Complex networks embedded in space: Dimension and scaling relations between mass, topological distance, and Euclidean distance

Many real networks are embedded in space, and often the distribution of the link lengths r follows a power law, p(r)∼r−δ. Indications that such systems can be characterized by the concept of dimension were found recently. Here, we present further support for this claim, based on extensive numerical simulations of model networks with a narrow degree distribution, embedded in lattices of dimensions de=1 and de=2. For networks with δ<de, d is infinity, while for δ>2de, d has the value of the embedding dimension de. In the intermediate regime of interest de≤δ<2de, our numerical results suggest that d decreases continuously from d=∞ to de, with d−de∝(2−δ′)/[δ′(δ′−1)] and δ′=δ/de. We also analyze how the mass M and the Euclidean distance r increase with the topological distance ℓ (minimum number of links between two sites in the network). Our results suggest that in the intermediate regime de≤δ<2de, M(ℓ) and r(ℓ) increase with ℓ as a stretched exponential, M(ℓ)∼exp[Adℓδ′(2−δ′)] and r(ℓ)∼exp[Aℓδ′(2−δ′)], such that M(ℓ)∼r(ℓ)d. For δ<de, M increases exponentially with ℓ (as known for δ=0), while r is constant and independent of ℓ. For δ≥2de, we find the expected power-law scaling, M(ℓ)∼ℓdℓ and r(ℓ)∼ℓ1/dmin, with dℓdmin=d. In de=1, we find the expected result, dℓ=dmin=1, while in de=2 we find surprisingly that although d=2, dℓ>2 and dmin<1, in contrast to regular lattices.