Critical Phase of Bond Percolations on Growing Networks

Abstract

The critical phase of bond percolation on the random growing tree is examined. It is shown that the root cluster grows with the system size $N$ as $N^{\psi }$ and the mean number of clusters with size $s$ per node follows a power function $n_s\propto s^{-\tau }$ in the whole range of open bond probability $p$. The exponent $\tau$ and the fractal exponent $\psi$ are also derived as a function of $p$ and the degree exponent $\gamma$, and are found to satisfy the scaling relation $\tau =1+{\psi }^{-1}$. Numerical results with several network sizes are quite well fitted by a finite size scaling for a wide range of $p$ and $\gamma$, which gives a clear evidence for the existence of a critical phase.

5 pages, 4 figures; accepted for publication in Physical Review E

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Cited by in corpus (11)

• Percolation on complex networks: Theory and application
• Generating-function approach for bond percolations in hierarchical networks
• Transition-type change between an inverted Berezinskii-Kosterlitz-Thouless transition and an abrupt transition in the bond percolation on a random hierarchical small-world network
• Hierarchical scale-free network is fragile against random failure
• Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network
• Percolation properties of growing networks under an Achlioptas process
• Analytic results for the percolation transitions of the enhanced binary tree
• Profile and scaling of the fractal exponent of percolations in complex networks
• Absence of the non-percolating phase for percolation on the non-planar Hanoi network
• Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree
• Renormalization-group theory of the abnormal singularities at the critical-order transition in bond percolation on pointed hierarchical graphs