Construction and Analysis of Random Networks with Explosive Percolation

Abstract

The existence of explosive phase transitions in random (Erdős Rényi-type) networks has been recently documented by Achlioptas et al. [Science 323, 1453[Science 323, (2009] via simulations. In this Letter we describe the underlying mechanism behind these first-order phase transitions and develop tools that allow us to identify (and predict) when a random network will exhibit an explosive transition. Several interesting new models displaying explosive transitions are also presented. PACS numbers: Valid PACS appear hereThe structure and dynamics of networked models and their application to social networks is an important and active area of research encompassing many fields ranging from physics [1, 2, 3] to sociology [4] to combinations thereof [5,6,7]. Ideas from statistical mechanics have contributed greatly to our understanding of such networks and their practical uses [8,9,10]. Of particular importance is the statistical mechanical notion of the order of a phase ('percolation') transition. Phase transitions in random network models are almost always second order [11,12] or higher [12,13]. Thus, it was surprising to many when Achlioptas et al. [14] reported recently that some models of interest in social networks can display first-order (discontinuous) transitions.In that work, they described several random graph models of the Erdős Rényi (ER) variety that exhibit firstorder or what they call "explosive" phase transitions. They provide convincing numerical evidence and a useful characterization of such transitions, but no details on the mechanisms underlying them. They describe several systems which display such transitions and a general class of systems which don't.In this paper we describe the underlying mechanisms behind explosive transitions in ER-type models. We show that, somewhat surprisingly, the key to explosive transitions is not the details of the edge-addition rules at work during the actual "explosion," but rather lies in the period preceding the explosion when a type of "powder keg" develops. In effect, the importance of the rules is to create an explosive situation, which can be detonated with almost any rule. In addition, our analysis provides an understanding of which random network models will have such transitions. This allows us to construct large classes of interesting models that display this behavior. (It also allows us to rule out many other models which will not display explosive transitions.)The prototypical network percolation example is that of pure (non-preferential) ER random graphs [11]. These begin with a set of n nodes, where n is large. Edges are then added to the graph, uniformly at random. As is well known, this system exhibits a phase transition as the number of edges τ increases. For τ < 0.5n all clusters are small (∼ log(n)) while for τ > 0.5n a large cluster (∼ n) appears. In the large-n limit this transition is a secondorder phase transition, i.e., letting s(τ ) be the size of the largest cluster after τ edges have been added, the graph of s(τ )/n against τ /n is continu...

Continuity of the explosive percolation transition

“…This growth rule, named as the Achlioptas process (AP), is then studied on the two-dimensional lattice [2,3] and on the scalefree networks [4][5][6] as well, yielding similar results. That suddenness has been widely believed to indicate a discontinuity at the percolation transition in the thermodynamic limit [7,8], and the similar explosiveness has been observed with the other growth rules proposed later [9][10][11][12][13]. These observations of the explosiveness have drawn much interest due to the striking difference from the wellknown continuous transition in the standard percolation models [14].…”

“…These observations of the explosiveness have drawn much interest due to the striking difference from the wellknown continuous transition in the standard percolation models [14]. However, in our point of view, the explosiveness has not been carefully investigated as yet enough to draw a decisive conclusion on the discontinuity, and possibly just represents an extremely steep but still continuous transition.Friedman and Landsberg [9] have suggested the argument of the powder keg as a circumstantial description to explain the apparent discontinuity of the explosive percolation transition. Meanwhile, da Costa et al [15] have reported that the explosive percolation is actually continuous for a modified version of the AP by analytically deriving the critical scaling relations based on numerical observations of power-law critical distribution of cluster size [16].…”

“…It may be natural to use the information above the transition point (t > t c ) in order to prove the (non-)existence of the gap or estimate the gap size. Thus, most of previous studies have focussed on this information [1][2][3][4][5][6][7][8][9][10][11][12][13]15], but could not provide a definitive answer due to the extremely slow convergence of the order parameter in system size. In this work, we took the opposite approach.…”

Crackling noise in fractional percolation

“…These jumps are discontinuous phase transitions. However, when such mechanisms are mixed, even weakly, with mechanisms that merge components purely at random then the transitions vanish, or become at most weakly discontinuous characterized by very small power law exponents [30][31][32][33][34][35][36][37][38][39][40][41] , see Supplementary Methods and Supplementary Figs S1-S3.…”

Transmission of packets on a hierarchical network: Statistics and explosive percolation

“…Recently, Achlioptas et al [39] studied percolation for the Erdos Renyi model using a product rule and found that the giant component emerged suddenly at the percolation threshold, and that the percolation transition was discontinuous. This discontinuous percolation transition appears when the growth of the largest cluster is systematically suppressed thereby promoting the formation of several large components that eventually merge in an explosive way [40]. Several aggregation models, based on percolation, have been developed to achieve this change in the nature of transition [19,39,[41][42][43][44][45][46][47].…”