Hybrid Percolation Transition in Cluster Merging Processes: Continuously Varying Exponents

TL;DR

This paper introduces a network growth model with a specific connection strategy that results in a hybrid percolation transition, characterized by both abrupt and gradual features, with a continuously varying critical exponent.

Contribution

It demonstrates a new hybrid percolation transition in cluster merging processes with a connection strategy involving a subset of nodes, revealing a condition for such transitions with varying exponents.

Findings

Hybrid transition exhibits both first- and second-order properties.

Cluster size distribution follows a power-law with a variable exponent.

The exponent varies between 2 and 2.5 depending on the subset size.

Abstract

Consider growing a network, in which every new connection is made between two disconnected nodes. At least one node is chosen randomly from a subset consisting of $g$ fraction of the entire population in the smallest clusters. Here we show that this simple strategy for improving connection exhibits a phase transition barely studied before, namely a hybrid percolation transition exhibiting the properties of both first-order and second-order phase transitions. The cluster size distribution of finite clusters at a transition point exhibits power-law behavior with a continuously varying exponent $\tau$ in the range $2 < \tau(g) \le 2.5$. This pattern reveals a necessary condition for a hybrid transition in cluster aggregation processes, which is comparable to the power-law behavior of the avalanche size distribution arising in models with link-deleting processes in interdependent networks.