhibit amazingly regular behavior through allometry relations (ARs) between their functional attributes and their size. An empirical allometry relation (EAR) between two properties of a complex system relates the average functionality 〈 Y 〉 and the average size 〈 X 〉 ; 〈 Y 〉 = a 〈 X 〉 b , where the allometry coefficient a and allometry exponent b are empirical constants fit to data. This EAR is a static relation and is found in every sub-discipline of Natural History. Herein we establish, both empirically and theoretically, that for some classes of evolving technology systems, the empirical allometry coefficient a is not constant, but is strongly dependent on the historical time at which the technology system originated. Specifically, we construct an EAR with a time-dependent coefficient, using the evolution of a broad class of military systems, over the last ten centuries, ranging from a medieval bowman to a modern rifleman, from a horse-drawn cannon to a tank. This time-dependent EAR is derived from fundamental considerations involving complexity, scaling, and renormalization group theory. The theory entails an information as well as complexity generalization of the traditional allometry coefficient that combines system size with the technology knowledge of the new system’s developers (time-dependence). It relates technology ARs to technology evolution relations, such as Moore’s Law, with implications for technology drawn from biological evolution analysis.