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 (minimum number of links between two sites in the network). Our results suggest that in the intermediate regime dₑ≤δ<2dₑ, M() and r() increase with as a stretched exponential, M()~exp[Ad^δ^'(2-δ^')] and r()~exp[A^δ^'(2-δ^')], such that M()~r()ᵈ. For δ<dₑ, M increases exponentially with (as known for δ=0), while r is constant and independent of . For δ≥2dₑ, we find the expected power-law scaling, M()~^d_ and r()~^1/dₘᵢₙ, with d_dₘᵢₙ=d. In dₑ=1, we find the expected result, d_=dₘᵢₙ=1, while in dₑ=2 we find surprisingly that although $d=2$, d_>2 and dₘᵢₙ<1, in contrast to regular lattices.