Kleinberg Navigation on Anisotropic Lattices

Figure 1

Simulations for lattice and angle anisotropies. A horizontal scale of 1/ln²𝐿 is used throughout. All curves approach 𝛼(∞), regardless of 𝑏. For the lattice case, there is a crossover effect where curves for 𝑏>1 dip below the 𝑏=1 curve. For the angular case, curves for 𝑏>1 approach the infinite limit at differing rates, while curves for 𝑏<1 eventually collapse onto the 𝑏=1 curve. These phenomena are further explored in Figure 3. See Figure 2 for the extrapolated 𝛼(∞).

(a) Lattice

(b) Angle