Two Types of Discontinuous Percolation Transitions in Cluster Merging Processes.

Y S Cho,B Kahng

doi:10.1038/srep11905

Abstract

Percolation is a paradigmatic model in disordered systems and has been applied to various natural phenomena. The percolation transition is known as one of the most robust continuous transitions. However, recent extensive studies have revealed that a few models exhibit a discontinuous percolation transition (DPT) in cluster merging processes. Unlike the case of continuous transitions, understanding the nature of discontinuous phase transitions requires a detailed study of the system at hand, which has not been undertaken yet for DPTs. Here we examine the cluster size distribution immediately before an abrupt increase in the order parameter of DPT models and find that DPTs induced by cluster merging kinetics can be classified into two types. Moreover, the type of DPT can be determined by the key characteristic of whether the cluster kinetic rule is homogeneous with respect to the cluster sizes. We also establish the necessary conditions for each type of DPT, which can be used effectively when the discontinuity of the order parameter is ambiguous, as in the explosive percolation model.

Highlights

The percolation transition (PT)[1], the emergence of a macroscopic-scale cluster at a finite threshold, has played a central role as a model for metal—insulator and sol—gel[2] transitions in physical systems as well as the spread of disease epidemics[3] and opinion formation in complex systems
We show that the two types of discontinuous percolation transition (DPT) have different origins
We propose the necessary conditions for each type of DPT as follows

Summary

Introduction

The percolation transition (PT)[1], the emergence of a macroscopic-scale cluster at a finite threshold, has played a central role as a model for metal—insulator and sol—gel[2] transitions in physical systems as well as the spread of disease epidemics[3] and opinion formation in complex systems. Recent extensive research[15,16,17,18] shows that the explosive percolation transition in a random graph is continuous in the thermodynamic limit. This result has reinforced the robustness of continuous PTs in CM processes. The number of edges added to the system at a certain time step divided by the system size N is defined as the time t, which serves as a control parameter in PTs. As time passes, the fraction of nodes belonging to the largest cluster in the system, denoted as G(t), increases from zero. It would be more interesting to investigate the origin of type-II DPTs because this type of DPT can occur in other models, for example, the k-core percolation model[27,28,29,30], discontinuous synchronization model[31,32], jamming transition model[33], and generalized epidemic process model[34]

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Results

Conclusion