Multiscaling in an $YX$ model of networks

We investigate a Hamiltonian model of networks. The model is a mirror formulation of the $XY$ model (hence the name)—instead of letting the $XY$ spins vary, keeping the coupling topology static, we keep the spins conserved and sample different underlying networks. Our numerical simulations show complex scaling behaviors with various exponents as the system grows and temperature approaches zero, but no finite-temperature universal critical behavior. The ground-state and low-order excitations for sparse, finite graphs are a fragmented set of isolated network clusters. Configurations of higher energy are typically more connected. The connected networks of lowest energy are stretched out giving the network large average distances. For the finite sizes we investigate, there are three regions—a low-energy regime of fragmented networks, an intermediate regime of stretched-out networks, and a high-energy regime of compact, disordered topologies. Scaling up the system size, the borders between these regimes approach zero temperature algebraically, but different network-structural quantities approach their $T=0$ values with different exponents. We argue this is a, perhaps rare, example of a statistical-physics model where finite sizes show a more interesting behavior than the thermodynamic limit.

Figure 1

(Color online) Three low-energy configurations of our $YX$ model with 100 vertices.

Figure 2

(Color online) The fraction of vertices not part of the largest connected cluster. In (a), we show the raw data as a function of temperature for four different system sizes. In (b), we determine the scaling exponents for low temperatures to $\alpha =1.61\pm 0.05$ and $\beta =0.22\pm 0.03$. Standard errors are smaller than the symbol size and omitted for clarity. Lines are guides for the eyes.

Figure 3

(Color online) The fraction of the second-largest cluster. (a) displays the raw data of $s_2$ as a function of temperature. In (b), we determine the scaling factor $\gamma =1.44\pm 0.02$. Standard errors are smaller than the symbol size and omitted for clarity. Lines are guides for the eyes.

Figure 4

(Color online) (a) Scaling of the diameter of the largest connected cluster $D$ as a function of temperature and (b) the determination of $D$’s scaling parameters $\delta$ and $ϵ$. We find $\delta =1.52\pm 0.02$ and $ϵ=-0.74\pm 0.02$. Standard errors are smaller than the symbol size and omitted for clarity. Lines are guides for the eyes.

Figure 5

(Color online) Figures corresponding to Fig. 4b for (a) $3N=2M$ and (b) $N=M$. The scaling exponents for (a) are determined to $\delta =1.17\pm 0.10$ and $ϵ=-0.52\pm 0.02$; for (b) they are $\delta =1.25\pm 0.10$ and $ϵ=-0.61\pm 0.02$. Standard errors are smaller than the symbol size and omitted for clarity. Lines are guides for the eyes.

Figure 6

(Color online) (a) The number of fragments as a function of temperature. The solid line is proportional to $T^{-1/2}$. (b) shows the determination of the scaling exponents $\zeta =1.94\pm 0.07$ and $\eta =0.96\pm 0.03$. Standard errors are smaller than the symbol size and omitted for clarity.

Figure 7

(Color online) The average density of vertices at a distance $r$ from a vertex for three different temperatures (indicated in the panels) and $N=1600$. To facilitate comparison, we let the panels cover the same ranges. The widths of the lines are at least the standard error. The inset in panel (c) is a blow up.

Figure 8

(Color online) The correlation function as a function of $r$ from for three different temperatures and $N=1600$, plotted in the same way as Fig. 7.