Trading Interactions for Topology in Scale-Free Networks

Scale-free networks with topology-dependent interactions are studied. It is shown that the universality classes of critical behavior, which conventionally depend only on topology, can also be explored by tuning the interactions. A mapping, ${\gamma }^{\prime }=(\gamma -\mu )/(1-\mu )$, describes how a shift of the standard exponent $\gamma$ of the degree distribution $P(q)$ can absorb the effect of degree-dependent pair interactions $J_{ij}\propto (q_i{q}_j{)}^{-\mu }$. The replica technique, cavity method, and Monte Carlo simulation support the physical picture suggested by Landau theory for the critical exponents and by the Bethe-Peierls approximation for the critical temperature. The equivalence of topology and interaction holds for equilibrium and nonequilibrium systems, and is illustrated with interdisciplinary applications.

Figure 1

$T_c$ versus $⟨q⟩$ for a scale-free network with $\gamma =3$ and $\mu =1/2$. BP estimates (solid squares) almost coincide with exact results (replica technique, open stars). The solid line is the lattice BP approximation for $Q=⟨q⟩$, and the dashed line is the mean-field approximation.

Figure 2

Specific heat from MCS of BA networks ($\gamma =3$), with $\mu =1/2$ and $⟨q⟩=10$. The critical temperature in the large-$N$ limit is $k_B{T}_c/J\approx 8.96$ [Eq. (6)]. For $N=5600$, MCS of the susceptibility maximum indicates $k_B{T}_c(N)/J\approx 8.65$, well above the temperature of the specific heat maximum.

Figure 3

Specific heat from the cavity method for BA networks, with $\mu =1/2$ and $⟨q⟩=4$. Shown are data for $N={10}^2$ (squares), $N={10}^4$ (triangles), and $N={10}^6$ (stars). Lines are guides to the eye. The finite-size critical temperature (apparent from the jump in $C$) rapidly converges for large $N$, to $k_B{T}_c/J\approx 2.92$ [Eq. (6)]. Clearly, the maximum in $C$ precedes $T_c$ and the jump in $C$ tends to close for large $N$.

Figure 4

Specific heat from the cavity method for BA networks, with $\mu =1/3$ and $⟨q⟩=4$, for $N={10}^2$ (circles), $N={10}^3$ (squares), and $N={10}^5$ (stars). A linear $T$ dependence develops for large $N$ and meets the high-$T$ background with a jump in slope at $T_c$. The finite-size critical temperature converges to $k_B{T}_c/J\approx 4.21$ [Eq. (6)].

Figure 5

Specific heat from MCS of BA networks with $N=5600$, for $\mu =1/3$ and $⟨q⟩=10$. The critical temperature in the large-$N$ limit is at $k_B{T}_c/J\approx 11.33$ [Eq. (6)].