Absorbing states of zero-temperature Glauber dynamics in random networks

We study zero-temperature Glauber dynamics for Ising-like spin variable models in quenched random networks with random zero-magnetization initial conditions. In particular, we focus on the absorbing states of finite systems. While it has quite often been observed that Glauber dynamics lets the system be stuck into an absorbing state distinct from its ground state in the thermodynamic limit, very little is known about the likelihood of each absorbing state. In order to explore the variety of absorbing states, we investigate the probability distribution profile of the active link density after saturation as the system size $N$ and $\langle k\rangle$ vary. As a result, we find that the distribution of absorbing states can be split into two self-averaging peaks whose positions are determined by $\langle k\rangle$, one slightly above the ground state and the other farther away. Moreover, we suggest that the latter peak accounts for a nonvanishing portion of samples when $N$ goes to infinity while $\langle k\rangle$ stays fixed. Finally, we discuss the possible implications of our results on opinion dynamics models.

Figure 1

An example of a system in which only blinkers are flippable. Active links are highlighted in boldface (red) and the blinker is marked with a question mark “?”. Successive flipping of spins at nodes $V_1$, $V_2,...,$$V_n$ leads to the merger of two domain walls and further relaxation. However, the time required for the process may be long (growing like $n^2$), giving a false impression of saturation. This matter is addressed by the acceleration algorithm described in Sec. II2b.

Figure 2

Dynamic scaling of ${\langle l_A\rangle }_{l_A\ne 0}$ averaged over surviving samples that have never reached the ground state until time $t$. Data collapse very well using the scaling function ${\langle l_A\rangle }_{l_A\ne 0}=f(t/\tau )$, where the relaxation time $\tau$ scales as $\tau \sim \ln N$. Numerical data are obtained from ${10}^5$ different network realizations for $\langle k\rangle =10$ as $N$ varies. The dip structure indicates that samples can be divided into two groups: the almost fully ordered case driving the initial fall and the quickly absorbed case causing the subsequent rise.

Figure 3

Probability distribution functions of $l_A$ in absorbing states (a) at $\langle k\rangle =1,2,3,4$ with $N=2^{10}$ and (b) at $N=2^9,2^{11},2^{13}$ with $\langle k\rangle =3$. Numerical data are obtained from ${10}^2$ zero-magnetization initial configurations and ${10}^5$ network realizations. The $l_A$ distribution is essentially bimodal, which implies that it is a mixture of two components that need to be analyzed separately.

Figure 4

The fraction of samples belonging to the right component, ${\phi }_R$, is calculated from the $l_A$ distribution. (a) If $N$ is fixed, the value of ${\phi }_R$ decreases with $\langle k\rangle$. (b) If $\langle k\rangle$ is fixed, ${\phi }_R$ eventually increases with $N$. Larger $N$ is required at greater $\langle k\rangle$ to observe the increasing trend. The lines are a guide to the eyes.

Figure 5

The means and widths of two peaks of $P\left(l_A\right)$ are measured. (a) The left-peak mean ${\left[\langle l_A\rangle \right]}_{{}_L}$ is plotted at $\langle k\rangle =3.25,...,4.25$ increased by $0.25$ from top to bottom. (b) The left-peak width $w_{{}_L}$ is plotted for the same parameter sets as (a). (c) The right-peak mean ${\left[\langle l_A\rangle \right]}_{{}_R}$ is plotted for $\langle k\rangle =1,...,7$ increased by 1 from bottom to top. (d) The right-peak width $w_{{}_R}$ is plotted for the same parameter sets as (c). Note that the curves at $\langle k\rangle =1,2$ have been vertically shifted by respective factors of 4 and 5 to avoid overlapping with other data. All the data are obtained from ${10}^3$ different initial configurations and ${10}^3$ network realizations. The lines are a guide to the eyes.

Figure 6

(a) The right-peak mean of $l_A$ measured for ${10}^7$ ER network samples (lower points) and ${10}^6$ regular random network samples (upper points). Lack of degree fluctuations in odd-degree regular random networks does not make much difference in the right-peak mean, while the right peak completely disappears in even-degree regular random networks. (b) The same quantity measured for different initial magnetization $m_0$ or initial active link density $l_0$ in ${10}^6$ ER network samples. The result suggests that for each value of $\langle k\rangle$, the right peak corresponds to the only stable spin configurations with approximately zero magnetization.

Figure 7

Sample-to-sample fluctuations caused by the contributions of network structure and initial spin configuration are measured for ${10}^2$ initializations of ${10}^5$ network realizations. (a) Variances of $l_A$ measured in two different ways (refer to the main text for details) agree well with each other, implying that the mean of $l_A$ distribution is unaffected by the exact structure of each particular ER network. (b) A similar result is observed for Binder cumulants of $l_A$. Here open symbols represent $({\sigma }^2,U_4)$ and solid lines represent $({\sigma }^{\prime 2},U_4^{\prime })$, respectively.

Figure 8

The $N$ dependence of three specific values of $\langle k\rangle$: $k_U\left(\diamond \right),k_{\sigma }\left(\circ \right),k_l\left(\times \right)$. They are possible indicators of the disappearance of the left peak (see the main text for their precise definitions).