  Quantifying the complexity of systems consisting of many interacting parts
has been an important challenge in the field of complex systems in both
abstract and applied contexts. One approach, the complexity profile, is a
measure of the information to describe a system as a function of the scale at
which it is observed. We present a new formulation of the complexity profile,
which expands its possible application to high-dimensional real-world and
mathematically defined systems. The new method is constructed from the pairwise
dependencies between components of the system. The pairwise approach may serve
as both a formulation in its own right and a computationally feasible
approximation to the original complexity profile. We compare it to the original
complexity profile by giving cases where they are equivalent, proving
properties common to both methods, and demonstrating where they differ. Both
formulations satisfy linear superposition for unrelated systems and
conservation of total degrees of freedom (sum rule). The new pairwise
formulation is also a monotonically non-increasing function of scale.
Furthermore, we show that the new formulation defines a class of related
complexity profile functions for a given system, demonstrating the generality
of the formalism.
