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

摘要

Many real networks are embedded in space, and often the distribution of the link lengths r follows a powernlaw, p(r) ∼ rn−δ . Indications that such systems can be characterized by the concept of dimension were foundnrecently. Here, we present further support for this claim, based on extensive numerical simulations of modelnnetworks with a narrow degree distribution, embedded in lattices of dimensions de = 1 and de = 2. For networksnwith δ < de, d is infinity, while for δ > 2de, d has the value of the embedding dimension de. In the intermediatenregime of interest de u0002 δ < 2de, our numerical results suggest that d decreases continuously from d =∞ tonde, with d − de ∝ (2 − δnu0005)/[δnu0005(δnu0005 − 1)] and δnu0005 = δ/de. We also analyze how the mass M and the Euclideanndistance r increase with the topological distance u0003 (minimum number of links between two sites in the network).nOur results suggest that in the intermediate regime de u0002 δ < 2de, M(u0003) and r(u0003) increase with u0003 as a stretchednexponential,M(u0003) ∼ exp[Adu0003δnu0005(2−δnu0005)] and r(u0003) ∼ exp[Au0003δnu0005(2−δnu0005)], such thatM(u0003) ∼ r(u0003)d. Forδ < de ,M increasesnexponentially with u0003 (as known for δ = 0), while r is constant and independent of u0003. For δ u0003 2de, we find thenexpected power-law scaling, M(u0003) ∼ u0003du0003 and r(u0003) ∼ u00031/dmin , with du0003dmin = d. In de = 1, we find the expectednresult, du0003 = dmin = 1, while in de = 2 we find surprisingly that although d = 2, du0003 > 2 and dmin < 1, in contrastnto regular lattices.