# Percolation on a maximally disassortative network

## TL;DR

This paper introduces a maximally disassortative network model to analyze how strong negative degree correlations influence percolation transitions, revealing differences in critical behavior based on degree distribution exponent.

## Contribution

The study provides an analytical framework for understanding percolation in maximally disassortative networks and highlights the impact of degree correlations on critical exponents.

## Key findings

- Critical exponent $eta$ matches uncorrelated networks for $ ext{γ}>3$.
- Strong degree correlations alter percolation behavior for $2< ext{γ}<3$.
- Analytical results are confirmed by finite-size scaling.

## Abstract

We propose a maximally disassortative (MD) network model which realizes a maximally negative degree-degree correlation, and study its percolation transition to discuss the effect of a strong degree-degree correlation on the percolation critical behaviors. Using the generating function method for bipartite networks, we analytically derive the percolation threshold and the order parameter critical exponent, $\beta$. For the MD scale-free networks, whose degree distribution is $P(k) \sim k^{-\gamma}$, we show that the exponent, $\beta$, for the MD networks and corresponding uncorrelated networks are same for $\gamma>3$ but are different for $2<\gamma<3$. A strong degree-degree correlation significantly affects the percolation critical behavior in heavy-tailed scale-free networks. Our analytical results for the critical exponents are numerically confirmed by a finite-size scaling argument.