For a given random graph, a connected component that contains a finite fraction of the entire graph’s vertices is called giant.

A well known result in random graphs is the existence and uniqueness of the giant component in Erdos-Rényi model: if $G\in G(n,p)$ with $p=c/n$ and $c>1$, then there is a unique giant component of size $\zeta$, the smallest positive solution to $$1-\zeta=e^{-c\zeta}$$.

This result has been generalized in many other settings :

- Starting with Molloy and Reed who proved a condition for the existence and uniqueness of a giant component for graph with a given degree sequence, under some technical requirements.
- Under other requirements by Bollobas and Riordan, or Janson and Luczak.
- Aiello et al. proved a similar result for Power-Law model.
- Some years ago, Joos et al. showed a condition for the existence of a giant component for any given degree sequence. The uniqueness is still an open problem (as far as I know).

My question is :

Is there any random graph model for which, under some supercritical regime, we do not have the uniqueness of the giant component, but rather multiple giant components with positive probability?

I would be quite surprised if there is one. But maybe some preferential attachment model, with a fitness parameter as per the Bianconi–Barabási model, or a more general Price's model, might have this property.