Infinite set of exponents describing physics on fractal networks

The generalised resistance between connected points a distance L apart on fractal networks of nonlinear (V approximately I alpha ) resistors scales as L zeta ( / alpha ) . It is shown that zeta ( alpha ) for alpha =- infinity , -1, 0 - , 0 + , 1 and infinity , describes physically relevant geometrical properties and d zeta /d alpha <or=0. For percolating clusters approximants are given for zeta for - infinity < alpha < infinity in 2-6 dimensions. For alpha <0 a family of solutions to Kirchhoff's equations exists, reminiscent of metastable states in spin glasses.