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We propose a model for evolving networks by merging building blocks represented as complete graphs, reminiscent of modules in biological system or communities in sociology. The model shows power-law degree distributions, power-law clustering spectra and high average clustering coefficients independent of network size. The analytical solutions indicate that a degree exponent is determined by the ratio of the number of merging nodes to that of all nodes in the blocks, demonstrating that the exponent is tunable, and are also applicable when the blocks are classical networks such as Erdős-Rényi or regular graphs. Our model becomes the same model as the Barabási-Albert model under a specific condition.

Propagation of chemical waves in the Belousov-Zhabotinsky reaction in a thin layer of ferroin-loaded cationexchange resin beads is investigated. Dispersion relations are obtained for waves propagating in layers of resin beads. The relations in layers of resin beads are different both in the trigger wave velocity and the refractory period, when mesh size and/or percentages of cross-linkage of resin beads are different. A system with a sharp boundary between two layers with different wave velocities is made using two different resin beads exhibiting different dispersion relations. Incident waves at any angle from the high-velocity layer are refracted at the boundary. Waves from the low-velocity layer also exhibit refraction when the angle of incidence is smaller than a critical value. Refraction obeys Snell's law. Incident waves from the low-velocity layer at angles larger than the critical value are reflected. The angle of reflection is equal to the critical angle and does not depend on the incident angle. When the period of incident waves from the high-velocity layer is shorter than the refractory period in the low-velocity layer, wave propagation across the boundary causes the change of frequency.

We propose the n-clique network as a powerful tool for understanding global structures of combined highly-interconnected subgraphs, and provide theoretical predictions for statistical properties of the n-clique networks embedded in a complex network using the degree distribution and the clustering spectrum. Furthermore, using our theoretical predictions, we find that the statistical properties are invariant between 3-clique networks and original networks for several observable real-world networks with the scale-free connectivity and the hierarchical modularity. The result implies that structural properties are identical between the 3-clique networks and the original networks.

We propose a growing network model that consists of two tunable mechanisms: growth by merging modules which are represented as complete graphs and a fitnessdriven preferential attachment. Our model exhibits the three prominent statistical properties are widely shared in real biological networks, for example gene regulatory, protein-protein interaction, and metabolic networks. They retain three power law relationships, such as the power laws of degree distribution, clustering spectrum, and degree-degree correlation corresponding to scale-free connectivity, hierarchical modularity, and disassortativity, respectively. After making comparisons of these properties between model networks and biological networks, we confirmed that our model has inference potential for evolutionary processes of biological networks.