Tricriticality in the $q$-neighbor Ising model on a partially duplex clique

We analyze a modified kinetic Ising model, a so-called $q$-neighbor Ising model, with Metropolis dynamics [Phys. Rev. E 92, 052105 (2015)] on a duplex clique and a partially duplex clique. In the $q$-neighbor Ising model each spin interacts only with $q$ spins randomly chosen from its whole neighborhood. In the case of a duplex clique the change of a spin is allowed only if both levels simultaneously induce this change. Due to the mean-field-like nature of the model we are able to derive the analytic form of transition probabilities and solve the corresponding master equation. The existence of the second level changes dramatically the character of the phase transition. In the case of the monoplex clique, the $q$-neighbor Ising model exhibits a continuous phase transition for $q=3$, discontinuous phase transition for $q\ge 4$, and for $q=1$ and $q=2$ the phase transition is not observed. On the other hand, in the case of the duplex clique continuous phase transitions are observed for all values of $q$, even for $q=1$ and $q=2$. Subsequently we introduce a partially duplex clique, parametrized by $r\in [0,1]$, which allows us to tune the network from monoplex ($r=0$) to duplex ($r=1$). Such a generalized topology, in which a fraction $r$ of all nodes appear on both levels, allows us to obtain the critical value of $r=r^{*}\left(q\right)$ at which a tricriticality (switch from continuous to discontinuous phase transition) appears.

Figure 1

Example of a partially duplex clique which consists of two complete graphs of size $N=6$ with interlayer connectivity $r=2/3$.

Figure 2

The average magnetization $\langle m\rangle$ as a function of the temperature $T$ for the duplex clique with $q$ changing from 1 to 6 (from left to right). Monte Carlo results (denoted by symbols) were obtained for the system of size $N={10}^4$ and averaged over $R=200$ realizations. Lines represent solutions of Eq. (9).

Figure 3

Analytical solutions of magnetization $m$ as a function of temperature for $q=2$ and (a) $r=0.486$, (b) $r=0.49$, (c) $r=0.499$, and (d) $r=0.5$. Solid lines represent stable solutions, and dashed lines show unstable ones.

Figure 4

Analytical solutions for magnetization $m$ as a function of temperature $T$ for $q=2$. Solid lines represent stable (attracting) solutions, dashed lines represent unstable (repellent) solutions: from left to right $r=0.51$, $r=0.6$, $r=0.7$, $r=0.8$, $r=0.9$, and $r=1$. The point where $m=0$ changes its character from an unstable to a stable solution is invisible due to overlapping of lines for different values of $r$.

Figure 5

The construction of the $m_1\left(T_1\right)$ (red arrow), i.e., the stable solution of an order parameter at the lower spinodal line shown for $r=0.501$. Inset: the decay of stable magnetization at lower spinodal line with parameter $r$ for $q=2$.

Figure 6

Analytical solutions for magnetization $m$ as a function of temperature $T$ for $q=4$ and (from left to right) $r=0$, $r=0.05$, $r=0.1$, $r=0.15r=0.18$, and $r=0.208$.

Figure 7

(a, b) The value of magnetization $m_1\left(T_1\right)$ for even (a) and odd (b) values of $q$. (c) The critical value $r^{*}$ for which the phase transition becomes continuous as a function of $q$. (d) Phase diagram for the $q$-neighbor Ising model on a partially duplex clique in a $(q,r)$ space. White regions indicate discontinuous phase transition and the gray ones continuous. The area where a phase transition is absent is marked with black. The small range for $q=2$ shown with orange represents the special case of a phase transition depicted in Fig. 3.

Figure 8

Critical temperature $T_c$ for the duplex clique as a function of $q$. Points represent solutions obtained using Eq. (11), and the solid line comes from a linear regression fit. To prevent visual overlap only every 10th data point is shown.

Figure 9

Finite-size effects for average absolute magnetization $\langle \left|m\right|\rangle$ vs temperature $T$ for $q=2$ and increasing network size: (a) $N=10000$, (b) $N=50000$, (c) $N=100000$, (d) $N=500000$. Symbols represent Monte Carlo simulations (squares, random initial conditions; circles,– ordered initial conditions), and solid and dashed lines give, respectively, stable and unstable solutions obtained from Eqs. (B2, B3, B4).