Absence of Phase Transition in the Biswas–Chatterjee–Sen Model on Directed Barabási–Albert Networks

Abstract

The phase transition in the Biswas–Chatterjee–Sen model on directed Barabási–Albert networks has been investigated by means of extensive Monte Carlo simulations with four and fifty neighbors. The relaxation time of the average opinion state of the network and its fourth-order Binder cumulant have been analyzed for several different network sizes. Contrary to what happens on regular lattices, the present results do not show any presence of critical points, or equivalently, an absence of a critical social temperature.

1. Introduction

Many-body systems with local interactions constitute one of the cornerstones of modern statistical physics, with the Ising model being one of the most paradigmatic examples for the study of critical phenomena and collective behavior [1]. In the classical Ising model, each lattice site is occupied by a binary spin $s_i=\pm 1$ that interacts ferromagnetically with its nearest neighbors. The competition between these interactions and thermal fluctuations leads to a continuous phase transition, characterized by the emergence of spontaneous magnetization below a critical temperature $T_c$ [2]. In one and two dimensions, the model is exactly solvable. While in the former case, there is no transition at finite temperatures, in the square lattice, the model presents a second-order transition at $k_B{T}_c/J={\displaystyle \frac{2}{ln(1+\sqrt{2})}}$, with universal critical exponents given by $\beta =1/8$, $\gamma =7/4$ and $\nu =1$ [3,4]. In three dimensions, although no analytical solution exists, recent up-to-date high-precision numerical studies determine $k_B{T}_c/J=4.5115232\left(1\right)$ and critical exponent $\nu =0.629912\left(86\right)$, which define the three-dimensional Ising universality class [5]. In these expressions, $k_B$ is the Boltzmann constant; $T_c$ the critical temperature; J the exchange interaction; and $\beta$, $\gamma$, and $\nu$ are the critical exponents of the magnetization, magnetic susceptibility, and correlation length, respectively.

Although originally proposed to study ferromagnetic materials, the importance of the Ising model now transcends magnetism, extending to systems as diverse as critical mixtures, neural networks, and information models [6]. Its conceptual robustness motivated the development of several generalizations, including nonequilibrium models, in which neither a well-defined Hamiltonian nor the principle of detailed balance exists. Among these models, the majority-vote model (MVM) stands out as a stochastic counterpart of the Ising model designed to describe collective decision-making processes [7]. In the MVM, each agent assumes a binary state $s_i=\pm 1$ and tends to align with the majority of its neighbors with probability $1-q$, while with probability q it adopts the opposite state, with q being interpreted as social noise or, mimicking a magnetic system, an effective temperature. Despite its nonequilibrium dynamics and explicit violation of the principle of detailed balance, the majority-vote model exhibits a continuous phase transition as a function of the parameter q, separating an ordered phase from a disordered phase [7,8] and displaying the same critical exponents as the Ising model in two and three dimensions.

Beyond the majority-vote model, other nonequilibrium kinetic models of binary spins have been proposed to investigate collective dynamics in complex systems. Among them, a kinetic opinion formation model introduced by Biswas, Chatterjee, and Sen, known as the BChS model [9], stands out as a stochastic generalization of majority-based consensus dynamics. In this model, state updates do not depend exclusively on the local majority, but rather on pairwise interactions incorporating additional probabilistic rules that allow for the coexistence of different influence mechanisms, resulting in richer and more flexible dynamics. The BChS model exhibits continuous phase transitions between ordered and disordered states, sharing universal features with the Ising model and the majority-vote model, while simultaneously introducing new critical behaviors associated with its specific microscopic dynamics [9,10,11]. In particular, Mukherjee et al. [10] showed that, as occurs for the MVM, the continuous version of the BChS model presents critical exponents identical to those of the Ising model in two and three dimensions.

Remarkably, both the MVM and the BChS model belong, on several regular topologies, to the Ising universality class, provided that spin-inversion symmetry is preserved. This result reinforces the central idea of universality theory: large-scale critical properties are essentially determined by symmetry and dimensionality, rather than by the specific details of the microscopic dynamics [12,13,14,15,16].

