## 4 Derivation of Network Parameters from Experimental and Theoretical Nanoparticle Patterns
The network shown here is not huge, but it contains typical topological parameters observed throughout the SEM images for analysis. In the \(C_{\mathrm{C}}-C_{\mathrm{B}}\) diagram, we recognize a triangle area in which the dots corresponding to the nodes scatter in less regularity. Several microscopic views give us further information on the topological roles of nodes. For instance, the nodes with the number 16 and 17 are located near the top of the triangle, around the maximum values of both \(C_{\mathrm{C}}\) and \(C_{\mathrm{B}}\). Near the left apex of the bottom side, the nodes with the numbers 1, 32 and 33 are located, which are far from the center of the structure with the largest distances. In contrast, although located on the bottom side, the node with the number 23 is around the right apex with a large value of \(C_{\mathrm{C}}\), which means that it is near the center, but at the top of a chain structure. Inside the triangle, nodes scatter according to edge distributions that include similar structures in every scale, which is the definition of fractals. In comparison, regularly-arranged patterns as a line (Figure 6b) and a square lattice (Figure 6c) show different dot distributions in the diagram. The line-shape linear network in Figure 6b includes 36 nodes, and all dots in the diagram are on one curve without dot scattering, which means that the roles of nodes as joints of the network are regular; as the sum of the distances from a given node (indicated by \(C_{\mathrm{C}}\)) decreases as the probability as a node on the shortcut between arbitrary pairs of nodes increases (indicated by \(C_{\mathrm{B}}\)). A similar regular arrangement is observed in the case of the square lattice, as shown in Figure 6c. The number of dots is smaller due to the symmetry in the square shape, and \(C_{\mathrm{C}}\) is larger since the dots are distributed in a compact space, but the dots are almost on one curved line. These facts indicate that the \(C_{\mathrm{C}}-C_{\mathrm{B}}\) diagram in Figure 6a properly represents the geometrical properties of the fractal-like structure observed in the nanoparticle network. Further analysis of experimentally-observed nanoparticle networks in Figure 4 is performed, as shown in Figure 7. In all of the \(C_{\mathrm{C}}-C_{\mathrm{B}}\) diagrams, similar triangles to Figure 6a form with slight changes of the dot distributions. If the number of nodes is larger, \(C_{\mathrm{C}}\) ranges in smaller values since the closeness is less found in a large network, and the values of \(C_{\mathrm{B}}\) are larger due to the enhanced importance as connected joints because we consider fractal structures that have winner-take-all crossing points. Except these size effects, we can find similar triangles in the \(C_{\mathrm{C}}-C_{\mathrm{B}}\) diagrams for fractal-like networks; there exist nodes in the center that are close to all points and edges that provide distributed nodes in every scale of one spatial structure. In comparison, theoretical DLA patterns used for simulating local electric fields in Figure 2 are also analyzed, as shown in Figure 9. We also find similar triangles, which indicate that the DLA model used in this study imitates network structures observed in the nanoparticles networks in Figures 6a and 7a-d. Using these diagrams, we can find the common rules in patterns that create local resonances. In each figure, we pick up intensified resonances with the surrounding nodes (in red and green circles) from Figure 2. The nodes form reversed 'U' shapes in all cases in the \(C_{\mathrm{C}}-C_{\mathrm{B}}\) diagrams, indicated by connected lines in the corresponding colors. The nodes near the end of the resonance structure have low values of \(C_{\mathrm{B}}\), and those in the center have larger \(C_{\mathrm{B}}\) values, while \(C_{\mathrm{C}}\) gradually changes along the resonance structure due to variation of distances to the center.