Explosive Percolation with Multiple Giant Components

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 $\alpha =1/2$. We show that multiple giant components appear simultaneously in a strongly discontinuous percolation transition and remain distinct. Furthermore, tuning the value of $\alpha$ determines the number of such components with smaller $\alpha$ 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.

Figure 1

The BFW model. (a) $C_1$ and $C_2$ versus edge density, $t/n$, for $n={10}^6$, showing the emergence of two stable giant components. Inset: $\Delta (1/2,0.5)/n$, $\Delta (1/2,0.35)/n$ for $C_1$ and $C_2$ versus $n$. (b) The lower and upper boundaries of $\Delta /n$ for both $C_1$ and $C_2$, where data points are averages over 200 to 2000 realizations and dashed lines are best fits.

Figure 2

(a) $\Delta C_{\max }$, the biggest single-edge increase in $C_1$ is independent of system size $n$ for BFW and for the restricted BFW process, indicating these transitions are strongly discontinuous. (b) Evolution of the distribution of $n(s)$, the fraction of components of size $s$, for the BFW model.

Figure 3

(a) Number of stable giant components of size greater than $0.1n$ versus $\alpha$ for $n={10}^6$. (b) When $\alpha =0.3$, three giant components emerge simultaneously. The inset shows convergence of the upper and lower boundaries of $\Delta /n$ for $C_1$, $C_2$, $C_3$, analogous to Fig. 1b.

Figure 4

The results of varying $\alpha$ on (a) the unrestricted BFW process in the regime where only one giant component emerges, and (b) the restricted BFW process. Both (a) and (b) show increasing delay and larger $\Delta C_{\max }$ for smaller $\alpha$.