Agglomerative percolation in two dimensions

Abstract

-We study a process termed agglomerative percolation (AP) in two dimensions. Instead of adding sites or bonds at random, in AP randomly chosen clusters are linked to all their neighbors. As a result the growth process involves a diverging length scale near a critical point. Picking target clusters with probability proportional to their mass leads to a runaway compact cluster. Choosing all clusters equally leads to a continuous transition in a new universality class for the square lattice, while the transition on the triangular lattice has the same critical exponents as ordinary percolation -violating blatantly the basic notion of universality.Percolation is a pervasive concept in statistical physics and an important branch of mathematics [1]. It typifies the emergence of long range connectivity in many systems such as the flow of liquids through porous media [2], transport in disordered media [3], spread of disease in populations [4], resilience of networks to attack [5], formation of gels [6] and even of social groups [7]. It also underlies a number of other critical phenomena -like the Ising order/disorder transition, which is a percolation transition on the set of spins with given sign [8].The phase transition in ordinary percolation (OP), where bonds or sites are added at random, represents a broad universality class. Recently Achlioptas et al.[9] made a simple modification by, at each step, selecting among two possibilities the link that leads to the slowest growth of large clusters. This global choice introduces a large length scale -the system size L -which can alter universality. Indeed, they concluded that an unusual, discontinuous transition (called "explosive percolation") emerges where a macroscopic cluster appears suddenly while at the same time scaling in other quantities is observed [10][11][12]. Various modifications of the rule have been made [13,14] -all finding evidence of a discontinuous transition. Although the claim for discontinuity in [9] was refuted later in [15], explosive percolation in that case does represent a new universality class.Here we discuss a percolation process that also contains a potentially large length scale in its definition, in this case the correlation length ξ. Our process has direct application to the study of complex networks. Instead of adding bonds randomly, we pick a random cluster and add bonds to its entire surface in order to link it to all adjacent clusters. Starting with the state where all clusters have size one, at each update t → t + 1 the process repeats until the entire lattice (or graph) is reduced to a single cluster. We call this "agglomerative percolation" (AP), in analogy with cluster growth by aggregation [16]. Thus if by chance a cluster of length scale ℓ is picked, links are added simultaneously at distances O(ℓ) apart.AP can be analyzed on any graph. It corresponds to random sequential renormalization [17] of a network, where a single cluster is identified as a 'super'-node that is a local coarse-graining of the graph. In this perspe...

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

“…Recent discoveries that widened enormously the scope of different behaviors at the percolation threshold include infinite order transitions in growing networks [1], supposedly first order transitions in Achlioptas processes [2] (that are actually continuous [3,4] but show very unusual finite size behavior [5]), and real first order transitions in interdependent networks [6][7][8][9]. Another class of "non-classical" percolation models, inspired by attempts to formulate a renormalization group for networks [10,11], was introduced in [12][13][14][15][16] and is called 'agglomerative percolation' (AP).…”

“…Although a similarly complete mathematical analysis is not possible on random graphs, both numerics and nonrigorous analytical arguments show that the same is true for 'critical' trees [12] and Erdös-Rényi graphs [16]. In contrast to these cases that establish AP as a novel phenomenon but do not present big surprises, the behavior on 2-d regular lattices [13] is extremely surprising: While AP on the triangular lattice is clearly in the OP universality class (with only some minor caveats), it behaves completely different on the square lattice. There the average cluster size at criticality diverges as the system size L increases (it stays finite for all realizations of OP on any regular lattice), the fractal dimension of the incipient giant cluster is D f = 2 (D f = 91/48 ≈ 1.90 for OP), and the cluster mass distribution obeys a power law with power τ = 2 (τ = 187/91 ≈ 2.055 for OP).…”

Percolation in Media with Columnar Disorder

“…Equation (11) implies that this bound is true whenever L > ln L z , in particular for any fixed aspect ratio L z /L. The bound (12) clearly shows that, for the region p (3) c < p B < p B,c (p z ), the system is in a Griffiths phase [29][30][31][32].…”

Agglomerative percolation on the Bethe lattice and the triangular cactus

“…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.…”

“…This means that in each growth process multiple bonds can be occupied at the same time. Therefore the natural control parameter in AP 2 is the number of clusters per site n instead of the fraction p of the occupied bonds or sites [9,12].…”