Anomalous Transport in Scale-Free Networks

To study transport properties of scale-free and Erdős-Rényi networks, we analyze the conductance $G$ between two arbitrarily chosen nodes of random scale-free networks with degree distribution $P(k)\sim k^{-\lambda }$ in which all links have unit resistance. We predict a broad range of values of $G$, with a power-law tail distribution ${\Phi }_{\mathrm{SF}}(G)\sim G^{-g_G}$, where $g_G=2\lambda -1$, and confirm our predictions by simulations. The power-law tail in ${\Phi }_{\mathrm{SF}}(G)$ leads to large values of $G$, signaling better transport in scale-free networks compared to Erdős-Rényi networks where the tail of the conductivity distribution decays exponentially. Based on a simple physical “transport backbone” picture we show that the conductances of scale-free and Erdős-Rényi networks are well approximated by $c{k}_A{k}_B/(k_A+k_B)$ for any pair of nodes $A$ and $B$ with degrees $k_A$ and $k_B$, where $c$ emerges as the main parameter characterizing network transport.

Figure 1

(a) Comparison for networks with $N=8000$ nodes between the cumulative distribution functions for the Erdős-Rényi and the scale-free cases (with $\lambda =2.5$ and 3.3). Each curve represents the cumulative distribution $F(G)$ vs $G$. The simulations have at least ${10}^6$ realizations. (b) Effect of system size on $F_{\mathrm{SF}}(G)$ vs $G$ for the case $\lambda =2.5$. The cutoff value of the maximum conductance $G_{\max }$ progressively increases as $N$ increases.

Figure 2

(a) PDF ${\Phi }_{\mathrm{SF}}(G|k_A,k_B)$ vs $G$ for $N=8000$, $\lambda =2.5$, and $k_A=750$ ($k_A$ is close to the typical maximum degree $k_{\max }=800$ for $N=8000$). (b) Most probable values $G^{*}$, estimated from the maxima of the distributions in Fig. 2a, as a function of the degree $k_B$. The data support a power-law behavior $G^{*}\sim k_B^{\alpha }$ with $\alpha =0.96\pm 0.05$. (c) Scaled most probable conductance $G^{*}/k_B$ vs scaled degree $x\equiv k_A/k_B$ for system size $N=8000$ and $\lambda =2.5$, for several values of $k_A$ and $k_B$: □ ($k_A=8$, $8\le k_B\le 750$), ◇ ($k_A=16$, $16\le k_B\le 750$), △ ($k_A=750$, $4\le k_B\le 128$), ◯ ($k_B=4$, $4\le k_A\le 750$), ▽ ($k_B=256$, $256\le k_A\le 750$), and ⊳ ($k_B=500$, $4\le k_A\le 128$). The curve crossing the symbols is the predicted function $G^{*}/k_B=f(x)=cx/(1+x)$ obtained from Eq. (5). We also show $G^{*}/k_B$ vs scaled degree $x\equiv k_A/k_B$ for Erdős-Rényi networks with $\overline{k}=2.92$, $4\le k_A\le 11$, and $k_B=4$ (●), the curve crossing the symbols representing the theoretical result according to Eq. (5), and an extension of this line to represent the limiting value of $G^{*}/k_B$ (dot-dashed line). The probability to obtain $k_A>11$ is extremely small in Erdős-Rényi networks, and thus we are unable to obtain significant statistics. Scaling function $f(x)$, as seen here, exhibits a crossover from a linear behavior to the constant $c$ ($c=0.87\pm 0.02$ for scale-free networks, horizontal dashed line, and $c=0.55\pm 0.01$ for Erdős-Rényi, dotted line). The inset shows a schematic of the “transport backbone” picture, where the circles labeled $A$ and $B$ denote nodes $A$ and $B$ and their associated links, which do not belong to the transport backbone.

Figure 3

(a) Simulation results for the cumulative distribution $F_{\mathrm{SF}}(G)$ for $\lambda$ between 2.5 and 3.5, consistent with the power law $F_{\mathrm{SF}}\sim G^{-(g_G-1)}$ [cf. Eq. (7)], showing the progressive change of the slope $g_G-1$. (b) The exponent $g_G-1$ from simulations (circles) with $2.5<\lambda <4.5$; shown also is a least squares fit $g_G-1=(1.97\pm 0.04)\lambda -(2.01\pm 0.13)$, consistent with the predicted expression $g_G-1=2\lambda -2$ [cf. Eq. (7)].