A contraction based, singular perturbation approach to near-decomposability in complex systems

We revisit the classical concept of near-decomposability in complex systems, introduced by Herbert Simon in his foundational article The Architecture of Complexity, by developing an explicit quantitative analysis based on singular perturbations and nonlinear contraction theory. Complex systems are often modular and hierarchic, and a central question is whether the whole system behaves approximately as the "sum of its parts", or whether feedbacks between modules modify qualitatively the modules behavior, and perhaps also generate instabilities. We show that, when the individual nonlinear modules are contracting (i.e., forget their initial conditions exponentially), a critical separation of timescales exists between the dynamics of the modules and that of the macro system, below which it behaves approximately as the stable sum of its parts. Our analysis is fully nonlinear and provides explicit conditions and error bounds, thus both quantifying and qualifying existing results on near-decomposability.