  Percolation in complex networks is viewed as both: a process that mimics
network degradation and a tool that reveals peculiarities of the underlying
network structure. During the course of percolation, networks undergo
non-trivial transformations that include a phase transition in the
connectivity, and in some special cases, multiple phase transitions. Here we
establish a generic analytic theory that describes how structure and sizes of
all connected components in the network are affected by simple and
colour-dependant bond percolations. This theory predicts all locations where
the phase transitions take place, existence of wide critical windows that do
not vanish in the thermodynamic limit, and a peculiar phenomenon of colour
switching that occurs in small connected components. These results may be used
to design percolation-like processes with desired properties, optimise network
response to percolation, and detect subtle signals that provide an early
warning of a network collapse.
