  Scaling phenomena have been intensively studied during the past decade in the
context of complex networks. As part of these works, recently novel methods
have appeared to measure the dimension of abstract and spatially embedded
networks. In this paper we propose a new dimension measurement method for
networks, which does not require global knowledge on the embedding of the
nodes, instead it exploits link-wise information (link lengths, link delays or
other physical quantities). Our method can be regarded as a generalization of
the spectral dimension, that grasps the network's large-scale structure through
local observations made by a random walker while traversing the links. We apply
the presented method to synthetic and real-world networks, including road maps,
the Internet infrastructure and the Gowalla geosocial network. We analyze the
theoretically and empirically designated case when the length distribution of
the links has the form P(r) ~ 1/r. We show that while previous dimension
concepts are not applicable in this case, the new dimension measure still
exhibits scaling with two distinct scaling regimes. Our observations suggest
that the link length distribution is not sufficient in itself to entirely
control the dimensionality of complex networks, and we show that the proposed
measure provides information that complements other known measures.
