# Recent Progress on Hybrid Percolation Transitions

## TL;DR

This paper reviews hybrid percolation transitions, where networks show both sudden jumps and critical behavior at the same threshold, and categorizes them into two types with shared scaling patterns.

## Contribution

The paper classifies hybrid percolation transitions into two distinct categories and outlines their universal scaling relations.

## Key findings

- Hybrid percolation transitions can be categorized into global suppression-induced and cascading failure-induced types.
- Critical behavior appears in cluster size distributions for global suppression-induced HPTs.
- Avalanche size distributions show critical behavior for cascading failure-induced HPTs.

## Abstract

Percolation describes the formation of a giant cluster once the average degree of a network exceeds a critical value. A hybrid percolation transition (HPT) denotes a phenomenon in which a discontinuous jump of the order parameter and the critical behavior, a basic pattern of a continuous transition, appear together at the same threshold. Such HPTs have been reported in many different systems. In this review, we present several representative examples of HPTs and classify them into two categories: global suppression-induced HPTs and cascading failure-induced HPTs. In the former class, critical behavior manifests itself in the distribution of cluster sizes, whereas in the latter it emerges in the distribution of avalanche sizes. We further outline the universal scaling relations shared by both types.