Strength reduction in electrical and elastic networks

Particular aspects of problems ranging from dielectric breakdown to metal-insulator transition can be studied using electrical or elastic networks. We present an expression for the mean breakdown strength of such networks. First, we introduce a method to evaluate the redistribution of current due to the removal of a finite number of elements from a hypercubic network of conductances. It is used to determine the reduction of breakdown strength due to a fracture of size $\kappa .$ Numerical analysis is used to show that the analogous reduction due to random removal of elements from electrical and elastic networks follow a similar form. One possible application, namely the use of bone density as a diagnostic tool for osteoporosis, is discussed.