Local Dimension of Complex Networks

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. ∗ [email protected]. † [email protected]. ‡ http://cyvision.ifsc.usp.br/cyvision/ 1 ar X iv :1 20 9. 24 76 v3 [ ph ys ic s. so cph ] 1 5 A ug 2 01 3 Dimension is one of the most basic concepts in Physics. Diffusion[1, 2], waves propagation[3], Brownian motion[4], as well as many other physical processes are highly influenced by the dimension in which those phenomena take place. Dimensionality also allows us to quantify the degrees of freedom in a system, as well to characterize the macroscopic dynamics on complex systems by means of statistical mechanics[5]. Representations of complex systems by complex networks[6–8] has proven to be generally successful to describe their various features without losing their intrinsic complexity. Important physical dynamics, like diffusion and information propagation can take place in such structures. Surprisingly, not much attention has been given to characterizing the dimensionality of complex networks. Some early attempts on describing the dimensionality of networks[9–11] took into account well-known regular lattices and small graphs. Characterization of the dimensionality of complex networks was first introduced by Csányi[12], and was further developed by Gastner and Newman[13]. They presented a flexible way to calculate the dimension of arbitrary networks in terms of the scaling property of the topological volume. In another work, Shanker[14] generalized this concept by developing a new and mathematically coherent definition of global dimension for complex networks based on the Riemann zeta function for graphs. He also proves that unbounded random and small-world networks present infinite dimension[15]. Recently, Daqing et al [16] introduced three novel methods to obtain the dimensionality of networks, which yield the same results even for distinct dynamics such as diffusion, random walks and percolation. They found that the dimension values depend neither on the size nor on the average degree of networks. The dimension measurements proposed recently[13, 16], provide good insights about the global dimensional structure of networks, but cannot characterize the nodes individually. In his paper, Gastner commented that the dimensionality may change considerably among the vertices of a network, therefore a local dimension measurement should be necessary to better characterize such systems, however no further works explored the dimensional features of individual vertices. Interdependent networks, for example, may encompass networks with distinct dimensions. In this case the measurement of global dimension does not represent all nodes in the network. In this paper we further investigate the use of dimensionality measurements to characterize two power grid networks, namely: the continental european network (EU power grid) and the western states power grid of the United States (US power grid). To characterize the