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Uniform Preferential Selection Model for Generating Scale-free Networks
Methodology and Computing in Applied Probability ( IF 0.9 ) Pub Date : 2021-04-29 , DOI: 10.1007/s11009-021-09854-w
Raheel Anwar , Muhammad Irfan Yousuf , Muhammad Abid

It has been observed in real networks that the fraction of nodes P(k) with degree k satisfies the power-law P(k) ∝ kγ for k > kmin > 0. However, the degree distribution of nodes in these networks before kmin varies slowly to the extent of being uniform as compared to the degree distribution after kmin. Most of the previous studies focus on the degree distribution after kmin and ignore the initial flatness in the distribution of degrees. In this paper, we propose a model that describes the degree distribution for the whole range of k > 0, i.e., before and after kmin. The network evolution is made up of two steps. In the first step, a new node is connected to the network through a preferential attachment method. In the second step, a certain number of edges between the existing nodes are added such that the end nodes of an edge are selected either uniformly or preferentially. The model has a parameter to control the uniform or preferential selection of nodes for creating edges in the network. We perform a comprehensive mathematical analysis of our proposed model in the discrete domain and prove that the model exhibits an asymptotically power-law degree distribution after kmin and a flat-ish distribution before kmin. We also develop an algorithm that guides us in determining the model parameters in order to fit the model output to the node degree distribution of a given real network. Our simulation results show that the degree distributions of the graphs generated by this model match well with those of the real-world graphs.



中文翻译:

生成无尺度网络的统一优先选择模型

它已在实际网络中观察到的节点的分数Pķ)与程度ķ满足幂律Pķ)α ķ - γķ > ķÑ > 0。然而,在节点的度分布与k m i n之后的度分布相比,这些网络在k m i n之前缓慢变化,达到均匀的程度。先前的大多数研究都集中于k m i n之后的学位分布。并忽略度分布中的初始平坦度。在本文中,我们提出了一个模型,该模型描述了k > 0的整个范围(即,在k m i n之前和之后)的度分布。网络演进由两个步骤组成。第一步,通过优先连接方法将新节点连接到网络。在第二步中,在现有节点之间添加一定数量的边缘,以使边缘的末端节点被均匀或优先选择。该模型具有一个参数,用于控制节点的均匀或优先选择,以在网络中创建边缘。我们在离散域中对我们提出的模型进行了全面的数学分析,证明了该模型在k m i n之后表现出渐近幂律度分布,在k m i n之前表现出平坦分布。。我们还开发了一种算法,指导我们确定模型参数,以使模型输出适合给定真实网络的节点度分布。我们的仿真结果表明,由该模型生成的图的度分布与实际图的度分布非常匹配。

更新日期:2021-04-30
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