Front Propagation on Oriented Graphs

The probabilities that a particle, starting from any node on the backbone, jumps to its downstream neighbor or to one of its three upstream neighbors are, respectively. 

To obtain the expression for the waiting time PDF we proceed as in Sect. 6.3, see the part corresponding to the comb structure with I = a. The Laplace transform of the waiting time PDF is given by 

The jump length distribution for a walker moving along the backbone of the network, segment AB in Fig. 6.6, is w(x) = Pout (- — 0 + Pm ix. + 1), where / is the distance between two consecutive nodes. Equations (5.30) and (4.46) yield the following expression for the front velocity  

Results for the front velocity, determined numerically from (6.45), are shown in Fig. 6.7, where the dimensionless front velocity is plotted as a function of the dimensionless growth rate rt. 

The front velocity is a monotonically increasing function of rr. As expected, it approaches asymptotically the advection velocity along the backbone l/r, since the front velocity cannot exceed the maximum particle velocity. The inset shows that the front velocity does not depend linearly on the bias b, which implies that the front 

---