Numerical demonstration of Lemma 1.

Using a seed network with large enough diameter to obtain statistically meaningful results (top left figure), we validate Lemma 1. As we can see at the top right figure the number of pair of nodes within distance r expected from Lemma 1 is equal to the one computed from the networks. Furthermore, the computed power-hop exponent is very close to the one expected from our lemma as well (bottom left figure). Finally, the actual scaling behavior for the networks after each Kronecker iteration are presented at the bottom right figure.