Ordinary percolation with discontinuous transitions

Abstract

Percolation on a one-dimensional lattice and fractals, such as the sierpinski gasket, is typically considered to be trivial, because they percolate only at full bond density. By dressing up such lattices with small-world bonds, a novel percolation transition with explosive cluster growth can emerge at a non-trivial critical point. There, the usual order parameter, describing the probability of any node to be part of the largest cluster, jumps instantly to a finite value. Here we provide a simple example in the form of a small-world network consisting of a one-dimensional lattice which, when combined with a hierarchy of long-range bonds, reveals many features of this transition in a mathematically rigorous manner.

Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network

“…Boettcher et al studied the bond percolation model, i.e., the one-state Potts model, in the same graph without backbone (K = 0) [12] and found a PF bifurcation of the RG FP similar to the present model (q = 2). They also calculated the maximum cluster size s max , which is a quantity corresponding to unconnected susceptibilityχ ≡ χ + Nm 2 because both quantities are defined as a summation of the two-point correlation.…”

“…A critical phase, if it exists, lies between a disordered phase withξ < 1/y d and an ordered phase withξ = ∞. Such a phase with a fractal exponent 0 < ψ < 1 is actually observed in the percolation transitions in enhanced trees [5], hyperbolic lattices [10], hierarchical graphs [11,12], and growing random networks [13].…”

“…This network is very useful because rigorous real-space renormalization is possible for various models in the simplest way. Furthermore various types of phase transition are observed depending on the model used, e.g., a discontinuous transition of the bond percolation model [12], equivalent to the one-state Potts model [14,15], and continuous transitions with a power-law singularity (PLS) or an essential singularity (ES) for the q-state Potts model with q 3 [16]. These are observed in other graphs [3][4][5]17,18].…”

“…If ln N is a proper cutoff length, μ should be unity. Equations (26) Table I, where we also show the Potts model with q = 1 [12] and q 3 [16]. See also the inset of Fig.…”

Hierarchical scale-free network is fragile against random failure

“…Here s max (N ; p) is the mean size of the largest cluster in a graph of size N at a given value of p. As known, p c1 = p c2 on Euclidean lattices, whereas, p c1 < p c2 on transitive nonamenable graphs [21,22]. Also, in complex networks, some growing network models [25][26][27][28][29] and hierarchical network models [23,[30][31][32][33] yield 0 = p c1 < p c2 for bond percolation, whereas, p c1 = p c2 for some static network models, such as uncorrelated networks…”