复杂系统、复杂网络自相似结构的涌现规律

The Law of Emergence of Self-similar Structures in Complex Systems and Complex Networks

在现实世界的复杂系统中,树状分形广泛存在。最近对于复杂网络的研究又揭示大量网络为自相似结构,随之而来的问题是:自相似结构是如何形成的。本文结合复杂系统与复杂网络,给出一个树状生长模型,通过论证揭示:生长过程及自相似结构的涌现,可集中由简单的幂律体现:幂律是自组织形成的临界状态,在它的支配下,系统得以保持有序演化发展,并涌现层层相似的自相似结构,其分形维数或相应的指数,是系统功能的度量。复杂寓于简单,简单形式的幂律蕴含了自组织演化发展并涌现自相似结构的全部过程的信息,是支配现实世界广泛现象的动力学。

In the real world,tree fractal structure exists in wide-spreading complex systems.Recently,the exploring of complex networks reveals that a vast amount of complex networks possess self-similar structure.The subsequent question is how the self-similar structure is formed and what the evolving process is.In this paper,considering both the complex systems and complex networks,we present a model of growing tree.From the argumentations given by this model,it shows that the evolving process and the emergence of self-similar structure can be completely expounded by a simple exponent law.This law is the self-organized critical state which governs the ordered evolving process and leads to the emergence of the layer-by-layer similar structure.The dimension of the tree fractal or the similar exponent is a measure of the function of system.Complexity resides in simplicity.The simple power law implies all of the information of the self-organized evolving process and the emergence of similar structure.It is an universal dynamics to govern the wide phenomena in the real world.