  The evolution of molecular interstellar clouds is a complex, multi-scale
process. The power-law density exponent describes the steepness of density
profiles, and it has been used to characterize the density structures of the
clouds yet its usage is usually limited to spherically symmetric systems.
Importing the Level-Set Method, we develop a new formalism that generates
robust maps of a generalized density exponent $k_{\rho}$ at every location for
complex density distributions. By applying it to high fidelity, high dynamical
range map of the Perseus molecular cloud constructed using data from the
Herschel and Planck satellites, we find that the density exponent exhibits a
surprisingly wide range of variation ($-3.5 \lesssim k_{\rho} \lesssim -0.5$).
Regions at later stages of gravitational collapse are associated with steeper
density profiles. Inside a region, gas located in the vicinities of dense
structures has very steep density profiles with $k_{\rho} \approx -3$, which
forms because of depletion. This density exponent analysis reveals diverse
density structures, forming a coherent picture that gravitational collapse
leads to a continued steepening of the density profile. We expect our method to
be effective in studying other power-law-like density structures, including
granular materials and the Large-Scale Structure of the Universe.
