  The internal organization of complex networks often has striking consequences
on either their response to external perturbations or on their dynamical
properties. In addition to small-world and scale-free properties, clustering is
the most common topological characteristic observed in many real networked
systems. In this paper, we report an extensive numerical study on the effects
of clustering on the structural properties of complex networks. Strong
clustering in heterogeneous networks induces the emergence of a core-periphery
organization that has a critical effect on the percolation properties of the
networks. We observe a novel double phase transition with an intermediate phase
in which only the core of the network is percolated and a final phase in which
the periphery percolates regardless of the core. This result implies breaking
of the same symmetry at two different values of the control parameter, in stark
contrast to the modern theory of continuous phase transitions. Inspired by this
core-periphery organization, we introduce a simple model that allows us to
analytically prove that such an anomalous phase transition is in fact possible.

Physics and Society;Disordered Systems and Neural Networks;Social and Information Networks