Subsequent studies investigated the dynamics of the BChS model on complex topologies and composite structures. Raquel et al. [17] analyzed the BChS model on Erdős–Rényi complex graphs and observed that both the critical noise value and the critical exponents explicitly depend on the average connectivity and the structural heterogeneity of the network, thus showing that the BChS model does not belong to the same universality class as the Ising model in such topologies. Studies on hierarchical networks and coupled structures also revealed regimes of prolonged metastability and slow relaxation, connecting the BChS model to nontrivial coarsening phenomena and long routes to consensus [18].

In the context of richer microscopic dynamics, Biswas and Sen introduced versions of the BChS model with extreme switching mechanisms, in which agents may flip their states independently of the local majority, resulting in new dynamical regimes and quantitative modifications of the phase diagram [19]. Despite these extensions, analyses based on finite-size scaling and Binder cumulants confirmed the persistence of continuous phase transitions compatible with the Ising universality class.

Recent contributions explored the BChS model on nontrivial topological structures, including Solomon-type networks and multilayer systems. These studies highlighted the role of effective dimensionality and interlayer connectivity in the stability of consensus and in the nature of the critical transition [20]. In addition, modern approaches based on machine learning have been employed to automatically identify critical regimes and dynamical patterns in the BChS model, offering new tools for the characterization of criticality in nonequilibrium systems [21].

Very recent works have further expanded the scope of the BChS model by considering systems composed of coupled groups. In particular, Suchecki et al. investigated the Biswas–Chatterjee–Sen model defined on two connected groups, demonstrating the emergence of symmetric and antisymmetric ordered states, as well as critical transitions associated with the competition between intra- and intergroup interactions [22]. These results reinforce the versatility of the BChS model and confirm its relevance as a natural extension of the majority-vote model and the Ising model to structured and modular nonequilibrium scenarios.

Sumour and Shabat [23] investigated Ising models on directed Barabási–Albert networks (DBANs) [24] with the usual Glauber dynamics. Interestingly, no spontaneous magnetization was found, in contrast to the case of undirected Barabási–Albert networks (UBANs) [25,26], where a spontaneous magnetization was found at a critical temperature which increases logarithmically with system size. This lack of a spontaneous magnetization is consistent with the fact that if, on directed networks, a spin $s_j$ influences spin $s_i$, then spin $s_i$ in turn does not influence $s_j$, and for Ising models, there may be no well-defined total energy.

In this work, a continuous version of the BChS model on DBANs is studied through extensive Monte Carlo simulations, in order to investigate the presence, or not, of a phase transition. It should be stressed here that although the BChS and Ising models on DBA networks are in the same universality class, it is not straightforward to infer that the continuous BChS model also has no phase transition, as in the Ising model.The reason for this is because the discrete version of the BChS model on DBA networks, when the opinion state has three distinct values {−1, 0, +1}, does undergo a second-order phase transition [27]. The paper is outlined as follows. In the next section, the model, the quantities used to describe the dynamics of the system, and some details of the Monte Carlo simulations are presented. The corresponding results are discussed in Section 3. Some final remarks are given in the last section.

2. Model and Simulations

In the BChS model, each individual can assume an opinion state s that, in the continuous version, lies in the interval $[-1,1]$, characterizing a one-dimensional political spectrum where the extremes $-1$ and 1 correspond to radical opinions, while 0 represents a neutral opinion.

The kinetic rules of the BChS model dynamics are simple and summarized below [9,10].

