1. Analytical solution for optimal navigation with total cost restriction
Yong Li, Yanqing Hu et al.

2. 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)

4. 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.

6. How to describe the navigation process mathematically?
---the dynamical equations of navigation under limited total cost are given below:

10. Future work: the 2-dimensional case

11. Thanks for your patience!