Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect

Abstract

A vast variety of real-life networks display the ubiquitous presence of scale-free phenomenon and small-world effect, both of which play a significant role in the dynamical processes running on networks. Although various dynamical processes have been investigated in scale-free small-world networks, analytical research about random walks on such networks is much less. In this paper, we will study analytically the scaling of the mean first-passage time (MFPT) for random walks on scale-free small-world networks. To this end, we first map the classical Koch fractal to a network, called Koch network. According to this proposed mapping, we present an iterative algorithm for generating the Koch network, based on which we derive closed-form expressions for the relevant topological features, such as degree distribution, clustering coefficient, average path length, and degree correlations. The obtained solutions show that the Koch network exhibits scale-free behavior and small-world effect. Then, we investigate the standard random walks and trapping issue on the Koch network. Through the recurrence relations derived from the structure of the Koch network, we obtain the exact scaling for the MFPT. We show that in the infinite network order limit, the MFPT grows linearly with the number of all nodes in the network. The obtained analytical results are corroborated by direct extensive numerical calculations. In addition, we also determine the scaling efficiency exponents characterizing random walks on the Koch network.

12 pages, 8 figures. Definitive version published in Physical Review E

References in corpus (11)

• Self-similar disk packings as model spatial scale-free networks
• Exact mean first-passage time on the T-graph
• Exact solution for mean first-passage time on a pseudofractal scale-free web
• Recursive graphs with small-world scale-free properties
• Random walks on complex trees
• Random walks on the Apollonian network with a single trap
• Evolving small-world scale-free networks consist of cliques
• Fractal scale-free networks resistant to disease spread
• Correlations in random Apollonian network
• Influences of degree inhomogeneity on average path length and random walks in disassortative scale-free networks
• The rigorous solution for the average distance of a Sierpinski network