Exact Solution for Optimal Navigation with Total Cost Restriction

TL;DR

This paper analytically determines the optimal power-law exponent for long-range connections in 1D networks under a total cost constraint, providing exact solutions and confirming results with simulations.

Contribution

It offers an exact analytical solution for the optimal navigation strategy in 1D networks with total cost restrictions, extending previous simulation-based findings.

Findings

Optimal exponent for 1D networks is 2

Exact solutions for time cost as a function of exponent

Analytical results match simulation data

Abstract

Recently, Li \textit{et al.} have concentrated on Kleinberg's navigation model with a certain total length constraint $\Lambda = cN$, where $N$ is the number of total nodes and $c$ is a constant. Their simulation results for the 1- and 2-dimensional cases indicate that the optimal choice for adding extra long-range connections between any two sites seems to be $\alpha=d+1$, where $d$ is the dimension of the lattice and $\alpha$ is the power-law exponent. In this paper, we prove analytically that for the 1-dimensional large networks, the optimal power-law exponent is $\alpha=2$ Further, we study the impact of the network size and provide exact solutions for time cost as a function of the power-law exponent $\alpha$. We also show that our analytical results are in excellent agreement with simulations.