We study $〈\sigma {(M}_B,r)〉,$ the average conductance of the backbone, defined by two points separated by Euclidean distance r, of mass $M_B$ on two-dimensional percolation clusters at the percolation threshold. We find that with increasing $M_B$ and for fixed $r,〈\sigma {(M}_B,r)〉$ asymptotically decreases to a constant, in contrast with the behavior of homogeneous systems and nonrandom fractals (such as the Sierpinski gasket) in which conductance increases with increasing $M_B.$ We explain this behavior by studying the distribution of shortest paths between the two points on clusters with a given $M_B.$ We also study the dependence of conductance on $M_B$ above the percolation threshold and find that (i) slightly above $p_c,$ the conductance first decreases and then increases with increasing $M_B$ and (ii) further above $p_c,$ the conductance increases monotonically for all values of $M_B,$ as is the case for homogeneous systems.

• Received 12 October 1999

DOI:https://doi.org/10.1103/PhysRevE.61.3435