• For each of the N nodes of the network, we assign an opinion variable $s_i$ in the continuous interval $[-1,1]$. The state of the network is given by the set ${\mathit{s}}^{\mathit{i}}=\left(s_1,s_2,\dots ,-s_i,\dots ,s_N\right)$. The dynamics of the system is initialized by randomly selecting the opinion states of each of the N nodes of the network, assigning them random numbers drawn from a uniform distribution in the interval $[-1,1]$.
• At each step of the dynamics, a node i of the network is randomly chosen to have its state updated.
• Next, a node j among the neighbors of the node i is randomly chosen to have its state updated as well. The affinity ${\mu }_{i,j}$ of the link between nodes i and j is randomly selected. The affinity parameter is an annealed random variable in the interval $[0,1]$, which can be negative with probability q. This affinity is also allowed to be either discrete or continuous.
• The two nodes i and j are updated according to the following expressions:

  $$\begin{array}{cc}\hfill & s_i(t+1)=s_i\left(t\right)+{\mu }_{i,j}{s}_j\left(t\right),\hfill \\ \hfill & s_j(t+1)=s_j\left(t\right)+{\mu }_{i,j}{s}_i\left(t\right),\hfill \end{array}$$

  (1)

  where $s_i\left(t\right)$ and $s_j\left(t\right)$ are the opinion states immediately before the update, and $s_i(t+1)$ and $s_j(t+1)$ are the updated opinion states.
• If any of the opinion states $s_{i,j}(t+1)$ of nodes i and j satisfy $s_{i,j}(t+1)>1$, they are set to $s_{i,j}(t+1)=1$ in order to keep the opinion states restricted to the interval $[-1,1]$. Similarly, if any of the updated opinion states satisfy $s_{i,j}(t+1)<-1$, the updated states become $s_{i,j}(t+1)=-1$. This constitutes a source of nonlinearity in the BChS model.

The rules above mean that the present BChS model has continuous opinion variables $s_i\left(t\right)$, while the affinities ${\mu }_{i,j}$ will be taken as discrete and also continuous to see the effects in both dynamics.

The main difference between the MVM and BChS models lies in the interaction mechanism between each node and its neighbors. While the opinion update in the MVM depends on the opinion states of all neighboring nodes, in the BChS model, the update depends only on a single, randomly selected, neighbor. This fundamental difference may be responsible for distinct critical behaviors of the two models on scale-free networks.

The fundamental observable of the consensus formation dynamics is the average opinion state of the network, m, which measures the level of consensus and is analogous to the magnetization in ferromagnetic systems. It is given by

$$m=\left|\sum _{i=1}^N{s}_i\left(t\right)\right|/N.$$

(2)

When computing the stationary configurations, the first $N_{\mathrm{term}}$ Monte Carlo steps have been discarded. Once the system has reached the stationary state, we begin to collect a time series consisting of $N_t$ configurations ${\mathit{s}}_{\mathbf{\ell }}$ ($\ell =0,1,\dots ,N_t$) of the system. In order to avoid statistical correlations among system configurations, we also discard a number of Monte Carlo steps between two successive elements of the configuration time series [28]. The interest here lies not only in collecting stationary configurations but also measuring a kind of the relaxation time of the average opinion state. When doing so, an ordered initial configuration with all opinion states $s_i=1$ is considered.

From the elements of the time series obtained for each random realization of the network, we can determine the order parameter M, the susceptibility $\chi$, and the Binder cumulant U, given by the following expressions [7]:

$$\begin{array}{ccc}M\left(q\right)\hfill & =\hfill & \left[\langle m\rangle \right],\hfill \\ \chi \left(q\right)\hfill & =\hfill & \left[N(\langle m^2\rangle -{\langle m\rangle }^2)\right],\hfill \\ U\left(q\right)\hfill & =\hfill & \left[1-{\displaystyle \frac{\langle m^4\rangle }{3{\langle m^2\rangle }^2}}\right],\hfill \end{array}$$

(3)

where the symbol $\langle \dots \rangle$ denotes an average taken over the time series, and the symbol $\left[\dots \right]$ represents the annealed average over the random realizations of the network. All observables are functions of the noise parameter q.

In this work, the above quantities, as a function of q, have been computed through extensive MC simulations on DBANs of finite sizes ranging from $N=576$, $1024$, $2304$, $4096$, $9216$, up to $\mathrm{16,384}$. In order to let the system reach its stationary state, the initial $N_{\mathrm{term}}=5\times {10}^5$ MC steps (MCS) have been discarded. The corresponding time averages have then been computed by taking the next $N_t=1\times {10}^6$ MCS. Here, one MCS consists of randomly choosing N sites of the network, as described above. For each set of network size N and parameter q, ${10}^3$ to ${10}^4$ different configurations have been considered to obtain the configurational averages. For all sets of parameters, we have generated 128 distinct networks.

