Modelo de Kuramoto com campos aleatórios em redes complexas

Mestrado em FísicaNeste trabalho é estudado o modelo de Kuramoto num grafo completo, em redes scale-free com uma distribuição de ligações P(q) ~ q-Y e na presença de campos aleatórios com magnitude constante e gaussiana. Para tal, foi considerado o método Ott-Antonsen e uma aproximação "annealed network". Num grafo completo, na presença de campos aleatórios gaussianos, e em redes scale-free com 2 5, encontraram-se transições de fase contínua (h √2). Para uma rede SF com y = 3, foi observada uma transição de fase de ordem infinita. Os resultados do modelo de Kuramoto num grafo completo e na presença de campos aleatórios com magnitude constante foram comparados aos de simulações, tendo-se verificado uma boa concordância. Verifica-se que, independentemente da topologia de rede, a constante de acoplamento crítico aumenta com a magnitude do campo considerado. Na topologia de rede scale-free, concluiu-se que o valor do acoplamento crítico diminui à medida que valor de y diminui e que o grau de sincronização aumenta com o aumento do número médio das ligações na rede. A presença de campos aleatórios com magnitude gaussiana num grafo completo e numa rede scale-free com y > 2 não destrói a transição de fase contínua e não altera o comportamento crítico do modelo de Kuramoto.In the present work, a random field Kuramoto model is studied in complete graphs and scale-free networks with the degree distribution P(q) ~ q-Y, taking into account constant random fields with constant magnitude as well as gaussian distributed. For this purpose, the Ott-Antonsen method and the annealed-network approximation are used. A continuous phase transition is found in the case of complete graph and gaussian random fields, and in the case of scale-free networks with 2 5, both first (h > √2) and second (h 2, gaussian random fields do not destroy the continuous phase transition and do not change critical behavior of the Kuramoto model

Statistical mechanics of topological phase transitions in networks

We provide a phenomenological theory for topological transitions in restructuring networks. In this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges. The associated microscopic dynamics satisfies the detailed balance condition and is equivalent to a lattice gas model on the edge-dual graph of a fully connected network. In our studies -- based on an exact enumeration method, Monte-Carlo simulations, and theoretical considerations -- we find a rich variety of topological phase transitions when the temperature is varied. These transitions signal singular changes in the essential features of the global structure of the network. Depending on the energy function chosen, the observed transitions can be best monitored using the order parameters Phi_s=s_{max}/M, i.e., the size of the largest connected component divided by the number of edges, or Phi_k=k_{max}/M, the largest degree in the network divided by the number of edges. If, for example the energy is chosen to be E=-s_{max}, the observed transition is analogous to the percolation phase transition of random graphs. For this choice of the energy, the phase-diagram in the [,T] plane is constructed. Single vertex energies of the form E=sum_i f(k_i), where k_i is the degree of vertex i, are also studied. Depending on the form of f(k_i), first order and continuous phase transitions can be observed. In case of f(k_i)=-(k_i+c)ln(k_i), the transition is continuous, and at the critical temperature scale-free graphs can be recovered.Comment: 12 pages, 12 figures, minor changes, added a new refernce, to appear in PR

척도 없는 네트워크에서의 에쉬킨-텔러 모델

학위논문 (석사)-- 서울대학교 대학원 : 물리·천문학부(물리학전공), 2014. 2. 강병남.The investigation of various spin models on complex networks has played a crucial role in the comprehension of collective behavior in natural phenomena. In particular, the Ising and the Potts model on networks have received attention in the last decade for such purposes. Recently, multiplex networks have been actively studied in detail, because many real-world networks are networks of networks. Despite these facts, only a few spin models that incorporate interactions between nodes on inter- and intra-networks have been studied yet. In this paper, we study the Ashkin-Teller (AT) model on scale-free random networks. In the AT model, spins on each site are of two types, and two spins of each type at the nearest neighbors interact with coupling strength $J_2$, and four spins of both types at the nearest neighbors interact with coupling strength $J_4$. In other words, $J_2$ and $J_4$ correspond to the interaction constants of the spins on the intra- and th inter-network, respectively. As was seen in the mean-field approximation, various phases emerge depending on the ratio $x=J_4/J_2$. Some examples of such phases are the paramagnetic phase, ferromagnetic phase, anti-ferromagnetic phase, the Baxter phase, and the sigma phase. We obtain the phase diagram on scale-free networks. While the phase transition between paramagnetic phase and the Baxter phase is discontinuous in the standard mean-field solution, it can be continuous depending on the degree exponent on scale-free network. In spirit of the Landau theory, we focus on this tricritical point that divides the phase space into regimes of the first- and second-order phase transition. Then we obtain the critical degree exponent as a function of $x$ using the analytical approach. For positive $x$, we can thus determine the type of order of the phase transition on scale-free networks in terms of $x$ and the degree exponent. In addition, we perform Monte Carlo simulation using the Metropolis algorithm to sketch the schematic phase diagram for $x<0$. Besides the diagram, a variety of thermodynamic quantities, such as magnetizations, heat capacity, susceptibility, the Binder cumulant, can come along for the ride. In this regime, anomalous behavior due to frustration can be observed. Finally, we examine the analytic and simulation results, and discuss the implications of the difference in the mean-field level between the phase diagrams of scale-free networks and homogeneous space.Contents Abstract i Contents iii List of Figures v Chapter 1 Introduction 1 Chapter 2 The Ashkin-Teller Model 4 Chapter 3 Mean-Field Approximation 9 3.1 Partition function and mean field hamiltonian 9 3.2 Mean field free energy 11 Chapter 4 The AT Model on Uncorrelated Scale-Free Networks 14 4.1 Mean field free energy in the infinite network limit 15 4.2 Equations of state 16 4.3 Example 1 : the Ising model 19 4.4 Example 2 : the 4-state Potts model 19 4.5 Critical degree exponent of the AT model 21 4.6 Brief simulation results 25 Chapter 5 Conclusion 29Maste