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Register a new userCombined update scheme in the Sznajd model
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We analyze the Sznajd opinion formation model, where a pair of neighboring individuals sharing the same opinion on a square lattice convince its six neighbors to adopt their opinions, when a fraction of the individuals is updated according to the usual random sequential updating rule (asynchronous updating), and the other fraction, the simultaneous updating (synchronous updating). This combined updating scheme provides that the bigger the synchronous frequency becomes, the more difficult the system reaches a consensus. Moreover, in the thermodynamic limit, the system needs only a small fraction of individuals following a different kind of updating rules to present a non-consensus state as a final state.
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We analyze the evolution of Sznajd Model with synchronous updating in several complex networks. Similar to the model on square lattice, we have found a transition between the state with no-consensus and the state with complete consensus in several complex networks. Furthermore, by adjusting the network parameters, we find that a large clustering coefficient favors development of a consensus. In particular, in the limit of large system size with the initial concentration p=0.5 of opinion +1, a consensus seems to be never reached for the Watts-Strogatz small-world network, when we fix the connectivity k and the rewiring probability p_s; nor for the scale-free network, when we fix the minimum node degree m and the triad formation step probability p_t.
We study the finite-temperature superfluid transition in a modified two-dimensional (2D) XY model with power-law distributed scratch-like bond disorder. As its exponent decreases, the disorder grows stronger and the mechanism driving the superfluid transition changes from conventional vortex-pair unbinding to a strong randomness criticality (termed scratched-XY criticality) characterized by a non-universal jump of the superfluid stiffness. The existence of the scratched-XY criticality at finite temperature and its description by an asymptotically exact semi-renormalization group theory, previously developed for the superfluid-insulator transition in one-dimensional disordered quantum systems, is numerically proven by designing a model with minimal finite size effects. Possible experimental implementations are discussed.
We study synchronization in the two-dimensional lattice of coupled phase oscillators with random intrinsic frequencies. When the coupling $K$ is larger than a threshold $K_E$, there is a macroscopic cluster of frequency-synchronized oscillators. We explain why the macroscopic cluster disappears at $K_E$. We view the system in terms of vortices, since cluster boundaries are delineated by the motion of these topological defects. In the entrained phase ($K>K_E$), vortices move in fixed paths around clusters, while in the unentrained phase ($K<K_E$), vortices sometimes wander off. These deviant vortices are responsible for the disappearance of the macroscopic cluster. The regularity of vortex motion is determined by whether clusters behave as single effective oscillators. The unentrained phase is also characterized by time-dependent cluster structure and the presence of chaos. Thus, the entrainment transition is actually an order-chaos transition. We present an analytical argument for the scaling $K_Esim K_L$ for small lattices, where $K_L$ is the threshold for phase-locking. By also deriving the scaling $K_Lsimlog N$, we thus show that $K_Esimlog N$ for small $N$, in agreement with numerics. In addition, we show how to use the linearized model to predict where vortices are generated.
We present a renormalization approach to solve the Sznajd opinion formation model on complex networks. For the case of two opinions, we present an expression of the probability of reaching consensus for a given opinion as a function of the initial fraction of agents with that opinion. The calculations reproduce the sharp transition of the model on a fixed network, as well as the recently observed smooth function for the model when simulated on a growing complex networks.
The Potts model is one of the most popular spin models of statistical physics. The prevailing majority of work done so far corresponds to the lattice version of the model. However, many natural or man-made systems are much better described by the topology of a network. We consider the q-state Potts model on an uncorrelated scale-free network for which the node-degree distribution manifests a power-law decay governed by the exponent lambda. We work within the mean-field approximation, since for systems on random uncorrelated scale-free networks this method is known to often give asymptotically exact results. Depending on particular values of q and lambda one observes either a first-order or a second-order phase transition or the system is ordered at any finite temperature. In a case study, we consider the limit q=1 (percolation) and find a correspondence between the magnetic exponents and those describing percolation on a scale-free network. Interestingly, logarithmic corrections to scaling appear at lambda=4 in this case.
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