Pair approximation for the $q$-voter model with independence on complex networks

We investigate the $q$-voter model with stochastic noise arising from independence on complex networks. Using the pair approximation, we provide a comprehensive, mathematical description of its behavior and derive a formula for the critical point. The analytical results are validated by carrying out Monte Carlo experiments. The pair approximation prediction exhibits substantial agreement with simulations, especially for networks with weak clustering and large average degree. Nonetheless, for the average degree close to $q$, some discrepancies originate. It is the first time we are aware of that the presented approach has been applied to the nonlinear voter dynamics with noise. Up till now, the analytical results have been obtained only for a complete graph. We show that in the limiting case the prediction of pair approximation coincides with the known solution on a fully connected network.

Figure 1

Typical sample trajectories of the up-spins concentrations and the corresponding active bonds concentrations when $q\le 5$. For the presented realizations $q=2$, and the network topology is a random 14-regular graph of the size $N=200$. The level of independence: (a) $p=0.2$ and (b) $p=0.4$. Spontaneous transitions between stationary states are noticeable below the critical point $p<p^{*}$ and only for $c$.

Figure 2

Average trajectories for a system of the size $N=2\times {10}^5$ on a random 14-regular graph. In both cases $q=2$. The value of independence: (a) $p=0.2$ and (b) $p=0.35$. Dots represent outcome of the Monte Carlo simulation; solid lines refer to the numerical solution of Eq. (26).

Figure 3

Average trajectories for a system placed on BA networks of the size $N=5\times {10}^5$ with $\langle k\rangle =40$ and $q=8$. The level of independence: (a) $p=0.03$, (b) $p=0.06$, and (c) $p=0.1$. Dots represent outcome of the Monte Carlo simulation; solid lines refer to the numerical solution of Eq. (26).

Figure 4

The relation between the active bonds and the up-spins concentrations for $q=2$ and different networks: (a) ER model with $\langle k\rangle =20$ and $p=0.3$, (b) and (c) random 14-regular graphs of different sizes and $p=0.2$. Lines refer to the theoretical result Eq. (31), dots correspond to simulated trajectories starting from the point $(c,b)=(1,0)$. Crosses in the right panels stand for the average value of $b$ for a given $c$. Paths consist of 150 MCS for ER and $2\times {10}^6$ MCS for RR. Fluctuations diminish with the growing system size.

Figure 5

Phase diagrams for (a) random regular and Barabási-Albert models and (b) scale-free networks with different exponents. Note that upper and bottom diagrams correspond to different structures, all of them consist of $N={10}^5$ vertices. The group of influence comprises $q=2$ agents. Lines indicate analytical prediction of PA and MFA, solid ones correspond to stable states, and dashed ones refer to unstable points. Markers represent outcomes of Monte Carlo simulations: $•\langle k\rangle =6$, $\blacksquare \langle k\rangle =8$, $▴\langle k\rangle =20$.

Figure 6

Critical behavior of the model on scale-free networks with average degree $\langle k\rangle =10$ and tail exponent $\lambda =2.43$. The group of influence comprises $q=2$ agents. (a) A finite-size scaling obtained for $\beta /\overline{\nu }=0.157$, $1/\overline{\nu }=0.314$, and $p^{*}=0.2855$. Critical decays of (b) the magnetization $m$ and (c) the fluctuations ${\chi }^{\prime }$ against the system size $N$. Markers refer to Monte Carlo simulations averaged over 1000 network realizations. Dashed lines, with corresponding slopes $\beta /\overline{\nu }=-0.157\left(6\right)$ and ${\gamma }^{\prime }/\overline{\nu }=0.691\left(9\right)$, come from the linear regression.

Figure 7

The phase diagrams for (a) random regular graph of the size $N={10}^5$ and (b) Erdős-Rényi network with $N=5\times {10}^5$ nodes and $\langle k\rangle =40$. The group of influence consists of $q=6$ agents for RR and $q=8$ for ER. Lines indicate analytical prediction of PA and MFA, solid ones correspond to stable states, and dashed ones refer to unstable points. Markers represent outcomes of Monte Carlo simulations. For the left panel: $•\langle k\rangle =14$, $\blacksquare \langle k\rangle =20$, $▴\langle k\rangle =40$. The hysteresis indicates a discontinuous phase transition and matches analytical result.

Figure 8

Phase diagrams for WS networks of the size $N=2\times {10}^5$ and with the average degree (a) $\langle k\rangle =8$ and (b) $\langle k\rangle =40$. In the case of a continuous phase transition $q=4$ and for discontinuous one $q=8$. With growing rewriting probability $p_r$, the average clustering coefficient $\langle C\rangle$ diminishes, and simulation results approach PA. Around $p_r\approx 0.6$ they match each other well. Lines indicate analytical predictions. Markers represent results of Monte Carlo simulations: $•p_r=0.1$, $\blacksquare p_r=0.2$, $▴p_r=0.7$.

Figure 9

A schematic representation of the pair approximation for the $q$-voter dynamics with stochastic noise. Darker lines correspond to the higher average degree of a network. The last, black line is an outcome of MFA. The model exhibits two types of phase transitions: (a) continuous and (b) discontinuous.