Spectral Anlysis on Explosive Percolation

We study the spectral properties of the process of explosive percolation. In particular, we explore how the maximum eigenvalue of the adjacency matrix of a network which governs the spreading efficiency evolves as the density of connection increases. Interestingly, for networks with connectivity that grow in an explosive way, information spreading and mass transport are found to be carried out inefficiently. In the conventional explosive percolation models that we studied, the sudden emergences of large-scale connectivity are found to come with relatively lowered efficiency of spreading. Nevertheless, the spreading efficiency of the explosive model can be increased by introducing heterogeneous structures into the networks.