Neighbor-independent [network substitution] rules

Even though the same replacement is performed at each node at each step, the networks produced are not homogeneous. In the first case shown, the picture produced after t steps has 4 × 3^{t - k - 1} regions with 3 × 2^k edges. In the limit t  ∞, the picture has the geometrical form of an Apollonian circle packing (see page 986). The number of nodes at distance up to r from a given node is at most 1 + Sum[c[i] + c[i - 1], {i, n}] where c[i_] := 2^DigitCount[i, 2]. In practice this number fluctuates greatly with r, making pictures like those on page 479 not exhibit smooth profiles. Averaged over all nodes, however, the number of nodes at distance up to r approximates r^Log[2, 3], implying an effective dimension of Log[2, 3]. Note that there is no upper limit on the dimension that can be obtained with appropriate neighbor-independent rules.