Fig. 3.

Transition from fractal to small-world networks. (A) (Left) A typical percolation module in network space. The colors identify submodules obtained by the box-covering algorithm with ℓ_B = 15. This fractal module contains 4,097 nodes with , ℓ_max = 139, and r_max = 136 mm. When a small fraction p_rew of the links are randomly rewired (8), the modular structure disappears together with the shrinking path length. The rewiring method starts by selecting a random link and cutting one of its edges. This edge is then rewired to another randomly selected node, and another random link starting from this node is selected. This is again cut and rewired to a new random node, and we repeat the process until we have rewired a fraction p_rew of links. The final link is then attached to the initially cut node, so that the degree of each node remains unchanged. (B) Small-world cannot coexist with modularity. The large diameter and modularity factor, Eq. 4 for ℓ_B = 15, of the fractal module in A (Left) diminish rapidly upon rewiring a tiny fraction p_rew ≈ 0.01 of links, while the clustering coefficient still remains quite large. (C) The transition from fractal to small-world to random structure is shown when we plot the mass versus the average distance for all modules for different p_rew values as indicated. The crossover from power-law fractal to exponential small-world/random is shown.