带权图的多重分形度量

分形维数及多重分形是分形理论的重要研究内容。复杂网络的多重分形已经得到了较为深入的研究,但对复杂网络多重分形的度量目前并没有可行的方法。带权图是复杂网络研究的重要对象,其中的节点权重及边权重可以为正实数、负实数、纯虚数及复数等多种不同的类型。除节点权重及边权重均为正实数的情形外,其他类型的带权图都具有多重分形特性,且均具有无穷多个复数形式的网络维数。通过对带权图多重分形的研究,文中给出了15种具有多重分形特性的带权图多重分形维数的模所构成的集合,并采用集合的势对带权图的多重分形特性进行度量。研究表明,15种带权图多重分形维数的模所构成的集合均是可数集,其中有2种集合是2重集合,另外13种集合是通常意义上的集合,而且所有的集合均是等势的,其势均为■0。

Measure for Multi-fractals of Weighted Graphs

Fractal dimension and multi-fractal are important research contents of fractal theory.The multi-fractal of complex networks has been studied in depth,while there is no feasible method to measure the multi-fractal of complex networks.Weighted graph is an important research object of complex network.Both node weight and edge weight in weighted graphs can be positive real number,negative real number,pure imaginary number and complex number,and so on.Among all types of weighted graphs,except the weighted graphs with both node weight and edge weight being positive real numbers,other types of weighted graphs share multi-fractals and append with infinity complex network dimensions.Through the study of multi-fractals of weighted graphs,this paper presents modulus of infinity complex network dimensions of all 15 weighted graphs that share multi-fractal,and measures multi-fractal of them by cardinality of sets obtained from modulus of infinity complex network dimensions of them.It shows that all sets obtained from modulus of infinity complex network dimensions of weighted graphs share multi-fractal are countable sets,while 2 are multisets,and the other 13 are ordinary sets.Moreover,all sets,regardless of multisets or ordinary sets,are equipotent with cardinality of ■0.