## 3 Experimental Results on Fractal-Like Metallic Nanoparticle Patterns  
To confirm these theoretical predictions, experiments were performed using similar equipment under similar parameters described in our previous reports [26]. In brief, the atmospheric-pressure plasmas in the gas flow of the Ar and NH\({}_{3}\) mixture yield N\({}_{2}\)H\({}_{4}\) with a concentration in the order of 10\({}^{21}\) m\({}^{-3}\). N\({}_{2}\)H\({}_{4}\) is a reductant agent, and so far, we have observed the formation of Ag nanoparticles in AgNO\({}_{3}\) aqueous solutions and their aggregation as a conductive layer; the structures were fractal-like ones in some range of experimental parameters, confirmed by the images from scanning electron microscopy (SEM). When we also observed Ag nanoparticle aggregates on Si substrates, some abnormal absorption spectra in the optical range were detected using the Fourier-transformed infrared (FT-IR) spectroscopy method (FT-720, Horiba), but its origins have not been identified yet. As a reference, in the case without N\({}_{2}\)H\({}_{4}\) treatment, absorption by Ag\({}^{+}\)NO\({}_{3}^{-}\)[31], which is a residue of the solutes, was more or less observed at the spectrum around 7600 nm. When we changed the AgNO\({}_{3}\) concentration \(C_{\text{AgNO}_{3}}\), several patterns emerged in the formation and aggregation of Ag particles. When \(C_{\text{AgNO}_{3}}\) was low (Figure 4a), no significant patterns existed and \(d_{\text{box}}\) was near one (shown in Figure 5). As increasing \(C_{\text{AgNO}_{3}}\) up to \(C_{\text{AgNO}_{3}}\) = 0.1 mol/L, fractal-like patterns were observed (Figure 4b-d), and \(d_{\text{box}}\) increased and got saturated around 1.8 to 1.9. Regarding the particle distance \(s_{\text{Ag}}\), measured from images in Figure 4 via particle allocation shown in Figures 6a and 7, \(s_{\text{Ag}}\) was almost constant for \(C_{\text{AgNO}_{3}}\) = 0.01 to 0.05 mol/L (although the particle size itself somewhat increased) and increased at \(C_{\text{AgNO}_{3}}\) = 0.1 mol/L (Figure 5). Figure 6 represents examples of the topological analysis proposed in this study for nanoparticle networks, and Figure 7 shows various results based on this analysis; the details of this analysis method will be described in Section 4. Figure 8 shows the transmittance spectra of these samples shown in Figure 4, measured by the FT-IR method as a function of \(\lambda\). While \(s_{\text{Ag}}\) was almost constant (700 to 900 nm) for \(C_{\text{AgNO}_{3}}\) = 0.01 to 0.05 mol/L, a significant abnormal absorption spectrum existed around 7300 nm. The wavelength was almost constant, and it showed twin peaks with the Ag\({}^{+}\)NO\({}_{3}^{-}\) absorption around 7600 nm. On the other hand, when \(C_{\text{AgNO}_{3}}\) was raised to 0.1 mol/L and \(s_{\text{Ag}}\) increased up to \(\sim\)1400 nm, the absorption spectra were located around 11,800 nm and 14,200 nm. The absorption spectra are in a different range from that of the conventional surface plasmon resonances around the visible light range [32] and also show narrow resonance features different from the broad ones [21]. As discussed in the theoretical predictions, resonances in fractal-like structures take place around 1/2\(\lambda\) of combined branches, composed of several particles. Rough consistency is found, as described in the following. In the cases with \(C_{\text{AgNO}_{3}}\) = 0.01 to 0.05 mol/L, for \(\lambda\)\(\sim\) 7300 nm, four particles with 800 nm per each distance \(s_{\text{Ag}}\) correspond to 1/2\(\lambda\), and for \(\lambda\)\(\sim\)10,400 nm, six particles with \(s_{\text{Ag}}\) = 800 nm correspond to 1/2\(\lambda\). In the case with \(C_{\text{AgNO}_{3}}\) = 0.1 mol/L, for \(\lambda\)\(\sim\) 11,800 nm, four particles with \(s_{\text{Ag}}\) = 1400 nm correspond to 1/2\(\lambda\), and for \(\lambda\)\(\sim\)14,200 nm, five particles with \(s_{\text{Ag}}\) = 1400 nm correspond to 1/2\(\lambda\).