The chain walking (CW) polymerization technique has the unique property of a movable catalyst synthesizing its own path by creating branch-on-branch structures. By successive attachment of monomers, the resulting architecture ranges from dendritic to linear growth depending on the walking rate, which is defined by the ratio of walking steps and reaction events of the catalyst. The transition regime is characterized by local dendritic sub-structures (dendritic blobs) and a global linear chain feature forming a dendritic bottle-brush. A scaling model for structures obtained by CW catalysis is presented and validated by computer simulation relating the extensions of CW structures to the catalyst's walking ability. The limiting case of linear (low walking rate) and dendritic growth (high walking rate) is recovered, and the latter is shown to bear analogies to the Barabási-Albert graph and Bernoulli growth random walk. We could quantify the size of the dendritic blob as a function of the walking rate by using spectral properties of the connectivity matrix of the simulated macromolecules. This allows us to fit the numerical constants in the scaling approach. We predict that independent of the underlying chemical process, all CW polymerization syntheses involving a highly mobile catalyst ultimately result in bottle-brush structures whose properties depend on a unique parameter: the walking rate.