Ising model on the Apollonian network with node-dependent interactions

Modified diffusive epidemic process on Apollonian networks

We present an analysis of an epidemic spreading process on an Apollonian network that can describe an epidemic spreading in a non-sedentary population. We studied the modified diffusive epidemic process using the Monte Carlo method by computational analysis. Our model may be helpful for modeling systems closer to reality consisting of two classes of individuals: susceptible (A) and infected (B). The individuals can diffuse in a network according to constant diffusion rates [Formula: see text] and [Formula: see text], for the classes A and B, respectively, and obeying three diffusive regimes, i.e., [Formula: see text], [Formula: see text], and [Formula: see text]. Into the same site i, the reaction occurs according to the dynamical rule based on Gillespie's algorithm. Finite-size scaling analysis has shown that our model exhibits continuous phase transition to an absorbing state with a set of critical exponents given by [Formula: see text], [Formula: see text], and [Formula: see text] familiar to every investigated regime. In summary, the continuous phase transition, characterized by this set of critical exponents, does not have the same exponents of the mean-field universality class in both regular lattices and complex networks.

Binary Apollonian networks

There is a well-known relationship between the binary Pascal's triangle and the Sierpinski triangle, in which the latter is obtained from the former by successive modulo 2 additions starting from a corner. Inspired by that, we define a binary Apollonian network and obtain two structures featuring a kind of dendritic growth. They are found to inherit the small-world and scale-free properties from the original network but display no clustering. Other key network properties are explored as well. Our results reveal that the structure contained in the Apollonian network may be employed to model an even wider class of real-world systems.

Ferromagnetic model on the Apollonian packing

This work investigates the influence of geometrical features of the Apollonian packing (AP) on the behavior of magnetic models. The proposed model differs from previous investigations on the Apollonian network (AN), where the magnetic coupling constants depend only on the properties of the network structure defined by the packing, but not on quantitative aspects of its geometry. In opposition to the exact scale invariance observed in the AN, the circle's sizes in the AP are scaled by different factors when one goes from one generation to the next, requiring a different approach for the evaluation of the model's properties. If the nearest-neighbors coupling constants are defined by J_{i,j}∼1/(r_{i}+r_{j})^{α}, where r_{i} indicates the radius of the circle i containing the node i, the results for the correlation length ξ indicate that the model's behavior depend on α. In the thermodynamic limit, the uniform model (α=0) is characterized by ξ→∞ for all T>0. Our results indicate that, on increasing α, the system changes to an uncorrelated pattern, with finite ξ at all T>0, at a value α_{c}≃0.743. For any fixed value of α, no finite temperature singularity in the specific heat is observed, indicating that changes in the magnetic ordering occur only when α is changed. This is corroborated by the results for the magnetization and magnetic susceptibility.

Complex networks from space-filling bearings

Two-dimensional space-filling bearings are dense packings of disks that can rotate without slip. We consider the entire first family of bearings for loops of four disks and propose a hierarchical construction of their contact network. We provide analytic expressions for the clustering coefficient and degree distribution, revealing bipartite scale-free behavior with a tunable degree exponent depending on the bearing parameters. We also analyze their average shortest path and percolation properties.

Annealed Ising model with site dilution on self-similar structures

We consider an Ising model on the triangular Apollonian network (AN), with a thermalized distribution of vacant sites. The statistical problem is formulated in a grand canonical ensemble, in terms of the temperature T and a chemical potential μ associated with the concentration of active magnetic sites. We use a well-known transfer-matrix method, with a number of adaptations, to write recursion relations between successive generations of this hierarchical structure. We also investigate the analogous model on the diamond hierarchical lattice (DHL). From the numerical analysis of the recursion relations, we obtain various thermodynamic quantities. In the μ→∞ limit, we reproduce the results for the uniform models: in the AN, the system is magnetically ordered at all temperatures, while in the DHL there is a ferromagnetic-paramagnetic transition at a finite value of T. Magnetic ordering, however, is shown to disappear for sufficiently large negative values of the chemical potential.

Ising models on the regularized Apollonian network

We investigate the critical properties of Ising models on a regularized Apollonian network (RAN), here defined as a kind of Apollonian network in which the connectivity asymmetry associated with its corners is removed. Different choices for the coupling constants between nearest neighbors are considered and two different order parameters are used to detect the critical behavior. While ordinary ferromagnetic and antiferromagnetic models on a RAN do not undergo a phase transition, some antiferrimagnetic models show an interesting infinite-order transition. All results are obtained by an exact analytical approach based on iterative partial tracing of the Boltzmann factor as an intermediate step for the calculation of the partition function and the order parameters.

