Continuous-time random walks on networks with vertex- and time-dependent forcing

We have investigated the transport of particles moving as random walks on the vertices of a network, subject to vertex- and time-dependent forcing. We have derived the generalized master equations for this transport using continuous time random walks, characterized by jump and waiting time densities, as the underlying stochastic process. The forcing is incorporated through a vertex- and time-dependent bias in the jump densities governing the random walking particles. As a particular case, we consider particle forcing proportional to the concentration of particles on adjacent vertices, analogous to self-chemotactic attraction in a spatial continuum. Our algebraic and numerical studies of this system reveal an interesting pair-aggregation pattern formation in which the steady state is composed of a high concentration of particles on a small number of isolated pairs of adjacent vertices. The steady states do not exhibit this pair aggregation if the transport is random on the vertices, i.e., without forcing. The manifestation of pair aggregation on a transport network may thus be a signature of self-chemotactic-like forcing.

Figure 1

(a) Concentration of particles at $t=10000$ on network and (b) distribution of concentration, for a 50 vertex Watts-Strogatz network with rewiring probability $p=0.05$ and exponential waiting time densities.

Figure 2

(a) Concentration of particles at $t=10000$ on network and (b) distribution of concentration, for a 50 vertex Watts-Strogatz network with rewiring probability $p=0.05$ and with vertex-dependent exponential waiting time densities.

Figure 3

Number of pairs of vertices with high concentration on a 50 vertex Watts-Strogatz network with rewiring probability $p=0.05$ and exponential waiting time densities.

Figure 4

(a) Concentration of particles at $t=400$ on network and (b) distribution of concentration, for a 50 vertex Watts-Strogatz network with rewiring probability $p=0.05$ and Mittag-Leffler waiting time densities. (c) The concentration as a function of time on vertices 4 (continuous line) and 6 (dashed line). These concentrations are normalized to the concentration at $t=10$.

Figure 5

(a) Concentration of particles at $t=40$ on network and (b) distribution of concentration, for a 50 vertex Watts-Strogatz network with rewiring probability $p=0.05$ with $\beta =0$. (c) The concentration as a function of time on vertices 4 (continuous line), 10 (small-dashed line), and 13 (large-dashed line). These concentrations are normalized to the concentration at $t=1$. Vertex 4 has an Mittag-Leffler waiting density, while the rest of the vertices have identical exponential waiting time densities.

Figure 6

(a) Concentration of particles at $t=400$ on network and (b) distribution of concentration, for a 50 vertex Watts-Strogatz network with rewiring probability $p=0.05$ with $\beta =1$. (c) The concentration as a function of time on vertices 4 (continuous line) and 5 (dashed line). These concentrations are normalized to the concentration at $t=10$. Vertex 4 has an Mittag-Leffler waiting density, while the rest of the vertices have identical exponential waiting time densities.