Juhász Róbert
Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O. Box 49, Hungary.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 2):066106. doi: 10.1103/PhysRevE.78.066106. Epub 2008 Dec 16.
A class of cubic networks composed of a regular one-dimensional lattice and a set of long-range links is introduced. Networks parametrized by a positive integer k are constructed by starting from a one-dimensional lattice and iteratively connecting each site of degree 2 with a kth neighboring site of degree 2. Specifying the way pairs of sites to be connected are selected, various random and regular networks are defined, all of which have a power-law edge-length distribution of the form P_{>}(l) approximately l;{-s} with the marginal exponent s=1 . In all these networks, lengths of shortest paths grow as a power of the distance and random walk is superdiffusive. Applying a renormalization group method, the corresponding shortest-path dimensions and random-walk dimensions are calculated exactly for k=1 networks and for k=2 regular networks; in other cases, they are estimated by numerical methods. Although, s=1 holds for all representatives of this class, the above quantities are found to depend on the details of the structure of networks controlled by k and other parameters.
引入了一类由规则的一维晶格和一组长程链接组成的立方网络。通过从一维晶格开始，并迭代地将每个度为2的位点与第k个相邻的度为2的位点相连，构建了由正整数k参数化的网络。通过指定选择要连接的位点对的方式，定义了各种随机和规则网络，所有这些网络都具有形式为(P_{>}(l)\approx l^{-s})的幂律边长度分布，其中边际指数(s = 1)。在所有这些网络中，最短路径的长度随着距离的幂次增长，并且随机游走是超扩散的。应用重整化群方法，精确计算了(k = 1)网络和(k = 2)规则网络的相应最短路径维度和随机游走维度；在其他情况下，通过数值方法进行估计。尽管对于这类的所有代表(s = 1)成立，但发现上述量取决于由k和其他参数控制的网络结构细节。