Conductivity of Coniglio-Klein clusters

We performed numerical simulations of the q-state Potts model to compute the reduced conductivity exponent t/ν for the critical Coniglio-Klein clusters in two dimensions, for values of q in the range [1,4]. At criticality, at least for q<4, the conductivity scales as C(L) ~ L(-t/ν), where t and ν are, respectively, the conductivity and correlation length exponents. For q=1, 2, 3, and 4, we followed two independent procedures to estimate t/ν. First, we computed directly the conductivity at criticality and obtained t/ν from the size dependence. Second, using the relation between conductivity and transport properties, we obtained t/ν from the diffusion of a random walk on the backbone of the cluster. From both methods, we estimated t/ν to be 0.986 ± 0.012, 0.877 ± 0.014, 0.785 ± 0.015, and 0.658 ± 0.030, for q=1, 2, 3, and 4, respectively. We also evaluated t/ν for noninteger values of q and propose the conjecture 40 gt/ν = 72 + 20 g - 3g(2) for the dependence of the reduced conductivity exponent on q, in the range 0 ≤ q ≤ 4, where g is the Coulomb gas coupling.

Nonequilibrium model on Apollonian networks

We investigate the majority-vote model with two states (-1,+1) and a noise parameter q on Apollonian networks. The main result found here is the presence of the phase transition as a function of the noise parameter q. Previous results on the Ising model in Apollonian networks have reported no presence of a phase transition. We also studied the effect of redirecting a fraction p of the links of the network. By means of Monte Carlo simulations, we obtained the exponent ratio γ/ν, β/ν, and 1/ν for several values of rewiring probability p. The critical noise q{c} and U were also calculated. Therefore, the results presented here demonstrate that the majority-vote model belongs to a different universality class than equilibrium Ising model on Apollonian network.

Fixed-point properties of the Ising ferromagnet on the Hanoi networks

The Ising model with ferromagnetic couplings on the Hanoi networks is analyzed with an exact renormalization group. In particular, the fixed points are determined and the renormalization- flow for certain initial conditions is analyzed. Hanoi networks combine a one-dimensional lattice structure with a hierarchy of long-range bonds to create a mix of geometric and small-world properties. Generically, those small-world bonds result in nonuniversal behavior, i.e., fixed points and scaling exponents that depend on temperature and the initial choice of coupling strengths. It is shown that a diversity of different behaviors can be observed with seemingly small changes in the structure of the networks. Defining interpolating families of such networks, we find tunable transitions between regimes with power-law and certain essential singularities in the critical scaling of the correlation length. These are similar to the so-called inverted Berezinskii-Kosterlitz-Thouless transition previously observed only in scale-free or dense networks.

q-state Potts model on the Apollonian network

The q-state Potts model is studied on the Apollonian network with Monte Carlo simulations and the transfer matrix method. The spontaneous magnetization, correlation length, entropy, and specific heat are analyzed as a function of temperature for different number of states, q. Different scaling functions in temperature and q are proposed. A quantitative agreement is found between results from both methods. No critical behavior is observed in the thermodynamic limit for any number of states.

Tactical voting in plurality elections

How often will elections end in landslides? What is the probability for a head-to-head race? Analyzing ballot results from several large countries rather anomalous and yet unexplained distributions have been observed. We identify tactical voting as the driving ingredient for the anomalies and introduce a model to study its effect on plurality elections, characterized by the relative strength of the feedback from polls and the pairwise interaction between individuals in the society. With this model it becomes possible to explain the polarization of votes between two candidates, understand the small margin of victories frequently observed for different elections, and analyze the polls' impact in American, Canadian, and Brazilian ballots. Moreover, the model reproduces, quantitatively, the distribution of votes obtained in the Brazilian mayor elections with two, three, and four candidates.

Bose-Einstein condensation in the Apollonian complex network

We demonstrate that a topology-induced Bose-Einstein condensation (BEC) takes place in a complex network. As a model topology, we consider the deterministic Apollonian network which exhibits scale-free, small-world, and hierarchical properties. Within a tight-binding approach for noninteracting bosons, we report that the BEC transition temperature and the gap between the ground and first excited states follow the same finite-size scaling law. An anomalous density dependence of the transition temperature is reported and linked to the structure of gaps and degeneracies of the energy spectrum. The specific heat is shown to be discontinuous at the transition, with low-temperature modulations as a consequence of the fragmented density of states.