# Structure of percolating clusters in random clustered networks

## TL;DR

This paper investigates the structure of the giant percolating cluster in random clustered networks, revealing how clustering and assortativity vary at the percolation threshold through analytical and numerical methods.

## Contribution

It introduces a detailed analytical framework for understanding the clustering and assortativity of percolating clusters in random clustered networks.

## Key findings

- Percolating clusters remain clustered at the percolation threshold in highly clustered networks.
- Assortativity of the percolating cluster depends on the network's degree and triangle distributions.
- Disassortativity is observed in Poisson-distributed networks, while uniform small-degree networks are assortative.

## Abstract

We examine the structure of the percolating cluster (PC) formed by site percolation on a random clustered network (RCN) model. Using the generating functions, we formulate the clustering coefficient and assortative coefficient of the PC. We analytically and numerically show that the PC in the highly clustered networks is clustered even at the percolation threshold. The assortativity of the PC depends on the details of the RCN. The PC at the percolation threshold is disassortative when the numbers of edges and triangles of each node are assigned by Poisson distributions, but assortative when each node in an RCN has the same small number of edges, most of which form triangles. This result seemingly contradicts the disassortativity of fractal networks, although the renormalization scheme unveils the disassortative nature of a fractal PC.