  Real networks are vulnerable to random failures and malicious attacks.
However, when a node is harmed or damaged, it may remain partially functional,
which helps to maintain the overall network structure and functionality. In
this paper, we study the network structure for a fractional percolation process
[Shang, Phys. Rev. E 89, 012813 (2014)], in which the state of a node can be
either fully functional (FF), partially functional (PF), or dysfunctional (D).
We develop new equations to calculate the relative size of the percolating
cluster of FF and PF nodes, that are in agreement with our stochastic
simulations. In addition, we find a regime in which the percolating cluster can
be described as a coarse-grained bipartite network, namely, as a set of finite
groups of FF nodes connected by PF nodes. Moreover, these groups behave as a
set of "supernodes" with a power-law degree distribution. Finally, we show how
this emergent structure explains the values of several critical exponents
around the percolation threshold.
