  No-exclaves percolation (NExP) is a nonlocal percolation process in which the
components are formed not only by the connected occupied nodes but also by the
agglomeration of empty nodes completely surrounded by the occupied nodes. It
has been studied in low dimensions, displaying such novel phenomena as the
discontinuous transition to complete percolation. However, its characteristics
in complex networks are still unexplored. In this paper, we study the NExP on
random networks by developing mean-field solutions using the generating
function formalism. Our theory allows us to determine the size of the giant
no-exclaves component as well as the percolation threshold, which are in
excellent agreements with Monte Carlo simulations on random networks and some
real-world networks. We show that on random networks NExP exhibits three phases
and two transitions between them: the phases are characterized by the presence
or absence of not only the giant NExP component but also the giant unoccupied
component, which is the giant connected component composed solely of unoccupied
nodes. This work offers theoretical understanding on the anatomy of phase
transitions in the NExP process.
