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Finite size scaling analysis of a nonequilibrium phase transition in the naming game model

We realize an extensive numerical study of the Naming Game model with a noise term which accounts for perturbations. This model displays a non-equilibrium phase transition between an absorbing ordered consensus state, which occurs for small noise, and a disordered phase with fragmented clusters characterized by heterogeneous memories, which emerges at strong noise levels. The nature of the phase transition is studied by means of a finite-size scaling analysis of the moments. We observe a scaling behavior typical of a discontinuous transition and we are able to estimate the thermodynamic limit. The scaling behavior of the clusters size seems also compatible with this kind of transition.


I. INTRODUCTION
The contributions of statistical physicists to the understanding of spreading of cultural traits, opinions or conventions of different nature are nowadays well established [1]. Several works focus on the general mechanisms responsible for the ordering dynamics that generates global consensus as an emergent phenomenon [2]. The introduction of minimal models is a standard practice of the area, which is intended to uncover the possible existence of some universal character shared by different real systems. Such idea of universal properties is in fact supported by the classical theory of phase transitions, which is expected to give at least a partial legacy for these non-equilibrium systems.
In this work we focus our attention to a model of conventions spreading successfully used for describing linguistic dynamics [3]. The model, usually called Naming Game [4], is characterized by a collective dynamics which implement a memory-based negotiation strategy, where a sequence of trials shapes and reshapes the system memories, allowing for intermediate individual states and feedback effects. These rules appear to be more realistic than simple imitation or local majority mechanisms, commonly implemented by the use of Ising-like dynamics. The introduction of a noise term, which can account for external or internal perturbations or agents' irresolute attitude, generates global consensus for small noise levels [4]. On the other hand, strong noise conducts the system to a stationary state characterized by several coexisting conventions. On the basis of some analytical considerations, supported by numerical simulations, it was suggested that the onset of the consensus state can be described as a non-equilibrium phase transition giving rise to order [5].
The purpose of this work is to study and clearly characterize the nature of this transition throughout an accurate finite-size scaling analysis. For this reason, we consider the implementation of the model on a 2D square lat-tice, where our Monte Carlo simulations are performed.
In section 2 we will describe the details of the model and we will introduce a comparison with prior results in the literature, in section 3 we will report the numerical analysis for the characterization of the phase transition and we will discuss our results.

II. THE MODEL
The dynamics rules defining the model are quite simple. Each player is characterized by an inventory which can contain an infinite number of conventions. In fact, it is structured by an array of potentially infinite cells where each cell is set on one of an infinite number of possible numerable states. In the initial state players start with an empty inventory. At each time step, a pair of agents is randomly selected. The first agent selects one of its conventions or, if its inventory is empty, it creates a new one. After that, the convention is transmitted to the second agent. If this last agent possesses the transmitted convention, with a probability β, the two agents update their inventories so as to keep only the convention involved in the interaction. Conversely, with probability 1 − β, no actions are performed by the couple of agents. Otherwise, if the second agent does not possess the transmitted convention, the interaction is a failure and it adds the new convention in its inventory.
The game is simulated on a regular 2D square lattice with L × L sites and periodic boundary conditions. This implementation defines a short-range interaction system, where agents communicate only with their four nearest neighbors. The special case where β = 1 corresponds to the original Naming Game embedded on a lowdimensional lattice. This model was extensively studied and it shown different convergence behaviors from the mean-field case. In fact, consensus is reached by means of a coarsening process which needs less agents' memory effort but longer convergence times than the mean-field model (the upper critical dimension is 4) [6].
The model with general values of β, which effectively generates the described transition, was studied only in one previous work [5]. That paper studied a special case of the mean-field model by means of an analytical approximation where agents can store a maximum of only two different conventions. In this specific approximation of the original model, it was shown that a shift from a consensus state to a polarized one exists at β c = 1/3. Simulations of the original model, i.e. with an unlimited number of conventions, suggested that the transition happen at the same value of β obtained for the analytical approximation [5]. This fact was inferred looking at the divergence of the convergence time near those values. In details, the convergence time required by the system to reach the consensus state (t conv ), present two different behaviors [5]: one for the simplified case with just two conventions, where t conv ∝ (β − β c ) −1 and one for the case of the original model, with an unlimited number of conventions, where t conv ∝ (β − β c ) −0.3 . Finally, using similar arguments, the authors shown that homogeneous random networks and heterogeneous topologies with power-law degree distributions present the same β value for the transition if the pair selection criterion consists in randomly choosing a link [5].

