In this paper, we proposed an ungrowing scale-free network model, indicating the growth may not be a necessary condition of the self-organization of a network in a scale-free structure. The analysis shows that the degree distributions of the present model can varying from the Poisson form to the power-law form with the decrease of a free parameter α. This model provides a possible mechanism for the evolution of some scale-free networks with fixed size, such as the friendship networks of school children and the functional networks of the human brain. [12] . The ubiquity of complex networks and the discoveries of the common network properties inspire scientists to construct general models. In the simplest way, these models can be categorized into two classes, the growing models and the ungrowing ones. In the growing models, the number of nodes in network grows continuously, while in ungrowing cases, the number of nodes is fixed. The Watts-Strogatz (WS) model is the pioneer and representative one of ungrowing models, which can be constructed by starting with a regular network and randomly moving one endpoint of each edge with probability p [11]. WS networks display small-world effect, that is, they are highly clustered and of small average distance. The most successful one of growing models is the Barabási-Albert (BA) model [12] , which suggests that two main ingredients of the self-organization of a network in a scale-free structure are growth and preferential attachment. These points to the facts that most networks grow continuously by adding new nodes, which are preferentially attached to existing nodes with a large number of neighbours. Thus far, various proposals for dynamical evolution of scale-free networks has been introduced. Roughly, these models can be classified into three main scenarios. One is under the preferential attachment mechanism [12] [13] [14] [15] [16] [17] [18] , and a related
Preferential attachment in rewiring has been studied on a network of fixed size with the interesting conclusion that a power law degree distribution can be achieved without a growing network REF .
Scale-free networks without growth
Physica A 387, 1683-1688 (2008)
Mathematics Physics