Cluster aggregation model for discontinuous percolation transition

Details Cluster aggregation model for discontinuous percolation transition Cluster aggregation model for discontinuous percolation transition

2009-11-20

arxiv.org/abs/0911.4001v2

Physical Review E 81, 030103(R) (2010)

Physics

Condensed Matter

Statistical Mechanics

5 pages, 5 figures

Scientific paper

The evolution of the Erd\H{o}s-R\'enyi (ER) network by adding edges can be viewed as a cluster aggregation process. Such ER processes can be described by a rate equation for the evolution of the cluster-size distribution with the connection kernel $K_{ij}\sim ij$, where $ij$ is the product of the sizes of two merging clusters. Here, we study more general cases in which $K_{ij}$ is sub-linear as $K_{ij}\sim (ij)^{\omega}$ with $0 \le \omega < 1/2$; we find that the percolation transition (PT) is discontinuous. Moreover, PT is also discontinuous when the ER dynamics evolves from proper initial conditions. The rate equation approach for such discontinuous PTs enables us to uncover the mechanism underlying the explosive PT under the Achlioptas process.

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