## 1 Introduction  
Geometrical effects of metallic components in two- or three-dimensional space are key issues to control electromagnetic waves [1,2]. In the research of metamaterials, the designs of the spatial shapes of metals are being explored not only for microwaves or terahertz waves, but also for lights in the optical range [1,3, 4, 5, 6, 7, 8, 9, 10]. A careful design of the structure in each metamaterial unit leads to extraordinary properties that are not realized in nature, such as negative-refractive-index states and cloaking phenomena. The structures proposed so far have been elaborately fabricated as spatial periodic patterns using a unit microstructure much smaller than the wavelength, and nano-size precision is required for control in optics. For future photonic devices that have a visible size, a large volume of metamaterial components with a huge number of nano-scale structures, in the millimeter scale for instance, is required; fabrication processes based on photo masks will cost too much to use such optical devices for daily uses. From this point of view, self-assembled configurations of nano-structures [11,12] are quite favorable for optical metamaterials or similar required media. As a self-assembled structure with potential functions similar to regular metamaterials, in this report, we focus on fractals or fractal-like shapes [13,14] created without designed fabrication processes. There exist a number of structures in nature that form without artificial designs, but according to certain geometrical principles. Fractals are one of them, observed in the outer edges of clouds and ocean coasts for instance, and they have self-similarity structures at every scale. Although their optical properties have been investigated from several points of view [15], clear evidence for frequency characteristics similar to metamaterials has not been confirmed in general self-assembled fractal patterns. Regarding regular fractals designed artificially, several previous reports revealed their effects as being similar to metamaterials with spatial periodic structures [16; 17]. Irregular fractal structures, which include patterns created by nano-particles [18; 19; 20; 21] and by nano-pores [22; 23; 24; 25], also indicate extraordinary optical properties, but the previous studies are related to resonances near the surface plasmons in which dipole coupling between particles is frequently important [18; 19; 20] or broad-band absorption [21], the property of which is far from the resonance features that are significant in metamaterials with negative permeability. Recently, we reported the effects of a fractal-like nanoparticle network structure with abnormal optical resonances [26], which triggers this study to reveal the potentials of self-assembled fractal structures for optical metamaterial alternatives. As widely investigated, fractal structures with seemingly random shapes have some geometrical quantities that are common and constant, such as fractal dimension [13]; after its definition, the concept of the dimension becomes one of the general measures for the topological invariants for material microstructures, beyond fractal structures [14]. Further recognitions of common topological features may be possible by complex network theory [27; 28; 29], which describes both microscopical properties of constituent particles and their statistical ones in a macroscopic point of view. If the application of network theory is successful for optical solutions, this analysis scheme described below will be a new contribution of complex network theory. In this report, we demonstrate theoretical analysis and experimental observations about the optical responses of two-dimensional (2D) fractal-like metallic structures. The absorption spectra have similar and constant resonance frequencies in all model structures with a fixed size of the individual unit and in every pattern created in the experimental procedure with a certain range of parameters.