Avoiding a Spanning Cluster in Percolation Models

Abstract

When dynamics in a system proceeds under suppressive external bias, the system can undergo an abrupt phase transition, as can happen when an epidemic spreads. Recently, an explosive percolation (EP) model was introduced to understand such phenomena. The order of the EP transition has not been clarified in a unified framework covering low-dimensional systems and the mean-field limit. We introduce a stochastic model in which a rule for dynamics is designed to avoid the formation of a spanning cluster through competitive selection in Euclidean space. We use heuristic arguments to show that in the thermodynamic limit and depending on a control parameter, the EP transition can be either continuous or discontinuous if d < d(c) and is always continuous if d ≥ d(c), where d(c) is the spatial dimension and d is the upper critical dimension.

Continuous Percolation Transition in Random Cluster Growth Model

“…13,14 Recently it is demonstrated analytically as well as numerically that EP transition can be either continuous or discontinuous depending on the bias and the dimensionality of the system. 15 In this paper a generalized random cluster growth model with different initial seed concentration ρ and tunable growth probability g is presented. For a given ρ, a critical growth probability g c is found to exists at which continuous percolation transition occurs.…”

Structural properties of networks grown via an Achlioptas process

“…The growth behavior of the giant cluster follows a power-law with a critical exponent β = 1 in ER random networks [9,10], hence undergoing continuous phase transition. The values of the critical link density and the exponent β have been the main research interest, igniting active discussion on the universality class of the continuous/discontinuous percolation transition in various model systems [10][11][12][13][14][15][16][17][18][19].…”

“…Recently, percolation phenomena have been investigated in various complex network structures, revealing various transition natures that depend on how the networks are built [11][12][13][14][15][16][17][18][19]. In particular, the celebrated Achlioptas process (AP) [11] has drawn enormous attention due to the existence of a very sharp transition, as the name 'explosive percolation' suggests.…”

Inverting the Achlioptas rule for explosive percolation

“…We develop a theory of this kind of percolation based on the Smoluchowski equation, find the percolation threshold, and describe the scaling properties of this continuous transition, namely, the critical exponents and amplitudes, and scaling functions. We show that, qualitatively, this transition is similar to the ordinary percolation one, though occurring in less connected systems.PACS numbers: 64.60.ah, 05.40.-a, 64.60.FAggregation processes based on progressive merging together the smallest clusters of a few randomly chosen (Achlioptas rule) have attracted much attention in the last years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. These specific processes and models show a number of unusual features [17] distinguishing them sharply from ordinary aggregation processes and standard percolation, in which random clusters merge together with probability proportional to their sizes [18][19][20].…”