III. RESULTS AND DISCUSSION
In the following we develop a finite size scaling analysis to clearly characterize the phase transition of the model implemented on a 2D square lattice. The system presents very slow relaxation time close to the transition. For this reason, we analyze the final state reached after running 4 × 10 10 Monte Carlo steps. Such very long simulations force us to adopt L values limited between 20 and 60.
The phase transition is marked by the passage from an active stationary state of disordered and fragmented clusters to an absorbing state of a single cluster represented by the same word. In fact, for small β convergence is not attained and small clusters, with one or more different words, characterize the system. In particular, for β values slightly smaller than the one which generates a single cluster (β c ), the fragmented state is characterized by the presence of just two conventions, in accordance with the mean-field behavior of the model [5] (see Figure 1).
For these reasons the relative size of the largest cluster present in the system is an ideal parameter to characterize the transition [7,8]. This is defined as the size of the largest cluster composed by agents sharing the same unique convention normalized over the system size: s max /L 2 . In Figure 2 we can observe its behavior for different values of L. In accordance with typical phase transitions, we can observe a clear scaling looking at the rise of the critical value of β c (L) and at the characteristic steeper transitions for larger system sizes.
For a clear characterization of the type of the transition (continuous or discontinuous) and a quantitative estimation of β c it is useful to evaluate the fluctuations of the size of the largest cluster: ) and its moment ratio (reduced cumulant) [9]: where < > stands for averages over different simulations.
As can be seen in Figure 2, these quantities peak around β c (L). In particular, the maxima of the fluctuations are characterized by higher values for increasing values of L. In fact, the use of the maxima of these quantities has recently demonstrated to be a very robust approach for performing a finite size scaling analysis of discontinuous phase transition into absorbing states [10]. In this case, the asymptotic transition point (asymptotic coexistence point) can be obtained looking at the convergence of the finite size transition points β c (L) as estimated by the localization of the maxima of the fluctuations or the maxima of the moment ratio. In both cases, the convergence is expected to follow an algebraic behavior: β c (L) = β c + aL −2 , which is the usual equilibrium scaling [10,11].
These scaling laws are well verified by our data. Looking at Figure 3 we can see that the maxima positions for χ and U 2 effectively decrease as 1/L 2 . An extrapolation for L → ∞ yields β c = 0.329 ± 0.001 for the two cases, an excellent agreement between them.
An alternative approach [12] for the estimation of the asymptotic transition point uses the location of the observed discontinuity in the normalized size of the largest cluster (the first value of β for which < s max > /L 2 is less than 1). Using this method the convergence is supposed to be exponential and the extrapolation for L → ∞ gives β c = 0.328 ± 0.002 (see Figure 4). In the same figure we can observe how the difference β c −β c (L) clearly presents the expected exponential behavior.
Additional consistency checks of the above results can be performed verifying if the measured quantities present the typical scaling of a discontinuous transition near the transition point, a standard procedure for equilibrium finite-size scaling analysis [13]. The scaling plot of < s max > /L 2 should be obtained simply considering the rescaled control parameter β * = (β − β c )L d , where d is the system dimension. In a similar fashion, the scaling plot of the fluctuations should be obtained considering the rescaled fluctuation χ · L −d and the rescaled parameter β * . As shown in details in Figure 4, it is possible to obtain a reasonable collapse which satisfies these relations. In fact, data roughly collapse to a single curve, strongly suggesting the validity of the finite-size scaling ansatz expected for a discontinuous transition.

FIG. 4. Top:
Convergence to the asymptotic transition point βc of the finite-size transition points βc(L) measured from the jump location in the normalized size of the largest cluster. The continuous line represents the best fitting function: βc(L) = 0.328−0.23 exp(−0.076L). In the inset, the semilogaritmic plot shows in details the expected exponential behavior of the difference βc − βc(L). The continuous line represents an exponential fitting. Bottom: Rescaled plot for smax/L 2 (on the left) and its fluctuations (on the right).
We conclude our study analyzing the behavior of the size distribution of the clusters present in the fragmented phase near the transition. Obviously we examine only the simulations which do not converge to a unique cluster and, among them, the domains characterized by agents sharing the same unique convention (the regions with memory containing more than one convention are not considered). The probability distribution of the clusters size s is presented in Figure 5. In the range of the system size we study, the distribution decays as a power law.
An accurate estimation of the exponent is difficult for the system size we are using. A data fitting allover the region of considered s gives an exponent near 1.8. However, the exponent is very close to 2 in the intermediate region of s, just before the finite size effects which generate an accumulation on the higher bins responsible for the exponent decrease (see Figure 5). For an unambiguous characterization of this behavior and for the identification of possible logarithmic corrections to a consistent power law slope, simulations on larger systems should be realized. We remember that the fact of finding a discontinuous phase transition with a power law clusters size distribution, and not an exponential or Gaussian one, is not an unusual result, as can be seen, for example, in the Axelrod's model for social influence [7].
In analogy with the approach used in percolation theory, we can measure the average cluster size S defined as: S = s n s s 2 s n s s where n s stands for the number of clusters of size s and the sum run over all possible values of s. For a discontinuous transition S is expected to scale with L 2 at the transition point [14]. This relation is reasonable verified for our system as can be stated looking at Figure  5.
In summary, we have presented a numerical study of the Naming Game model with a noise term on a regular 2D lattice with the intent of clearly characterize, for the first time, the nature of the transition generated by this system.
Our analysis has shown that the model effectively displays a non-equilibrium phase transition between a consensus state and a phase with coexisting conventions in dependence of the control parameter which represents the efficiency of the communication process. The nature of the phase transition has been studied by means of a finitesize scaling analysis. The variance and the moment ratio show a scaling behavior which can be associated with a discontinuous transition and allow the estimation of the transition point at the thermodynamic limit. Additional confirmations of these results come from the collapse of the scaling plot of < s max > /L 2 and its fluctuations which have been obtained using the scaling law expected for a discontinuous transition.
Finally, we have presented some results for the behavior of the clusters distribution near the transition point. The distribution displays a power law behavior, while the average cluster size scaling with the system size is compatible with a discontinuous transition.
These results are not relevant only for the naming game community, but, more in general, they can be interesting for the community of the statistical physicists which work with discontinuous nonequilibrium phase transitions into absorbing states. In particular, they are related to systems used in the description of opinion formation and which are characterized by the presence of a large number of possible different absorbing states. Considering systems embedded in a 2D space, we can mention the Axelrods model. In such a model, the consensusfragmentation phase transition is discontinuous if the number of cultural features is greater than two [7]. A discontinuous behavior was also described in another version of the Naming Game model, characterized by the presence of an open-ended reservoir of words [8].