中文摘要：

最近，Li等研究了在Kleinberg导航模型中引入总能量／l=cⅣ约束后的最优导航问题，其中4为网络中所有长程连边的长度之和，C为正常数，Ⅳ为网络节点总数．他们通过在1维和2维导航模型中的模拟结果推测，在有限能量约束下Kleinberg导航模型中按照幂律方式添加长程连边的最优幂指数应该是α=d＋1，其中d为导航模型的维数．本文在平均场理论下，建立了2维有限能量约束下的导航过程的动态微分方程，通过对该方程进行数学分析以及数值求解，从理论上证明了当网络规模足够大且总能量相对较小时，2维有限能量约束下的最优导航幂指数确实为α=3，这一结果证实了Li等之前的推测．

英文摘要：

Recently, a certain total energy constraint A = cN was introduced into the Kleinberg＇s navigation model, where A 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 α = d＋ 1, where d is the dimension of the underlying lattice. Based on the mean field theory, the navigation process on the two-dimensional energy constrained navigation model is described by dynamical equations in this paper. 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 α = 3 for the two-dimensional case. Our results can perfectly correspond to simulations reported previously.