  Consider a multiplex network formed by two layers indicating social
interactions: the first layer is a friendship network and the second layer is a
network of business relations. In this duplex network each pair of individuals
can be connected in different ways: they can be connected by a friendship but
not connected by a business relation, they can be connected by a business
relation without being friends, or they can be simultaneously friends and in a
business relation. In the latter case we say that the links in different layers
overlap. These three types of connections are called multilinks and the
multidegree indicates the sum of multilinks of a given type that are incident
to a given node. Previous models of opinion formation on multilayer networks
have mostly neglected the effect of link overlap. Here we show that link
overlap can have important effects in opinion formation. Indeed, emerging
pattern of opinion formation can be significantly influenced by the statistical
properties of multilinks, and in particular by the multidegree distribution. To
quantitatively address this problem, we study a simple spin model, called the
Ashkin-Teller model including 2-body and 4-body interactions between nodes in
different layers. Here we fully investigate the rich phase diagram of this
model which includes a large variety of phase transitions (PTs). Indeed the
phase digram or the model displays continuous, discontinuous, successive
discontinuous, and hybrid PTs.
