Cohen Reuven, Havlin Shlomo
Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan, Israel.
Phys Rev Lett. 2003 Feb 7;90(5):058701. doi: 10.1103/PhysRevLett.90.058701. Epub 2003 Feb 4.
We study the diameter, or the mean distance between sites, in a scale-free network, having N sites and degree distribution p(k) proportional, variant k(-lambda), i.e., the probability of having k links outgoing from a site. In contrast to the diameter of regular random networks or small-world networks, which is known to be d approximately ln(N, we show, using analytical arguments, that scale-free networks with 2<lambda<3 have a much smaller diameter, behaving as d approximately ln(ln(N. For lambda=3, our analysis yields d approximately ln(N/ln(ln(N, as obtained by Bollobas and Riordan, while for lambda>3, d approximately ln(N. We also show that, for any lambda>2, one can construct a deterministic scale-free network with d approximately ln(ln(N, which is the lowest possible diameter.
我们研究无标度网络中节点之间的直径或平均距离，该网络有(N)个节点，度分布(p(k))与(k^{-\lambda})成正比，即从一个节点向外有(k)条边的概率。与规则随机网络或小世界网络的直径已知约为(d\approx\ln(N))不同，我们通过分析论证表明，(2\lt\lambda\lt3)的无标度网络直径要小得多，表现为(d\approx\ln(\ln(N)))。对于(\lambda = 3)，我们的分析得出(d\approx\ln(N)/\ln(\ln(N)))，这与博洛巴斯和里奥丹得到的结果一致，而对于(\lambda\gt3)，(d\approx\ln(N))。我们还表明，对于任何(\lambda\gt2)，都可以构建一个确定性的无标度网络，其(d\approx\ln(\ln(N)))，这是可能的最小直径。
