Jung S, Kim S, Kahng B
Nonlinear and Complex Systems Laboratory, Department of Physics, Pohang University of Science and Technology, Pohang, Kyongbuk 790-784, Korea.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 May;65(5 Pt 2):056101. doi: 10.1103/PhysRevE.65.056101. Epub 2002 Apr 15.
We introduce a deterministic model for scale-free networks, whose degree distribution follows a power law with the exponent gamma. At each time step, each vertex generates its offspring, whose number is proportional to the degree of that vertex with proportionality constant m-1 (m>1). We consider the two cases: First, each offspring is connected to its parent vertex only, forming a tree structure. Second, it is connected to both its parent and grandparent vertices, forming a loop structure. We find that both models exhibit power-law behaviors in their degree distributions with the exponent gamma = 1+ln(2m-1)/ln m. Thus, by tuning m, the degree exponent can be adjusted in the range, 2 < gamma < 3. We also solve analytically a mean shortest-path distance d between two vertices for the tree structure, showing the small-world behavior, that is, d approximately ln N/ln K macro, where N is system size, and k macro is the mean degree. Finally, we consider the case that the number of offspring is the same for all vertices, and find that the degree distribution exhibits an exponential-decay behavior.
我们引入了一种用于无标度网络的确定性模型，其度分布遵循幂律，幂指数为γ。在每个时间步，每个顶点产生其后代，后代数量与该顶点的度成正比，比例常数为m - 1（m > 1）。我们考虑两种情况：第一种，每个后代仅与其父顶点相连，形成树形结构。第二种，它与父顶点和祖父顶点都相连，形成环形结构。我们发现这两种模型在度分布上都呈现幂律行为，幂指数γ = 1 + ln(2m - 1)/ln m。因此，通过调整m，度指数可在2 < γ < 3的范围内进行调节。我们还解析求解了树形结构中两个顶点之间的平均最短路径距离d，显示出小世界行为，即d约为ln N/ln K宏，其中N是系统规模，K宏是平均度。最后，我们考虑所有顶点后代数量相同的情况，发现度分布呈现指数衰减行为。
