Origin of the mixed-order transition in multiplex networks: the   Ashkin-Teller model
  Recently, diverse phase transition (PT) types have been obtained in multiplex networks, such as discontinuous, continuous, and mixed-order PTs. However, they emerge from individual systems, and there is no theoretical understanding of such PTs in a single framework. Here, we study a spin model called the Ashkin-Teller (AT) model in a mono-layer scale-free network; this can be regarded as a model of two species of Ising spin placed on each layer of a double-layer network. The four-spin interaction in the AT model represents the inter-layer interaction in the multiplex network. Diverse PTs emerge depending on the inter-layer coupling strength and network structure. Especially, we find that mixed-order PTs occur at the critical end points. The origin of such behavior is explained in the framework of Landau-Ginzburg theory. 
S. Jang, J.S. Lee, S. Hwang, and B. Kahng
cond-mat.stat-mech