Emergence of biconnected clusters in explosive percolation

By introducing a simple competition mechanism for bond insertion in random graphs, explosive percolation exhibits a sharp phase transition with rich critical phenomena. We investigate high-order connectivity in explosive percolation using an event-based ensemble, focusing on biconnected clusters, where any two sites are connected by at least two independent paths. Our numerical analysis confirms that explosive percolation with different intra-cluster bond competition rules shares the same percolation threshold and universality, with biconnected clusters percolating simultaneously with simply connected clusters. However, the volume fractal dimension $d_{f}'$ of biconnected clusters varies depending on the competition rules of intra-cluster bonds. The size distribution of biconnected clusters exhibits double-scaling behavior: large clusters follow the standard Fisher exponent derived from the hyperscaling relation $\tau'=1+1/d_{f}'$, while small clusters display a modified Fisher exponent $\tau_0<\tau'$. These findings provide insights into the intricate nature of connectivity in explosive percolation.