Percolation and synchronization are two phase transitions that have been extensively studied since already long ago. A classic result is that, in the vast majority of cases, these transitions are of the second-order type, i.e. continuous and reversible. Recently, however, explosive phenomena have been reported in complex networks' structure and dynamics, which rather remind first-order (discontinuous and irreversible) transitions. Explosive percolation, which was discovered in 2009, corresponds to an abrupt change in the network's structure, and explosive synchronization (which is concerned, instead, with the abrupt emergence of a collective state in the networks' dynamics) was studied as early as the first models of globally coupled phase oscillators were taken into consideration. The two phenomena have stimulated investigations and debates, attracting attention in many relevant fields. So far, various substantial contributions and progresses (including experimental verifications) have been made, which have provided insights on what structural and dynamical properties are needed for inducing such abrupt transformations, as well as have greatly enhanced our understanding of phase transitions in networked systems. Our intention is to offer here a monographic review on the main-stream literature, with the twofold aim of summarizing the existing results and pointing out possible directions for future research.

We present a new approach to determine quantitatively the phase content of multiphase polymers by applying a spin-diffusion pulse sequence (filter) prior to cross-polarization (CP) in a 13C-CPAS experiment. The technique is exemplified for two rather different polymer systems, a high-density polyethylene (HDPE) and four styrene−isoprene diblock copolymers, all of known phase contents. In principle, the technique should be applicable to any multiphase system in which spin-diffusion between different regions/phases exists and where the magnetization of phases can selectively be filtered out.