# Interevent time distribution, burst, and hybrid percolation transition

## TL;DR

This paper develops a theoretical framework for hybrid percolation transitions, revealing that inter-event times between cluster crossing events follow power-law distributions and signal imminent criticality.

## Contribution

It introduces a restricted percolation model that captures the dynamics of cluster merging and crossing events, advancing understanding of hybrid percolation transitions.

## Key findings

- Inter-event times follow power-law distributions.
- Crossing event bursts signal upcoming transitions.
- The model explains critical dynamics of HPT.

## Abstract

Critical phenomena of a second-order percolation transition are known to be independent of cluster merging or pruning process. However, those of a hybrid percolation transition (HPT), mixed properties of both first-order and second-order transitions, depend on the processes. The HPT induced by cluster merging is more intrigue and little understood than the other. Here, we construct a theoretical framework using the so-called restricted percolation model. In this model, clusters are ranked by size and partitioned into small- and large-cluster sets. As the cluster rankings are updated by cluster coalescence, clusters may move back and forth across the set boundary. The inter-event time (IET) between two consecutive crossing times have two distributions with power-law decays, which in turn characterize the criticality of the HPT. A burst of such crossing events occurs and signals the upcoming transition. We discuss a related phenomenon to this critical dynamics.