Agglomerative percolation on bipartite networks: Nonuniversal behavior due to spontaneous symmetry breaking at the percolation threshold

Abstract

Ordinary bond percolation (OP) can be viewed as a process where clusters grow by joining them pairwise, by adding links chosen randomly one by one from a set of predefined 'virtual' links. In contrast, in agglomerative percolation (AP) clusters grow by choosing randomly a 'target cluster' and joining it with all its neighbors, as defined by the same set of virtual links. Previous studies showed that AP is in different universality classes from OP for several types of (virtual) networks (linear chains, trees, Erdös-Rényi networks), but most surprising were the results for 2-d lattices: While AP on the triangular lattice was found to be in the OP universality class, it behaved completely differently on the square lattice. In the present paper we explain this striking violation of universality by invoking bipartivity. While the square lattice is a bipartite graph, the triangular lattice is not. In conformity with this we show that AP on the honeycomb and simple cubic (3-d) lattices -both of which are bipartite -are also not in the OP universality classes. More precisely, we claim that this violation of universality is basically due to a Z2 symmetry that is spontaneously broken at the percolation threshold. We also discuss AP on bipartite random networks and suitable generalizations of AP on k−partite graphs.

Agglomerative percolation on the Bethe lattice and the triangular cactus

“…The numerical simulation study on the exact bipartite random graph earned only the critical exponent ν and the fractal dimension D of the giant cluster as ν = 4.7(2) and D = 0.567 (6), which are close to those for ER network but differ by more than one standard deviation [12]. Therefore, at the present stage, MFT for AP on the bipartite graphs are far from completion.…”

“…Then the selected cluster merges all the nearest neighboring clusters to form a new cluster. The phase transition in AP is shown to be continuous, but belongs to a new universality class different from the class of the random percolation if the base structure of AP is bipartite [12]. On the bipartite structure like a two-dimensional square lattice the merging process spontaneously breaks the Z 2 symmetry at the transition threshold, which is the origin of the new universality class [12].…”

“…The phase transition in AP is shown to be continuous, but belongs to a new universality class different from the class of the random percolation if the base structure of AP is bipartite [12]. On the bipartite structure like a two-dimensional square lattice the merging process spontaneously breaks the Z 2 symmetry at the transition threshold, which is the origin of the new universality class [12]. In contrast, the universality of the transition of AP on the triangular lattice, which is not bipartite, is the same as that of the random percolation [12].…”

“…On the bipartite structure like a two-dimensional square lattice the merging process spontaneously breaks the Z 2 symmetry at the transition threshold, which is the origin of the new universality class [12]. In contrast, the universality of the transition of AP on the triangular lattice, which is not bipartite, is the same as that of the random percolation [12]. Using analytical methods and numerical simulations, APs on the one-dimensional ring [8], the two-dimensional square lattice and triangular lattice [9], critical tree [10], and complex network [11] were studied.…”

“…Another kind of studies was on the agglomerative percolation (AP) [8][9][10][11][12]. In AP one cluster is randomly selected instead of a bond or a site.…”