Scaling electrical percolation networks based on renormalization group theory

Many natural disordered systems such as percolation metal films may be approximated as fractals. Probing their properties can be difficult depending on the length scale involved. Often, characterizing the system at a convenient length scale and building models for extrapolating the measured data to other length scales is preferred. In such situations, a general algorithm for scaling the model network while preserving its statistical equivalence is required. Here, we provide an algorithm that draws inspiration from renormalization group theory for scaling disordered fractal networks. This algorithm includes three steps: expand, map, and reduce resolution, where the mapping is the only computationally expensive step. We describe a way to minimize the computational burden and accurately scale the model network. We experimentally validate the algorithm in a percolating electrical network formed by an ultra-thin gold film on a glass substrate. By measuring the resistance between many pairs of pads separated by a given length, we accurately predict the mean and standard deviation of the resistance distribution measured across pads separated by twice the original distance. The algorithm presented here is general and may be applied to any disordered fractal system.