Majority-vote model on triangular, honeycomb and Kagomé lattices

# Majority-vote model on triangular, honeycomb and Kagome lattices

## Abstract

On Archimedean lattices, the Ising model exhibits spontaneous ordering. Three examples of these lattices of the majority-vote model with noise are considered and studied through extensive Monte Carlo simulations. The order/disorder phase transition is observed in this system. The calculated values of the critical noise parameter are , , and for honeycomb, Kagomé and triangular lattices, respectively. The critical exponents , and for this model are , , and ; , , and ; , , and for honeycomb, Kagomé and triangular lattices, respectively. These results differs from the usual Ising model results and the majority-vote model on so-far studied regular lattices or complex networks. The effective dimensionalities of the system (honeycomb), (Kagomé), and (triangular) for these networks are just compatible to the embedding dimension two.

Monte Carlo simulation, critical exponents, phase transition, non-equilibrium
###### pacs:
05.10.Ln, 05.70.Fh, 64.60.Fr

## I Introduction

The majority-vote model (MVM) (1) defined on two-dimensional regular lattices shows second-order phase transition with critical exponents , ,  — which characterize the system in the vicinity of the phase transition — identical (1); (2); (3) with those of equilibrium Ising model (4); (5).

On the other hand MVM on the complex networks exhibit different behavior (6); (7); (8); (9); (10); (12); (11). Campos et al. investigated MVM on undirected small-world network (6). This network was constructed using the square lattice (SL) by the rewiring procedure. Campos et al. found that the critical exponents and are different from those of the Ising model (5) and depend on the rewiring probability. Luz and Lima studied MVM on directed small-world network (7) constructed using the same process described by Sánchez et al. (13). They also found that the critical exponents and are different from these of the Ising model on square lattice, but contrary to results of Campos et al. (6) for MVM the exponents do not depend on the rewiring probability. Pereira et al. (8) studied MVM on undirected Erdős–Rényi’s (ERU) classical random graphs (14), and Lima et al. (9) also studied this model on directed Erdős–Rényi’s (ERD) and their results obtained for critical exponents agree with the results of Pereira et al. (8), within the error bars. Lima et al. (10) also studied this model on random Voronoy–Delaunay lattice (15) with periodic boundary conditions. Lima also (11) studied the MVM on directed Albert–Barabási (ABD) network (16) and contrary to the Ising model on these networks (17), the order/disorder phase transition was observed in this system. However, the calculated and exponents for MVM on ABD and ABU networks are different from those for the Ising model (5) and depend on the mean value of connectivity of ABD and ABU network. Lima and Malarz (18) studied the MVM on and Archimedean lattices (AL). They remark that the critical exponents , and for MVM on AL are different from the Ising model (5) and differ from those for so-far studied regular two-dimensional lattices (1); (2), but for AL, the critical exponents are much closer to those known analytically for SL Ising model.

The results presented in Refs. (6); (7); (8); (9); (10); (12); (11) show that the MVM on various complex topologies belongs to different universality classes. Moreover, contrary for MVM on regular lattices (1); (2), the obtained critical exponents are different from those of the equilibrium Ising model (5). Very recently, Yang and Kim (19) showed that also for -dimensional hypercube lattices () critical exponents for MVM differ from those for SL Ising model. The same situation occurs on hyperbolic lattices (20).

In this paper we study the MVM on three AL, namely on triangular , honeycomb , and Kagomé lattices.

The AL are vertex transitive graphs that can be embedded in a plane such that every face is a regular polygon. The AL are labeled according to the sizes of faces incident to a given vertex. The face sizes are sorted, starting from the face for which the list is the smallest in lexicographical order. In this way, the triangular lattice gets the name , abbreviated to , honeycomb lattice is called and Kagomé lattice is . Critical properties of these lattices were investigated in terms of site percolation (21) and Ising model (22).

Our main goal is to check the hypothesis of Grinstein et al. (23) — i.e., that non-equilibrium stochastic spin systems with up-down symmetry fall in the universality class of the equilibrium Ising model — for systems in-between ordinary, regular lattices (like SL (1)) and complex spin systems (like spins on ERU and ERD (8); (9) or ABU and ABD (12); (11)).

With extensive Monte Carlo simulation we show that MVM on , and AL exhibits second-order phase transitions with effective dimensionality , 1.92 and and has critical exponents that do not fall into universality class of the equilibrium Ising model.

## Ii Model and simulation

We consider the MVM (1) defined by a set of “voters” or spin variables taking the values or , situated on every node of the , and AL with sites for and , and sites for . The evolution is governed by single spin-flip like dynamics with a probability of -th spin to flip is given by

 wi=12[1−(1−2q)σi⋅sign(z∑j=1σj)], (1)

and the sum runs over the number (for and lattices) and (for lattice) of nearest neighbors of -th spin. The control parameter plays the role of the temperature in equilibrium systems and measures the probability of aligning against the majority of neighbors. It means, that a given spin adopts the majority sign of its neighbors with probability and the minority sign with probability (1); (8); (9); (10); (11); (12).

