Figure 1

(Color online) Schematic representation of our model network. Nodes are distributed uniformly on the unit square. Their connectivity (represented by node diameter) follows a power-law distribution. Flow through the network derives from a potential $P$ that varies between 1 (left boundary) and 0 (right boundary).Reuse & Permissions

Figure 2

(Color online) (a) Pressure distributions, $\Phi \left(P\right)$, for scale-free networks with different degree exponents, $\gamma$, as well as for an E-R network. The dashed line is a Gaussian fit with mean $\overline{P}=0.5$ and standard deviation ${\sigma }_P\approx 0.08$. We use networks of size $N=8100$ nodes, and the results are averaged over 250 realizations. (b) Velocity distributions, $\Phi \left(u\right)$, for different network types. Inset: exponent $\nu$ of the velocity distribution power-law tail plotted against the connectivity exponent $\gamma ∊\left[2,3.2\right]$; shown also is a least-squares linear interpolation $\nu =\left(0.99\pm 0.02\right)\gamma +\left(1.03\pm 0.05\right)$.Reuse & Permissions

Figure 3

(Color online) The particle MSD follows a power law in time during early times, before it is influenced by boundary effects (inset). The main plot shows the scaling exponent $\beta$ of the particle MSD, for advection-dominated transport $\left(A⪢1\right)$ in scale-free networks, plotted against the connectivity degree exponent $\gamma$. Transport is anomalous and falls in the superdiffusive regime $\left(\beta >1\right)$. Shown also are the values of $\beta$ for an E-R network—for which transport is also superdiffusive—and a regular lattice–for which transport is normal, $\beta =1$. We use networks with $N=8100$ nodes and set $A=7$ and $\epsilon =0.0014$. We use 3000 particles and we average over 100 realizations. Within each realization, we construct the network, solve the flow potential equation, and advance particles by advection and diffusion, according to the flow field from the potential solution. This makes our model significantly more computationally expensive than the diffusive random-walk algorithms of [17, 18], in which a walker jumps directly from node to node. We have confirmed that our results are independent of network size for $N>6000$.Reuse & Permissions

Figure 4

(Color online) Purely diffusive transport $\left(A⪡1\right)$ in scale-free networks leads to linear scaling of the mean-square displacement $\left(\beta \approx 1\right)$. As $A$ increases, we observe a transition from normal transport to anomalous transport, with scaling exponents, $\beta$, that reach different asymptotic values in the superdiffusive regime, depending on the network topology (Fig. 3). We use networks with $N=8100$ nodes, fixed $\epsilon =0.04$, and 3000 particles. Simulation results are averaged over 100 realizations.Reuse & Permissions

Figure 5

(Color online) (a) Marginal distribution of waiting times for advection-dominated transport and different scale-free networks. Insets: joint distribution $\psi \left(d,\tau \right)$ for two different scale-free networks ($\gamma =2.0$ and 4.8). Bright red color corresponds to high value of $\psi \left(d,\tau \right)$. Link length and transition time exhibit strong coupling for $\gamma =2.0$, for which the joint distribution collapses around the scaling $\tau \sim d^2$. The results are for network size $N=8100$ nodes and over 300 realizations. (b) MSD for coupled and unidirectional CTRW, using data from different network types. Bottom-right inset: MSD for coupled and uncoupled biased, 1D CTRW. Top-left inset: MSD from network simulations compared with the results from 1D coupled CTRW, and 1D RW with back-flow and linear interpolation along the jumps. By backflow we mean that the velocity along links in the network simulations is predominantly in the positive $x$ direction, but the flow is “backward” for some links. The 1D random-walk simulations with backflow are based on taking a fraction of the jumps in the negative $x$ direction, to better approximate the actual 2D network simulations. In the classical CTRW framework, particles are assumed to ‘wait’ at a given location until they jump to a new one. By interpolation we mean that we post-compute the MSD at a given time as if the particles had experienced a constant velocity between jumps—a behavior that more closely resembles the setting of the 2D particle tracking network simulations.Reuse & Permissions