Abstract: An exclusively mathematical approach is proposed to address the scale-free property of selforganized networks of cities. Defining a spatial correlation function based on the triangular lattice model of cities stemming from central place networks, a scaling relationship is established between yardstick (r) and city-city correlation density (C(r)) such as C(r)∝r^D, where D is the scaling factor. The scaling factor is in fact the spatial correlation dimension if only 1≤D≤2. A self-organized network of cities is regarded as scale-free system provided there exists a linear range on the logr-logC(r) plot. The Henan cities of China are taken as an example to show how to fit the observed data to the correlation function for calculating the scaling factor, and to illustrate the process of spatial complication of urban evolution. The results indicate that urban systems and the traffic networks based on urban systems are scale-free networks, and the network of cities can be characterized with the power-law of yardstick and correlation density relation.

Key words: urban system, transport network, fractal cities, spatial complexity, scale-free network, symmetry breaking

摘要： 基于城市体系的三角点阵模型提出了自组织城市网络空间结构无标度性的数学定义和描述方法：假定区域城市是三角点阵格局上的无规分布，构造城市两两之间的空间关联函数；只要城市关联密度与码尺之间满足幂指数关系，就表明城市体系是无标度网络，可以用标度指数进行度量。以河南省为研究区，对城市空间关联进行了实证分析，发现在双对数坐标图上存在明确的无标度区，从而在揭示自组织城市网络内在复杂规律的同时，证实了上述方法刻画无标度城市网络的有效性。

关键词: 城市体系, 交通网络, 分形城市, 空间复杂性, 无标度网络, 对称破缺