In a second-order phase transition, the dynamical quantities in Equation (3) are expected to obey the following finite-size scaling relations:

$$\begin{array}{ccc}M\hfill & =\hfill & N^{-\beta /\nu }{f}_M\left(N^{1/\nu }\left(q-q_c\right)\right),\hfill \\ \chi \hfill & =\hfill & N^{\gamma /\nu }{f}_{\chi }\left(N^{1/\nu }\left(q-q_c\right)\right),\hfill \\ U\hfill & =\hfill & f_U\left(N^{1/\nu }\left(q-q_c\right)\right),\hfill \end{array}$$

(4)

where $1/\nu$, $\beta /\nu$, and $\gamma /\nu$ are the critical exponents, similar to the magnetic ones; $q_c$ is the critical noise; and $f_{M,\chi ,U}$ are the respective scaling functions. With the knowledge of $q_c$, the equations above allow for the computation of the desired critical exponent ratio.

3. Results

3.1. Square Lattice

First of all, it is convenient to reproduce the results for the continuous version of the BChS model on a regular square lattice as a test of the simulations. Finite lattices of sizes $L=12,16,24,32,48,64,96$ have been used with helicoidal boundary conditions in order to reduce the computational time [29]. In Figure 1a,b, the dependence of the reduced Binder cumulant $U_4$ and the susceptibility $\chi$ are shown as a function of the noise parameter q for several finite lattices of size L with $N=L\times L$ considering continuous affinity ${\mu }_{ij}$. From these figures, one can see that (i) the cumulant crossings attest indeed that we have a second-order phase transition in the system, with a critical noise given by $q_c=0.2266\left(3\right)$; (ii) the susceptibility exhibits a singularity close to $q_c$ when the system size increases, reflecting the fact that in the thermodynamic limit, i.e., when $N\to \infty$, the peak value of susceptibility tends towards infinity [28]; (iii) the logarithm behavior of the maximum value of the susceptibility and order parameter at $q_c$, both as a function of the logarithm of the size L shown in Figure 1c, give the corresponding exponent ratio $\gamma /\nu =1.75\left(1\right)$ and $\beta /\nu =0.126\left(1\right)$. These values are in quite good agreement with those from the two-dimensional Ising universality class, as expected.

Regarding the Binder cumulant crossings, they do not intersect at a single point, but rather within a finite interval. When taking lattices $(L,L^{\prime })$, with $L^{\prime }\le L$, the crossings do not follow a scaling law that allows us to determine the critical noise parameter in the thermodynamic limit as usual. Nevertheless, the interval is quite small, of the order of ${10}^{-4}$. We considered $16\le L\le 96$ and, from the 21 pieces of data available, computed $q_c$ above and the corresponding error bar.The corresponding cumulant at the transition was estimated as $U_4^{*}=0.601\left(8\right)$. This quantity depends on the boundary conditions [30]; however, the present value is closer to $U_4^{*}=0.61069$… [31] with periodic boundary conditions than $U_4^{*}=0.396\left(2\right)$, obtained with open boundary conditions [30].

3.2. Directed Barabási–Albert Networks

Figure 2 and Figure 3 show the fourth-order Binder cumulant of the continuous version of the BChS model on DBANs for continuous (a) and discrete (b) values of ${\mu }_{ij}$ for the network connectivities $z=4$ and $z=50$. Unlike the results obtained for square lattices, we see that the curves of the fourth-order Binder cumulant $U_4$ do not intersect at a specific point $q_c$, suggesting no phase transition at a non-zero value of q for any version of the model.

