## 4 Derivation of Network Parameters from Experimental and Theoretical Nanoparticle Patterns  
So far, we have investigated the effects of self-assembled fractal-like aggregates of metallic nanoparticles with extraordinary optical features for electromagnetic waves. The results have been analyzed mainly according to the empirical scaling. Here, to derive more common and general features that such nano-structures possess macroscopically and statistically, we analyze the fractal-like network properties of nanoparticles using centrality measures [29]. For several previous decades, besides the general definition of dimensions in the fractal structures [13; 14] used in Figure 5, measures for network topology have been already proposed in bonding structures and reaction paths in chemical reactions, like the Wiener index and topological efficiency index [33; 34; 35], which might also work for the characterization of observed structures. Here, however, we use others, developed in network theory [29; 36], closely relevant to and based on electric circuit models [37], by considering the importance of electric conductivity over nanoparticle network structures in our study. Since nanoparticles make contact with each other in fractal-like patterns, we define the nodes of a network by nanoparticles and edges by their adjacency. Although this analysis is based on network theory [29], another point this approach can clarify is the prediction of the possible presence of optical resonance structures in the observed nanoparticle patterns, as described below. Examples of results obtained in this analysis are shown in Figure 6. The following is the procedure for analyzing nanoparticle networks. Picking up an isolated cluster and allocating nodes and edges to the observed nanoparticles, we calculate two network indices: closeness centrality and betweenness centrality [29]. The measure of closeness centrality for a given node \(C_{\mathrm{C}}\) is defined as an the inverse of the sum of path lengths between the node and the others; the measure becomes large if the node is located in the center of the network structure. The measure of betweenness centrality for a given node \(C_{\mathrm{B}}\) is defined as a relative probability of the occupation of a position along the shortest path among all nodes; \(C_{\mathrm{B}}\) becomes large when the node is located near the center and the channel of the network becomes narrow. In relative perspectives, \(C_{\mathrm{C}}\) is sensitive to the aspect ratio of the entire  
Figure 8: Transmittance spectra of Ag nanoparticle patterns detected by the FT-IR method. From the top, target corresponds to cases of AgNO\({}_{3}\) concentration \(C_{\mathrm{AgNO_{3}}}\) = 0.001 mol/L, 0.01 mol/L, 0.02 mol/L, 0.05 mol/L and 0.1 mol/L in Figure 4. Arrows point out extraordinary absorption, and hatched regions indicate artifact signals mixed in the calibration procedure for removing background signals (H\({}_{2}\)O, etc.). The spectrum with \(C_{\mathrm{AgNO_{3}}}\) = 0.02 mol/L is from [26]. network configuration (e.g., square or rectangular shapes), whereas \(C_{\mathrm{B}}\) is sensitive to the deformation of the network configuration into singular patterns. We plot these two measures for all nodes in one 2D diagram, since correlation between \(C_{\mathrm{C}}\) and \(C_{\mathrm{B}}\) will clarify topological properties of nanoparticle networks. In the general analysis of complex networks, other statistical parameters like degree distributions and cluster coefficients also become measures of network topology, but a nanoparticle network in our cases has a restriction of 2D patterns in which the maximum degree or number of edges for each node is four at most, and shortcuts to remote nodes are prohibited; correlation between \(C_{\mathrm{C}}\) and \(C_{\mathrm{B}}\) will be favorable instead. Figure 6a shows the case of the nanoparticle network with 33 nodes found in the SEM image shown in Figure 4c, with \(C_{\mathrm{AgNO_{3}}}\) at 0.02 mol/L.