Open AccessTheoretical analysis on optimal navigation with total energy restriction in a two-dimensional lattice
Author(s) -
Yong Li,
Dou Fei-Ling,
Ying Fan,
Zengru Di
Publication year - 2012
Publication title -
wuli xuebao
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.199
H-Index - 47
ISSN - 1000-3290
DOI - 10.7498/aps.61.228902
Subject(s) - exponent , lattice (music) , constraint (computer aided design) , statistical physics , dimension (graph theory) , power law , total energy , physics , range (aeronautics) , computer science , mathematics , combinatorics , geometry , statistics , psychology , philosophy , linguistics , materials science , acoustics , displacement (psychology) , composite material , psychotherapist
Recently, a certain total energy constraint =cN was introduced into the Kleinberg's navigation model, where is the total length of the long-range connections, c is a positive constant and N is the network size. The simulation results obtained in the one and two-dimensional cases indicate that with total cost restricted the optimal power-law exponent for adding extra long-range links between any two nodes seems to be =d+1, where d is the dimension of the underlying lattice in this paper. Based on mean field theory, the navigation process on the 2-dimensional cost constrained navigation model can be described by dynamical equations. Based on our theoretical analysis and the numerical results of the dynamical equations, we prove that for large networks and comparatively small total energy, the optimal power-law exponent is =3 for the two-dimensional case. Our results can perfectly correspond to simulations reported previously.