  Dimensionality is one of the most important properties of complex physical
systems. However, only recently this concept has been considered in the context
of complex networks. In this paper we further develop the previously introduced
definitions of dimension in complex networks by presenting a new method to
characterize the dimensionality of individual nodes. The methodology consists
in obtaining patterns of dimensionality at different scales for each node,
which can be used to detect regions with distinct dimensional structures as
well as borders. We also apply this technique to power grid networks, showing,
quantitatively, that the continental European power grid is substantially more
planar than the network covering the western states of US, which present
topological dimension higher than their intrinsic embedding space dimension.
Local dimension also successfully revealed how distinct regions of network
topologies spreads along the degrees of freedom when it is embedded in a metric
space.
