Dynamical Properties of the Kleinberg networks

A great number of systems defined as complex consist of
interconnected parts or individual components performing a network or
graph. Communication between the parts is essential for their
existence so that it is necessary a better understanding of their
ability to communicate depending on the amount of information that
transits. The dynamics of package transport in these systems and the
emergence of congestion are problems of high scientific and economic
interest. In this work we investigate the dynamical properties of
transport of packages (informations) between sources and previously
defined destinations, considering different models of spatially
embbeded networks such as lattice and Kleinberg. More precisely, we
study a second-order continuous phase transition from a phase of free
transport to a congestion phase, when the packages are accumulated in
certain regions of the network. By means of a Finite Size Scaling, we
describe this phase transition characterizing its critical exponents.
For 1D and 2D lattice networks, we observe that the critical parameter
$p_c$ scales with exponents approximately $-1$ and $-0.5$ with respect
to the system size. In the case of Kleinberg newtorks where shortcuts
between two nodes $i$ and $j$ are added to the network according to a
probability distibution given by $P(r_ {ij}) \sim r_{ij}^{-\alpha}$,
we show that the best scenario occurs when $\alpha = d$, where $d$ is
the dimention of the topology structure. In this regime, package
traffic were shown to be more resilient to the increase of number of
packages in the network. The confirmation of our result is obtained
not only from direct measure of order parameter, that is, the ratio between undelivered and generated packets, but is
also supported by our analysis of finite size.