To study the critical behavior of the model we define the variable . In particular, we are interested in the magnetization , susceptibility and the reduced fourth-order cumulant

 M(q)≡⟨|m|⟩, (2a) χ(q)≡N(⟨m2⟩−⟨m⟩2), (2b) U(q)≡1−⟨m4⟩3⟨m2⟩2, (2c)

where stands for a thermodynamics average. The results are averaged over the independent simulations.

These quantities are functions of the noise parameter and obey the finite-size scaling relations

 M=L−β/νfm(x), (3a) χ=Lγ/νfχ(x), (3b) dUdq=L1/νfU(x), (3c) where ν, β, and γ are the usual critical exponents, fm,χ,U(x) are the finite size scaling functions with x=(q−qc)L1/ν (3d)

being the scaling variable. Therefore, from the size dependence of and we obtained the exponents and , respectively. The maximum value of susceptibility also scales as . Moreover, the value of for which has a maximum is expected to scale with the system size as

 q∗=qc+bL−1/ν with b≈1. (4)

Therefore, the relations (3c) and (4) may be used to get the exponent . We evaluate also the effective dimensionality, , from the hyper-scaling hypothesis

 2β/ν+γ/ν=Deff. (5)

We performed Monte Carlo simulation on the , and AL with various systems of size , , , , , and for and AL and , , , , , and for . It takes Monte Carlo steps (MCS) to make the system reach the steady state, and then the time averages are estimated over the next MCS. One MCS is accomplished after all the spins are investigated whether they flip or not. The results are averaged over independent simulation runs for each lattice and for given set of parameters .

## Iii Results and Discussion

In Fig. 1 we show the dependence of the magnetization , Binder cumulant , and the susceptibility on the noise parameter , obtained from simulations on , and AL with ranging from to sites. The shape of , , and curve, for a given value of , suggests the presence of the second-order phase transition in the system. The phase transition occurs at the value of the critical noise parameter . The critical noise parameter is estimated as the point where the curves for different system sizes intercept each other (24). Then, we obtain and ; and ; and for , and AL, respectively.

In Fig. 3 we plot the dependence of the magnetization vs. the linear system size . The slopes of curves correspond to the exponent ratio according to Eq. (3a). The obtained exponents are , , and , respectively for , and AL.

The exponents ratio at are obtained from the slopes of the straight lines with for , for , and for , as presented in Fig. 4. The exponents ratio at are for , for , and for , as presented in Fig. 5.

To obtain the critical exponent , we used the scaling relation (4). The calculated values of the exponents are for (circles), for (squares), and for (diamonds) (see Fig. 2). Eq. (5) yields effective dimensionality of systems for , for , and for . The MVM on those three AL has the effective dimensionality close to two contrary to ER classical random graphs () (8) or directed AB networks () (12) with roughly the same nodes connectivity () as for and , and () AL.

The results of simulations are collected in Tab. 1.

## Iv Conclusion

We presented a very simple non-equilibrium MVM on , and AL. On these lattices, the MVM shows a second-order phase transition. Our Monte Carlo simulations demonstrate that the effective dimensionality is close to two, i.e. that hyper-scaling may be valid.

Finally, we remark that the critical exponents , and for MVM on regular , and AL are similar to the MVM model on regular and (18) and are different from the Ising model (5) and differ from those for so-far studied regular lattices (1); (2) and for the directed and undirected ER random graphs (8); (9) and for the directed and undirected AB networks (11); (12). However, in the latter cases (8); (9); (12); (11) the scaling relations (3) must involve the number of sites instead of linear system size as these networks in natural way do not posses such characteristic which allow for dependence 4. For , and AL some critical exponents are much closer to those known analytically for square lattice Ising model, i.e. , and , but except for they differ for more than three numerically estimated uncertainties.

###### Acknowledgements.
Authors are grateful to Dietrich Stauffer for stimulating discussions and for critical reading of the manuscript. J.C.S. and F.W.S.L. acknowledge the support the system SGI Altix 1350 the computational park CENAPAD, UNICAMP-USP, SP-BRASIL and also the agency FAPEPI for the financial support. K.M. acknowledges the machine time on SGI Altix 3700 in AGH University of Science and Technology, Academic Computer Center CYFRONET (grant No. MEiN/SGI3700/AGH/024/2006).

### Footnotes

1. obtained using at
2. obtained using at
3. obtained using ratio given by dependence at
4. The linear dimension of such networks, i.e. its diameter — defined as an average node-to-node distance — grows usually logarithmically with the system size (25).

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