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Z. Naturforsch. 67a, 225 – 229 (2012)
doi:10.5560/ZNA.2012-0012
Percolation in a Hierarchical Lattice
Yilun Shang
Institute for Cyber Security, University of Texas at San Antonio, Texas 78249, USA
Received August 19, 2011 / revised November 23, 2011 / published online May 2, 2012
Reprint requests to: Y. S.; E-mail: shylmath@hotmail.com
We study the percolation in the hierarchical lattice of order N where the probability of connection between two nodes separated by a distance k is of the form min {αβk, 1}, α ≥ 0 and β > 0. We focus on the vertex degrees of the resulting percolation graph and on whether there exists an infinite component. For fixed β, we show that the critical percolation value αc(β) is non-trivial, i.e., αc(β) ϵ (0,∞ ), if and only if β ϵ (N, N2).
Key words: Percolation; Random Graph; Degree; Hierarchical Lattice; Phase Transition.
PACS numbers: 64.60.ah; 02.50.Cw; 02.10.Ox
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