Solution of the explosive percolation quest. II. Infinite-order transition produced by the initial distributions of clusters

Abstract

We describe the effect of power-law initial distributions of clusters on ordinary percolation and its generalizations, specifically, models of explosive percolation processes based on local optimization. These aggregation processes were shown to exhibit continuous phase transitions if the evolution starts from a set of disconnected nodes. Since the critical exponents of the order parameter in explosive percolation transitions turned out to be very small, these transitions were first believed to be discontinuous. In this article we analyze the evolution starting from clusters of nodes whose sizes are distributed according to a power law. We show that these initial distributions change dramatically the position and order of the phase transitions in these problems. We find a particular initial power-law distribution producing a peculiar effect on explosive percolation, namely before the emergence of the percolation cluster, the system is in a "critical phase" with an infinite generalized susceptibility. This critical phase is absent in ordinary percolation models with any power-law initial conditions. The transition from the critical phase is an infinite order phase transition, which resembles the scenario of the Berezinskii-Kosterlitz-Thouless phase transition. We obtain the critical singularity of susceptibility at this peculiar infinite-order transition in explosive percolation. It turns out that the susceptibility in this situation does not obey the Curie-Weiss law.

arXiv comment17 pages, 5 figures

References in corpus (27)

• Epidemic spreading in scale-free networks
• Critical phenomena in complex networks
• Are randomly grown graphs really random?
• "Explosive Percolation" Transition is Actually Continuous
• Self-organized adaptation of a simple neural circuit enables complex robot behaviour
• Impact of Single Links in Competitive Percolation -- How complex networks grow under competition
• Explosive Percolation is Continuous, but with Unusual Finite Size Behavior
• Infinite-Order Percolation and Giant Fluctuations in a Protein Interaction Network
• Explosive percolation in scale-free networks
• Percolation Transitions in Scale-Free Networks under Achlioptas Process
• Construction and Analysis of Random Networks with Explosive Percolation
• Anomalous percolating properties of growing networks
• Explosive percolation: a numerical analysis
• Local cluster aggregation models of explosive percolation
• Scaling behavior of explosive percolation on the square lattice
• Cluster aggregation model for discontinuous percolation transition
• Tricritical point in explosive percolation
• A new route to Explosive Percolation
• Continuity of the Explosive Percolation Transition
• Phase Transition with the Berezinskii--Kosterlitz--Thouless Singularity in the Ising Model on a Growing Network
• Solution of the explosive percolation quest: Scaling functions and critical exponents
• BKT-like transition in the Potts model on an inhomogeneous annealed network
• Transition-type change between an inverted Berezinskii-Kosterlitz-Thouless transition and an abrupt transition in the bond percolation on a random hierarchical small-world network
• Critical exponents of the explosive percolation transition
• Explosive percolation in graphs
• From explosive to infinite-order transitions on a hyperbolic network
• Critical Phase in Complex Networks: a Numerical Study

Cited by in corpus (5)

• Percolation on complex networks: Theory and application
• Hybrid Percolation Transition in Cluster Merging Processes: Continuously Varying Exponents
• Inverting the Achlioptas rule for explosive percolation
• A hybrid percolation transition at a finite transition point in scale-free networks
• Percolation Transitions in Growing Networks Under Achlioptas Processes: Analytic Solutions