Figure 3 | Scientific Reports

Figure 3

From: The Polynomial Volume Law of Complex Networks in the Context of Local and Global Optimization

Figure 3

Hierarchical spatial networks. In (bc) and (ef) the arithmetic means of the volumes for 10,000 arbitrary chosen nodes are depicted. The estimated dimension including the standard deviation is given. (a) Mocnik model with 13 nodes and ρ = 1.5. (b) Volumes of the undirected network associated to a Mocnik model with 10,000 nodes in two-dimensional space and ρ = 1.8. (c) Volumes in the public transport network of Sweden41, which is a multi-modal and hierarchical network. The data is fitted by the polynomial volume law. (d) Hierarchical Mocnik model, with the base layer depicted in grey and layer 1, in black. (e) Volumes of the undirected network associated to two-dimensional hierarchical Mocnik models with ρ = 1.8. (In fact, the value of ρ is below 2 for most real-world networks.) The following hierarchies are depicted: no hierarchy (N0 = 10000), flat hierarchy (N0 = 10000, N1 = 1000), steep hierarchy (N0 = 10000, N1 = 100), and two-layered hierarchy (N0 = 10000, N1 = 1000, N2 = 100). (f) Volumes of the undirected network associated to two-dimensional weighted hierarchical Mocnik models with ρ = 1.8. The following hierarchies are depicted: no hierarchy (N0 = 10000; w0 = 1), flat hierarchy (N0 = 10000, N1 = 3000; w0 = 1, w1 = 0.375), steep hierarchy (N0 = 10000, N1 = 100; w0 = 1, w1 = 0.25), and two-layered hierarchy (N0 = 10000, N1 = 3000, N2 = 100; w0 = 1, w1 = 0.375, w2 = 0.25).

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