Totally Asymmetric Simple Exclusion Process on Networks

We study the totally asymmetric simple exclusion process (TASEP) on complex networks, as a paradigmatic model for transport subject to excluded volume interactions. Building on TASEP phenomenology on a single segment and borrowing ideas from random networks we investigate the effect of connectivity on transport. In particular, we argue that the presence of disorder in the topology of vertices crucially modifies the transport features of a network: irregular networks involve homogeneous segments and have a bimodal distribution of edge densities, whereas regular networks are dominated by shocks leading to a unimodal density distribution. The proposed numerical approach of solving for mean-field transport on networks provides a general framework for studying TASEP on large networks, and is expected to generalize to other transport processes.

Figure 1

An example of a Poissonian network of $M=30$ junctions with average connectivity $c=1$ and its strongly connected component (bold). Every segment consists of $L$ sites on which particles undergo TASEP dynamics. At the junctions the particles choose one of $k_v^{\mathrm{out}}$ outlets.

Figure 2

The average segment current $J$ as a function of the overall density $\rho$ on a Bethe network of degree $c$ superposing numerical mean-field predictions (closed symbols), explicit simulations ($N_V=80$ junctions, $L=100$ sites per edge, open symbols), and a one-vertex analytical result Eq. (6) (solid lines). The dashed lines delimit the phases for $c=2$.

Figure 3

The current-density relation of the strongly connected component of a Poissonian network of average connectivity $c$, $N_V=200$ junctions. Simulations (markers, segments of $L=100$ sites) coincide with mean field results for a given graph instance, resolving sample to sample fluctuations (of the order of the symbols size). For comparison, the Bethe result Eq. (6) is shown for the same $c$ (dotted lines).

Figure 4

Distribution of segment properties for networks of connectivity $c=10$, comparing a Poisson network (black) to a Bethe network (dashed red). Symbols show simulation results, lines are mean-field predictions. (a) Distribution of segment densities $W_S({\rho }_s)$ for an overall density $\rho =0.3$ (inset: $\rho =0.7$). Convergence of the density peak to a delta peak is very slow in simulations due to collective fluctuations in the shock phase, closed markers correspond to fourfold longer runs than open ones. (b) Distribution of segment currents $W(J_s)$ for $\rho =0.3$. (c) Fraction of segments in high ($n_{\mathrm{HD}}$) and low ($n_{\mathrm{LD}}$) density segments as a function of the overall density $\rho$.