Diffusion with a topological bias on random structures with a power-law distribution of dangling ends

We study diffusion with a topological bias on random structures having dangling ends whose length L is chosen from a power-law distribution P(L)∼$L^{\mathrm{-}(\alpha +1)}$. We find that the mean-square displacement 〈$x^2$〉 of a random walker on the backbone varies asymptotically as 〈$x^2$〉∼(logt${)}^{2\alpha }$, slower than any power of t, in contrast with 〈x〉∼t, the conventional result for a nonrandom lattice. Our predictions are confirmed by numerical simulations for percolation and for the random comb.