Nonlinear resistor fractal networks, topological distances, singly connected bonds and fluctuations

We consider a fractal network of nonlinear resistors, with the voltage V behaving as a power of;he current I, V= RI^α. The resistance between two points at a distance L is R(L)δ L^ξ(α). We prove that ξ(δ) describes the scaling of the topological-chemical distance, while i(m) describes that of the number of singly connected ‘red’ bonds. -For random resistors, y e also consider the width of the resistance distribution, ΔRδ L^ξ₂^(α). Values for ξ and ξ₂ are explicitly derived for two model fractals, and ΔR/R is found to grow with L for the Sierpinski gasket and a > 1.612. The relevance of the results to percolation clusters is discussed.