Identifying highly influential nodes in multilayer networks based on global propagation

Based on percolation theory and the independent cascade model, this paper considers the selection of the optimal propagation source when the propagation probability is greater than the percolation threshold. First, based on the percolation characteristics of real networks, this paper presents an iterative algorithm of linear complexity to solve the probability of the propagation source transmitting information to the network's giant component, that is, the global propagation probability. Compared with the previous multiple local simulation algorithm, this algorithm eliminates random errors and significantly reduces the operation time. A sufficient and necessary condition is provided, and it is proved that the final propagation range of the propagation source obeys the bimodal distribution. Based on this sufficient and necessary condition, we extend the efficient iterative algorithm proposed in this article to multi-layer networks and find that for two-layer networks, the final propagation range of the propagation source follows a four-peak distribution. Through iterations and calculations, the probability of each peak and the number of nodes included can be directly obtained, and the propagation expectations of the nodes in the multi-layer network can then be calculated, which can result in a better ranking of the propagation influence of the nodes. In addition, to maximize the influence of multi-propagation sources, this paper also presents a de-overlapping method, which has evident advantages over traditional methods.