Explosive Percolation with Multiple Giant Components

Abstract

We generalize the random graph evolution process of Bohman, Frieze, and Wormald [T. Bohman, A. Frieze, and N. C. Wormald, Random Struct. Algorithms, 25, 432 (2004)]. Potential edges, sampled uniformly at random from the complete graph, are considered one at a time and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function asymptotically approaching the value α = 1/2. We show that multiple giant components appear simultaneously in a strongly discontinuous percolation transition and remain distinct. Furthermore, tuning the value of α determines the number of such components with smaller α leading to an increasingly delayed and more explosive transition. The location of the critical point and strongly discontinuous nature are not affected if only edges which span components are sampled. The percolation phase transition models the onset of large-scale connectivity in lattices or networks, in systems ranging from porous media, to resistor networks, to epidemic spreading [2][3][4][5]. Percolation was considered a robust second-order transition until a variant with a choice between edges was shown to result in a seemingly discontinuous transition [6]. Subsequent studies have shown similar results for scale-free networks [7,8], lattices [9, 10], local cluster aggregation models [11], singleedge addition models [12,13], and models which control only the largest component [14]. It seems a fundamental requirement that in the sub-critical regime the evolution mechanism produces many clusters which are relatively large, though sublinear, in size [11,12,15]. Most recently, the notions of "strongly" versus "weakly" discontinuous transitions have been introduced [16], with the model studied in [6] showing weakly discontinuous characteristics, while an idealized deterministic "most explosive" model [15,17] is strongly discontinuous. Here we analyze and extend a related model by Bohman, Frieze and Wormald (BFW) [1], which predates the more recent work, and show that surprisingly [18], multiple stable giant components can coexist and that the percolation transition is strongly discontinuous.The "most explosive" deterministic process [15][16][17] begins with n isolated nodes, with n set to a power of two for convenience. Edges that connect pairs of isolated nodes are added sequentially, creating components of size k = 2, until no isolated nodes remain. The cutoff k is then doubled and edges leading to components of size k = 4 are added sequentially, until all components have size k = 4. k is then doubled yet again and the process iterated. By the end of the phase k = n/2, only two components remain, each with size n/2. The addition of the next edge connects those two components during which the size of * Electronic address: chwei@ucdavis.edu † Electronic address: raissa@cse.ucdavis.edu the largest component jumps in value by n/2. Letting t denote the number of edges added to the graph, we define the critical t c as the single edge who's addition produces the l...

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.…”

Formation mechanism and size features of multiple giant clusters in generic percolation processes

“…It has been recently analyzed by Chen and D'Souza [30], showing a strongly discontinuous percolation with multiple stable giant clusters. The detailed rule of the BFW model is as follows.…”

“…In an earlier paper, percolation with multiple giant clusters was also discussed in a process of aggregation with freezing [31]. According to the analysis of Chen et al [30], the key to formation and coexisting of multiple giant clusters in BFW model is the high probability of sampling internal-cluster links in the supercritical region, and any link which would lead to merging two giant clusters will be rejected. Later, the BFW model was further studied on the lattice by K. J. Schrenk et al [32].…”

“…The so called BFW model [29] was analyzed by Chen and D'Souza [30], which exhibits a strongly discontinuous percolation and two stable giant clusters coexisting after the transition. In an earlier paper, percolation with multiple giant clusters was also discussed in a process of aggregation with freezing [31].…”

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