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Investigating analytical solution for navigation networks with cost constraint. Presenting exact solution with power-law exponent findings. Navigation dynamics equations discussed. Future work includes exploring 2-dimensional case.
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Analytical solution for optimal navigation with total cost restriction Yong Li, Yanqing Hu et al.
Navigation is important --- Kleinberg’s navigation model has been widely studied. ---S. Carmi et al., PRL,(2009) ---C. Caretta Cartozo et al., PRL,(2009) • Limited total cost is much more realistic --- Cost constraint is a crucial ingredient in the design and development of efficient navigation networks. ---G. Li et al. , PRL, (2010)
G. Li et al. simulate Kleinberg navigation model with limited total cost in the 1-dimensional and 2-dimensional cases. They suggest that the optimal power law exponent is a=d+1. • So far, analytical solution for optimal navigation under limited total cost has not been investigated. • We give the exact solution and prove that the optimal power-law exponent is -2 in the 1-dimensional case.
How to describe the navigation process mathematically? ---the dynamical equations of navigation under limited total cost are given below:
