Ashkin-Teller model and diverse opinion phase transitions on multiplex networks

Q-neighbor Ising model on multiplex networks with partial overlap of nodes

The q-neighbor Ising model for the opinion formation on multiplex networks with two layers in the form of random graphs (duplex networks), the partial overlap of nodes, and LOCAL&AND spin update rule was investigated by means of the pair approximation and approximate master equations as well as Monte Carlo simulations. Both analytic and numerical results show that for different fixed sizes of the q-neighborhood and finite mean degrees of nodes within the layers the model exhibits qualitatively similar critical behavior as the analogous model on multiplex networks with layers in the form of complete graphs. However, as the mean degree of nodes is decreased the discontinuous ferromagnetic transition, the tricritical point separating it from the continuous transition, and the possible coexistence of the paramagnetic and ferromagnetic phases at zero temperature occur for smaller relative sizes of the overlap. Predictions of the simple homogeneous pair approximation concerning the critical behavior of the model under study show good qualitative agreement with numerical results; predictions based on the approximate master equations are usually quantitatively more accurate but yet not exact. Two versions of the heterogeneous pair approximation are also derived for the model under study, which, surprisingly, yield predictions only marginally different or even identical to those of the simple homogeneous pair approximation. In general, predictions of all approximations show better agreement with the results of Monte Carlo simulations in the case of continuous than discontinuous ferromagnetic transition.

Collapse and the interplay between essentiality and impact in socioecological systems

We study conditions leading to collapse on a nonequilibrium toy model introduced here for the interaction dynamics between a social and an ecological system based on the concept of essentiality of services and goods. One key difference from previous models is the separation between purely environmental collapse and that caused by an imbalance in the population consumption of essential goods. By studying different regimes defined by phenomenological parameters, we identify sustainable and unsustainable phases as well as the likelihood of collapse. The behavior of the stochastic version of the model is analyzed with a combination of analytical and computational techniques introduced here and shown to be consistent with key features of such processes in real life.

Link overlap influences opinion dynamics on multiplex networks of Ashkin-Teller spins

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 opinion models on multilayer networks have mostly neglected the effect of link overlap. Here we show that link overlap can have important effects in the formation of a majority opinion. Indeed, the formation of a majority opinion 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 two-body and four-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. Indeed, the phase diagram or the model displays continuous, discontinuous, and hybrid phase transitions, and successive jumps of the order parameters within the Baxter phase.

Pair approximation for the q-voter model with independence on multiplex networks

The q-voter model with independence is investigated on multiplex networks with full overlap of nodes in the layers. The layers are various complex networks corresponding to different levels of social influence. Detailed studies are performed for the model on multiplex networks with two layers with identical degree distributions, obeying the LOCAL&AND and GLOBAL&AND spin update rules differing by the way in which the q-lobbies of neighbors within different layers exert their joint influence on the opinion of a given agent. Homogeneous pair approximation is derived for a general case of a two-state spin model on a multiplex network and its predictions are compared with results of mean-field approximation and Monte Carlo simulations of the above-mentioned q-voter model with independence for a broad range of parameters. As the parameter controlling the level of agents' independence is changed ferromagnetic phase transition occurs which can be first- or second-order, depending on the size of the lobby q. Details of this transition, e.g., position of the critical points, critical exponents and the width of the possible hysteresis loop, depend on the topology and other features of the layers, in particular on the mean degree of nodes in the layers which is directly predicted by the homogeneous pair approximation. If the mean degree of nodes is substantially larger than the size of the q-lobby good agreement is obtained between numerical results and theoretical predictions based on the homogeneous pair approximation concerning the order and details of the ferromagnetic transition. In the case of the model on multiplex networks with layers in the form of homogeneous Erdős-Rényi and random regular graphs as well as weakly heterogeneous scale-free networks this agreement is quantitative, while in the case of layers in the form of strongly heterogeneous scale-free networks it is only qualitative. If the mean degree of nodes is small and comparable with q predictions of the homogeneous pair approximation are in general even qualitatively wrong.

Effective-potential approach to hybrid synchronization transitions

The Kuramoto model exhibits different types of synchronization transitions depending on the type of natural frequency distribution. To obtain these results, the Kuramoto self-consistency equation (SCE) approach has been used successfully. However, this approach affords only limited understanding of more detailed properties such as the stability. Here we extend the SCE approach by introducing an effective potential, that is, an integral version of the SCE. We examine the landscape of this effective potential for second-order, first-order, and hybrid synchronization transitions in the thermodynamic limit. In particular, for the hybrid transition, we find that the minimum of effective potential displays a plateau across the region in which the order parameter jumps. This result suggests that the effective potential can be used to determine a type of synchronization transition.

Tricriticality in the q-neighbor Ising model on a partially duplex clique

We analyze a modified kinetic Ising model, a so-called q-neighbor Ising model, with Metropolis dynamics [Phys. Rev. E 92, 052105 (2015)PLEEE81539-375510.1103/PhysRevE.92.052105] on a duplex clique and a partially duplex clique. In the q-neighbor Ising model each spin interacts only with q spins randomly chosen from its whole neighborhood. In the case of a duplex clique the change of a spin is allowed only if both levels simultaneously induce this change. Due to the mean-field-like nature of the model we are able to derive the analytic form of transition probabilities and solve the corresponding master equation. The existence of the second level changes dramatically the character of the phase transition. In the case of the monoplex clique, the q-neighbor Ising model exhibits a continuous phase transition for q=3, discontinuous phase transition for q≥4, and for q=1 and q=2 the phase transition is not observed. On the other hand, in the case of the duplex clique continuous phase transitions are observed for all values of q, even for q=1 and q=2. Subsequently we introduce a partially duplex clique, parametrized by r∈[0,1], which allows us to tune the network from monoplex (r=0) to duplex (r=1). Such a generalized topology, in which a fraction r of all nodes appear on both levels, allows us to obtain the critical value of r=r^{*}(q) at which a tricriticality (switch from continuous to discontinuous phase transition) appears.

Mixed-order phase transition in a minimal, diffusion-based spin model

In this paper we exactly solve, within the grand canonical ensemble, a minimal spin model with the hybrid phase transition. We call the model diffusion based because its Hamiltonian can be recovered from a simple dynamic procedure, which can be seen as an equilibrium statistical mechanics representation of a biased random walk. We outline the derivation of the phase diagram of the model, in which the triple point has the hallmarks of the hybrid transition: discontinuity in the average magnetization and algebraically diverging susceptibilities. At this point, two second-order transition curves meet in equilibrium with the first-order curve, resulting in a prototypical mixed-order behavior.

Layer-layer competition in multiplex complex networks

The coexistence of multiple types of interactions within social, technological and biological networks has moved the focus of the physics of complex systems towards a multiplex description of the interactions between their constituents. This novel approach has unveiled that the multiplex nature of complex systems has strong influence in the emergence of collective states and their critical properties. Here we address an important issue that is intrinsic to the coexistence of multiple means of interactions within a network: their competition. To this aim, we study a two-layer multiplex in which the activity of users can be localized in each of the layers or shared between them, favouring that neighbouring nodes within a layer focus their activity on the same layer. This framework mimics the coexistence and competition of multiple communication channels, in a way that the prevalence of a particular communication platform emerges as a result of the localization of user activity in one single interaction layer. Our results indicate that there is a transition from localization (use of a preferred layer) to delocalization (combined usage of both layers) and that the prevalence of a particular layer (in the localized state) depends on the structural properties.