Ising model in clustered scale-free networks

The Ising model in clustered scale-free networks has been studied by Monte Carlo simulations. These networks are characterized by a degree distribution of the form $P\left(k\right)\sim k^{-\gamma }$ for large $k$. Clustering is introduced in the networks by inserting triangles, i.e., triads of connected nodes. The transition from a ferromagnetic (FM) to a paramagnetic (PM) phase has been studied as a function of the exponent $\gamma$ and the triangle density. For $\gamma >3$ our results are in line with earlier simulations, and a phase transition appears at a temperature $T_c\left(\gamma \right)$ in the thermodynamic limit (system size $N\to \infty$). For $\gamma \le 3$, a FM-PM crossover appears at a size-dependent temperature $T_{\mathrm{co}}$, so the system remains in a FM state at any finite temperature in the limit $N\to \infty$. Thus, for $\gamma =3, T_{\mathrm{co}}$ scales as $lnN$, whereas for $\gamma <3$, we find $T_{\mathrm{co}}\sim J{N}^z$, where the exponent $z$ decreases for increasing $\gamma$. Adding motifs (triangles in our case) to the networks causes an increase in the transition (or crossover) temperature for exponent $\gamma >3$ (or $\le 3$). For $\gamma >3$, this increase is due to changes in the mean values $\langle k\rangle$ and $\langle k^2\rangle$, i.e., the transition is controlled by the degree distribution (nearest-neighbor connectivities). For $\gamma \le 3$, however, we find that clustered and unclustered networks with the same size and distribution $P\left(k\right)$ have different crossover temperature, i.e., clustering favors FM correlations, thus increasing the temperature $T_{\mathrm{co}}$. The effect of a degree cutoff $k_{\mathrm{cut}}$ on the asymptotic behavior of $T_{\mathrm{co}}$ is discussed.

Figure 1

Schematic representation of a typical network considered in this work for which one separately specifies the number of single links and triangles attached to each node. Triangles are indicated by asterisks (*).

Figure 2

Probability density as a function of the degree $k$ for networks with $\gamma =3$, minimum degree $k_0$ = 3, and $N={10}^5$ nodes. The data shown are an average over 200 network realizations for each value of the parameter $\nu$ = 0, 1, and 2.

Figure 3

Mean degree $\langle k\rangle$ as a function of $N^{-1/2}$ for networks with $\gamma =3, k_0=3$, and three values of $\nu$: 0 (squares), 1 (circles), and 2 (diamonds). Error bars of the simulations results are less than the symbol size. Lines are fits to the expression $\langle k\rangle =a+b{N}^{-1/2}$.

Figure 4

Transition temperature $T_c$ for networks with $\gamma =5$ as a function of the minimum degree $k_0$. Open symbols represent results of MC simulations, as obtained from Binder's cumulant for $\nu$ = 0 (squares), 1 (circles), and 2 (diamonds). Error bars are less than the symbol size. Solid lines were derived from the analytical expression given in Eq. (11), using values of $\langle k\rangle$ and $\langle k^2\rangle$ obtained from Eqs. (4) and (14). Solid squares are data obtained in Ref. [38] for unclustered networks ($\nu$ = 0). The dashed line is a guide to the eye.

Figure 5

Transition temperature $T_c$ for scale-free networks with $\gamma =5$ as a function of the mean degree $\langle k\rangle$. Data points are results for networks with $\nu$ = 0 (open squares) and $\nu$ = 2 (solid diamonds). Error bars are less than the symbol size. Solid and dashed-dotted lines are analytical predictions from Eq. (11) for $\nu$ = 0 and 2, respectively. The dashed line represents the mean-field result given by Eq. (15).

Figure 6

Crossover temperature $T_{\mathrm{co}}/J$ for scale-free networks with $\gamma =3$ and minimum degree $k_0=3$ as a function of the system size $N$. Squares, $\nu =0$; circles, $\nu =1$; diamonds, $\nu =2$. Lines are least-squares fits to the data points for $N\ge 4000$ nodes.

Figure 7

(a) Crossover temperature as a function of the mean degree $\langle k\rangle$ for networks with $\gamma =3$ and $N=8000$ nodes. Symbols represent results for three $\nu$ values: 0 (squares), 1 (circles), and 2 (diamonds). The data shown were obtained for networks with several minimum degrees, $k_0\ge 3$. (b) Ratio $T_{\mathrm{co}}/J\langle k\rangle$ vs $\langle k\rangle$ for the same kind of networks as in (a). Lines are guides to the eye.

Figure 8

Crossover temperature $T_{\mathrm{co}}/J$ for scale-free networks with $\gamma =3$ and $N=8000$ nodes, as a function of the triangle density $\nu$. Each set of data corresponds to a fixed value of the mean degree: $\langle k\rangle =14$ (circles) and $\langle k\rangle =18$ (squares). Dashed lines are least-squares fits to the data points.

Figure 9

Crossover temperature as a function of system size $N$ for networks with $\gamma =3, \nu =2$, and minimum degree $k_0=3$. Circles and squares represent results of MC simulation for networks with and without triangles but with the same degree distribution $P\left(k\right)$. Error bars are less than the symbol size. The solid line is the analytical prediction, Eq. (11), obtained from the mean values $\langle k\rangle$ and $\langle k^2\rangle$.

Figure 10

Crossover temperature $T_{\mathrm{co}}/J$ for scale-free networks with $\nu =0$ and $k_0=3$ as a function of system size $N$ in a logarithmic plot. Symbols represent results for three values of the parameter $\gamma$: 2 (squares), 2.5 (circles), and 3 (diamonds). Error bars are less than the symbol size. Lines are guides to the eye.

Figure 11

Crossover temperature $T_{\mathrm{co}}/J$ as a function of system size $N$ for scale-free networks with minimum degree $k_0=3$ and three values of the triangle density $\nu$: 0 (squares), 1 (circles), and 2 (diamonds). (a) Networks with $\gamma =2$; (b) networks with $\gamma =2.5$. Lines are guides to the eye.

Figure 12

Crossover temperature as a function of system size $N$ for networks with $\gamma =2, \nu =2$, and minimum degree $k_0=3$. Circles and squares represent results of MC simulations for networks with and without triangles, respectively, but with the same degree distribution $P\left(k\right)$. Error bars are less than the symbol size. The solid line indicates the result derived from $\langle k\rangle$ and $\langle k^2\rangle$ by using Eq. (11).