Figure 4 shows, as an example, the behavior of the order parameter M as a function of the Monte Carlo steps for the model with continuous affinity ${\mu }_{ij}$ for $z=4$ and two different values of the noise parameter. For this particular case, an even larger lattice with ${10}^6$ nodes has been used and an average over nine different runs has been performed. The simulations have been done with the initial opinion state in all nodes of the network in the +1 state. In both cases, we can see that M oscillates around zero. However, in (a), where $q=0.36$, M decays to negative values longer than for $q=0.42$ in (b).

Defining the value of the Monte Carlo step when the order parameter first becomes negative as a kind of relaxation time $\tau$, it is possible to have a view of what happens when q varies. This is shown in Figure 5a, where we have the inverse of $\tau$ as a function of q for $z=4$ and both continuous and discrete affinity. The full lines are linear fits that do not cross the noise parameter axis. Similar behavior is obtained for other values of z. The results in Figure 5a give additional evidence of the absence of a critical transition of the continuous BChS model in the DBA networks. However, the fluctuations of the order parameter as a function of the Monte Carlo steps are quite large and the system converges to a kind of fluctuating stationary state instead of undergoing a phase transition.

As a matter of verification, we have in Figure 5b this very same procedure applied to the model on a regular square lattice. It can be seen that in this case, a clear non-zero value of the critical noise is obtained. In fact, a linear fit is not applicable in this case to obtain a more precise value of $q_c$. Moreover, it should be noticed that $\tau$ here is not a true relaxation time in the sense of having an exponential decay with the Monte Carlo steps. So, it would not be convenient to extract a precise value of the critical exponent $\zeta$. Even so, a linear fit of the data with the expression $\tau \sim {(q-q_c)}^{-\zeta }$ gives a rough estimate of $\zeta \sim 0.8\left(3\right)$. This value is different from the Ising universality class $\zeta =2$ but, at the same time, one should bear in mind that the dynamical exponent does depend on the type of the dynamics (which is different in the present model).

Upon closer inspection, Figure 2 and Figure 3 reveal that there is still a finite-size effect on the fourth-order Binder cumulants because they do not converge to the value of the larger network. Such behavior can also be noticed in the order parameter susceptibility. Figure 6 shows $\chi$ as a function of q for different network sizes for the specific case $z=4$ with continuous affinity in (a) and $z=50$ with discrete affinity in (b). The same qualitative behavior is obtained for other values of the model parameters. It can be clearly seen that the susceptibilities also do not converge themselves to the value of the larger lattice. Furthermore, the peak of the susceptibility increases with the network size, suggesting here a possible phase transition. However, there are some details that go indeed in the opposite conclusion. First, the peaks of $\chi$ are slightly rounded and not so pronounced as those in Figure 1b for the regular square lattice, mainly for larger N. Second, the position of the peaks seems either quite close to the value of $q_{max}$ or oscillates slightly around $q_{max}$, as is shown in Figure 7a. Note in this figure that there is not a clear scaling behavior of $q_{max}$ with N, as is usual in second-order transitions. Finally, there is a clear scaling of the maximum value of the susceptibility with the network sizes, as is shown in Figure 7b, with a definite value for the critical exponent ratio $\gamma /\nu$. Nevertheless, as has been previously reported for other models in this type of network [27], even when the model does undergo a phase transition, such an exponent ratio is a spurious result because the true value of $\gamma /\nu$ comes from the scaling of the susceptibility at the true critical point instead. The above arguments suggest that although the results for the present model are not as fully convincing as one should expect, the analyzed quantities suggest indeed an absence of any transition as the noise parameter varies.

4. Conclusions

The continuous version of the nonequilibrium kinetic Biswas–Chatterjee–Sen model has been studied through extensive Monte Carlo simulations on directed Barabási–Albert networks. Although the BChS and Ising models belong to the same universality class, they do not behave in the same way as with the very same topology. While the Ising model has no phase transition, the continuous BChS model (with continuous and discrete affinities) also has no phase transition, in contrast to the discrete version, where a second-order phase transition takes place on DBA networks. However, the fluctuations of the order parameter as a function of Monte Carlo steps are quite large and the system converges to a kind of fluctuating stationary state.