The evolution of molecular interstellar clouds, during which stars form, is a complex, multi-scale process. The power-law density exponent describes the steepness of density profiles in the log-log space, and it has been used to characterize the density structures of the clouds. Its effectiveness results from the widespread emergence of power-law-like density structures in complex systems that have reached intermediate asymptotic states. However, 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 kp at every location for complex density distributions. By applying it to a 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 < kp < -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 kp ~ -3, which form because of depletion. This density exponent analysis reveals diverse density structures in a molecular cloud, forming a coherent picture that gravitational collapse and accretion contribute to a continued steepening of the density profile. We expect our method to be effective in studying other power-law-like density structures, including the density structure of granular materials and the Large-Scale Structure of the Universe.