Article,
Decentralized navigation of multiple packages on transportation networks
PHYSICAL REVIEW E, (2018)
DOI: 10.1103/PhysRevE.98.032306
Abstract
We investigate by numerical simulation and finite-size analysis the
impact of long-range shortcuts on a spatially embedded transportation network. Our networks are built from two-dimensional (d = 2) square
lattices to be improved by the addition of long-range shortcuts added
with probability P (r(ij)) similar to r(ij)(-alpha) J. M. Kleinberg,
Nature 406, 845 (2000). Considering those improved networks, we
performed numerical simulation of multiple discrete package navigation
and found a limit for the amount of packages flowing through the
network. Such a limit is characterized by a critical probability of
creating packages p(c) where above this value a transition to a
congested state occurs. Moreover, p(c) is found to follow a power law,
p(c) similar to L-gamma, where L is the network size. Our results
indicate the presence of an optimal value of alpha(min) approximate to
1.7, where the parameter gamma reaches its minimum value and the
networks are more resilient to congestion for larger system sizes.
Interestingly, this value is close to the analytically found value of
alpha for the optimal navigation of single packages in spatially embedded networks, where alpha(opt) = d. Finally, we show that the power
spectrum for the number of packages navigating the network at a given
time step t, which is related to the divergence of the expected delivery
time, follows a universal Lorentzian function, regardless of the
topological details of the networks.
