Growing spin model in deterministic and stochastic trees

We solve the growing asymmetric Ising model [J. Sienkiewicz, K. Suchecki, and J. A. Hołyst, Phys. Rev. E 89, 012105 (2014)] in the topologies of deterministic and stochastic (random) scale-free trees predicting its nonmonotonous behavior for external fields smaller than the coupling constant $J$. In both cases, we indicate that the crossover temperature corresponding to maximal magnetization decays approximately as ${(lnlnN)}^{-1}$, where $N$ is the number of nodes in the tree.

Figure 1

A schematic plot of a deterministic tree with $z=3$ and $L=3$.

Figure 2

Graphical representation of the weighting of spins at a given sept of the tree.

Figure 3

Average spin ${\langle s\rangle }^t$ in a tree $(z=3,L=12)$ as a function of temperature for different values of the external magnetic field: $h=0.1$ (squares), $h=0.5$ (circles), $h=0.9$ (upward triangles), $h=1$ (downward triangles), and $h=2$ (diamonds). Solid lines come from Eqs. (9) and (10). All data points have been averaged over $M={10}^4$ realizations.

Figure 4

(a) Crossover temperature $T_c$ vs the depth of the tree $L$. Symbols are numerical solutions of Eq. (9) while the solid line comes from $T_c=2{(lnL)}^{-1}$. In all cases, $z=10$. (b) The logarithm of the crossover temperature $T_c$ vs the double logarithm of the number of nodes in a tree $N$. Symbols (squares, $z=2$; circles, $z=5$; triangles, $z=10$) are numerical solutions of Eq. (9), while solid lines come from Eq. (11).

Figure 5

(a) Average spin ${\langle s\rangle }^{\text{sf}}$ in a random scale-free tree $\left(N={10}^5\right)$ as a function of temperature for different values of the external magnetic field: $h=0.1$ (squares), $h=0.5$ (circles), $h=0.9$ (upward triangles), $h=1$ (downward triangles), and $h=2$ (diamonds). Solid lines come from Eqs. (15) and (16). All data points have been averaged over $M={10}^5$ realizations. (b) Crossover temperature $T_c$ vs the number of nodes $N$. Symbols are numerical solutions of Eq. (15), the solid line comes from Eq. (17), while the dashed line comes from Eq. (B4) (see Appendix pp2 for details).

Figure 6

Sum of the squares of displacement between the numerical and theoretical results of the crossover temperature for random scale-free trees vs parameter $a$ [see Eq. (B6)].