Riascos A P, Mateos José L
Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, D.F., México.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Sep;90(3):032809. doi: 10.1103/PhysRevE.90.032809. Epub 2014 Sep 17.
We introduce a formalism of fractional diffusion on networks based on a fractional Laplacian matrix that can be constructed directly from the eigenvalues and eigenvectors of the Laplacian matrix. This fractional approach allows random walks with long-range dynamics providing a general framework for anomalous diffusion and navigation, and inducing dynamically the small-world property on any network. We obtained exact results for the stationary probability distribution, the average fractional return probability, and a global time, showing that the efficiency to navigate the network is greater if we use a fractional random walk in comparison to a normal random walk. For the case of a ring, we obtain exact analytical results showing that the fractional transition and return probabilities follow a long-range power-law decay, leading to the emergence of Lévy flights on networks. Our general fractional diffusion formalism applies to regular, random, and complex networks and can be implemented from the spectral properties of the Laplacian matrix, providing an important tool to analyze anomalous diffusion on networks.
我们基于分数阶拉普拉斯矩阵引入了一种网络上的分数阶扩散形式体系，该矩阵可直接由拉普拉斯矩阵的特征值和特征向量构建。这种分数阶方法允许具有长程动力学的随机游走，为反常扩散和导航提供了一个通用框架，并能在任何网络上动态诱导出小世界特性。我们得到了平稳概率分布、平均分数阶返回概率和一个全局时间的精确结果，表明与正常随机游走相比，使用分数阶随机游走时网络导航效率更高。对于环的情况，我们得到了精确的解析结果，表明分数阶转移概率和返回概率遵循长程幂律衰减，从而导致网络上出现列维飞行。我们的一般分数阶扩散形式体系适用于规则、随机和复杂网络，并且可以从拉普拉斯矩阵的谱性质实现，为分析网络上的反常扩散提供了一个重要工具。
