  Percolation theory concerns the emergence of connected clusters that
percolate through a networked system. Previous studies ignored the effect that
a node outside the percolating cluster may actively induce its inside
neighbours to exit the percolating cluster. Here we study this inducing effect
on the classical site percolation and K-core percolation, showing that the
inducing effect always causes a discontinuous percolation transition. We
precisely predict the percolation threshold and core size for uncorrelated
random networks with arbitrary degree distributions. For low-dimensional
lattices the percolation threshold fluctuates considerably over realizations,
yet we can still predict the core size once the percolation occurs. The core
sizes of real-world networks can also be well predicted using degree
distribution as the only input. Our work therefore provides a theoretical
framework for quantitatively understanding discontinuous breakdown phenomena in
various complex systems.
