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Impact of initial lattice structures on networks generated by traces of random walks

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  • Ikeda, Nobutoshi

Abstract

We show that the platform stage of network evolution plays a principal role in the topology of resulting networks generated by short-cuts stimulated by the movements of a random walker, the mechanism of which tends to produce power-law degree distributions. To examine the numerical results, we have developed a statistical method which relates the power-law exponent γ to random properties of the subgraph developed in the platform stage. As a result, we find that an important exponent in the network evolution is α, which characterizes the size of the subgraph in the form V∼tα, where V and t denote the number of vertices in the subgraph and the time variable, respectively. 2D lattices can impose specific limitations on the walker’s diffusion, which keeps the value of α within a moderate range and provides typical properties of complex networks. 1D and 3D cases correspond to different ends of the spectrum for α, with 2D cases in between. Especially for 2D square lattices, a discontinuous change of the network structure is observed, which varies according to whether γ is greater or less than 2. For 1D cases, we show that emergence of nearly complete subgraphs is guaranteed by α<1/2, although the transient power-law is permitted at low increase rates of edges. Additionally, the model exhibits a spontaneous emergence of highly clustered structures regardless of its initial structure.

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  • Ikeda, Nobutoshi, 2010. "Impact of initial lattice structures on networks generated by traces of random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(16), pages 3336-3347.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:16:p:3336-3347
    DOI: 10.1016/j.physa.2010.03.027
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    References listed on IDEAS

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    1. Ikeda, N., 2007. "Network formed by traces of random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(2), pages 701-713.
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    1. Ikeda, Nobutoshi, 2021. "Stratified structure of fractal scale-free networks generated by local rules," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 583(C).
    2. Ikeda, Nobutoshi, 2019. "Growth model for fractal scale-free networks generated by a random walk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 424-434.
    3. Ikeda, Nobutoshi, 2017. "Topology of growing networks accelerated by intermediary process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 484(C), pages 378-393.
    4. Ikeda, Nobutoshi, 2020. "Fractal networks induced by movements of random walkers on a tree graph," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).

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