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The $k$-core percolation is a fundamental structural transition in complexnetworks. Through the analysis of the size jump behaviors of $k$-core in theevolution process of networks, we confirm that $k$-core percolation iscontinuous phase transition when $k=1,2$ while it is a hybridfirst-order-second-order phase transition when $k\ge 3$. $2$-core percolationbelongs to different universality class from that of $1$-core (giant component)percolation. The discontinuity of $k$-core percolation with $k\ge 3$ can beconcluded from largest size jump of $k$-core which will not disappear in thethermodynamic limit while its continuous characteristic is reflected by secondlargest size jump which converges to zero in power law as $N\to \infty$.Furthermore, along with the previously known exponent $\beta=0.5$, we obtain aset of exponents which are independent of $k$ when $k\ge 3$ and also differentfrom those critical exponents of $1$-core and $2$-core percolation.

Revealing the phase transition behaviors of k-core percolation in random networks