Vital layer nodes of multiplex networks for immunization and attack

When dealing with the optimal prevention of epidemics or destruction of network structures, one important question that can be asked is the location of vital nodes which need to be immunized or removed first. In the last decade, the vital nodes identification has attracted increasing attentions. However, the majority of the existing achievements are limited to single networks, how to identify the vital nodes of multiplex networks need further exploration. The nodes of multiplex networks can be divided into two categories: multiplex node (MN) and layer node (LN). In this paper, we focus on identifying the vital LNs of multiplex networks for immunization or attack. We extend several indexes or algorithms from single networks to multiplex networks, including high degree, high betweeness and their variations based on adaptive strategies, and the collective influence, explosive immunization and simulated annealing, to identify the vital LNs. By performing them on different kinds of multiplex networks, we find the explosive immunization is always the best for the identification of vital LNs. Particularly, the performances of the proposed indexes and algorithms could be improved considerably when the greedy reinserting strategy is considered except the explosive immunization, which however still performs the best. Our work offers a deeper understanding for the vital nodes identification in multiplex network and provides novel insights for further studies of the immunization and attack on multiplex networks.