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The dimension of a system is one of the most fundamental quantities to characterize its structure and basic physical properties. Diffusion 1 and vibrational excitations 2 , for example, as well as the universal features of a system near a critical point depend crucially on its dimension 3,4 . However, in the theory of complex networks the concept of dimension has been rarely discussed. Here we study models for spatially embedded networks and show how their dimension can be determined. Our results indicate that networks characterized by a broad distribution of link lengths have a dimension higher than that of the embedding space. We illustrate our findings using the global airline network and the Internet and argue that although these networks are embedded in two-dimensional space they should be regarded as systems with dimension close to 3 and 4.5, respectively. We show that the network dimension is a key concept to understand not only network topology, but also dynamical processes on networks, such as diffusion and critical phenomena including percolation.Networks consist of entities (nodes) and their connections (links) 5 . Usually, networks are embedded either in two-or in threedimensional space. For example, airline and Internet networks, as well as friendship networks where the nodes are the residences of friends, are embedded in the two-dimensional surface of the earth, whereas the neuronal network in the brain is embedded in a complex three-dimensional structure. If in a d-dimensional lattice, the links connect only neighbouring nodes (in space), then the dimension of the network is trivially identical to the dimension of the embedding space. In most cases, however, links are not short ranged, their length distribution is broad, connecting also distant nodes. The question we pose here is: Is there a finite dimension that characterizes such a spatially embedded network and how can we determine it? The knowledge of the dimension 6,7 is not only important for a structural characterization of the network, but is also crucial for understanding the function of the network, as the dimension governs the dynamical processes in the network.In network theory, the dimension of a network has been rarely considered. Research 8-13 has focused on two types of networks where the links between the nodes are either short range, connecting only nearby nodes (like a lattice), or long range, connecting any two nodes with the same probability. In the short range case, the dimension d of the network is identical to the dimension of the embedding space, whereas in the long range case the embedding space is irrelevant and the network can be regarded as having an infinite dimension. As a consequence, the mean distance between two nodes scales with the number of nodes N as N 1/d in the short range case, and as log N (ref. 14) or log log N (ref. Many real networks, however, do not fall into these categories, and the lengths of their links are characterized by a broad power law distribution. For example, in a mobile phone communicati...

We study the effects of spatial constraints on the structural properties of networks embedded in one or two dimensional space. When nodes are embedded in space, they have a well defined Euclidean distance r between any pair. We assume that nodes at distance r have a link with probability p(r) ∼ r −δ . We study the mean topological distance l and the clustering coefficient C of these networks and find that they both exhibit phase transitions for some critical value of the control parameter δ depending on the dimensionality d of the embedding space. We have identified three regimes. When δ < d, the networks are not affected at all by the spatial constraints. They are "small-worlds" l ∼ log N with zero clustering at the thermodynamic limit. In the intermediate regime d < δ < 2d, the networks are affected by the space and the distance increases and becomes a power of log N , and have non-zero clustering. When δ > 2d the networks are "large" worlds l ∼ N 1/d with high clustering. Our results indicate that spatial constrains have a significant impact on the network properties, a fact that should be taken into account when modeling complex networks.

We have re-examined the random release of particles from fractal polymer matrices using Monte Carlo simulations, a problem originally studied by Bunde et al. [J. Chem. Phys. 83, 5909 (1985)]. A certain population of particles diffuses on a fractal structure, and as particles reach the boundaries of the structure they are removed from the system. We find that the number of particles that escape from the matrix as a function of time can be approximated by a Weibull (stretched exponential) function, similar to the case of release from Euclidean matrices. The earlier result that fractal release rates are described by power laws is correct only at the initial stage of the release, but it has to be modified if one is to describe in one picture the entire process for a finite system. These results pertain to the release of drugs, chemicals, agrochemicals, etc., from delivery systems.