染色问题的网络特性

Network Characters of Coloring Problems

染色问题是约束满足问题的一个经典问题。通过分析染色问题的经典实例,发现染色问题具备复杂网络中常见的"小世界"特性,即染色问题所构成的网络中,任意两个节点之间的平均路径长度很小,整个系统呈现出高聚集度的特性,以及节点度的特异分布。这些特性是随机图所不具备的,因此,以随机图做染色问题测试集的传统方法是不完善的。在实验中同时发现,染色问题中最小染色数的大小与系统聚集度的大小密切相关,随聚集度的增大,呈指数关系增大。

Coloring Problem is a classical Constrain Satisfaction Problem. By analyzing the benchmark examples of Coloring Problem, 'Small World' characters are found, which are quite common in complex networks. In Small World networks, the average path lengths between each two nodes are small, while the whole system express high cluster tendency and the distributions of node degrees are skew. All these properties are lacked by Random Graphs, and thus we questioned completeness of the traditional method of utilizing random graph as benchmarks. Through the experimental results, the Least Coloring Number is found to have close relationships with the clustering coefficient of the network. With the raise of the clustering coefficient, the least coloring number increases exponentially.