Radicchi Filippo, Vilone Daniele, Meyer-Ortmanns Hildegard
School of Engineering and Science, International University Bremen, P. O. Box 750561, D-28725 Bremen, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Feb;75(2 Pt 1):021118. doi: 10.1103/PhysRevE.75.021118. Epub 2007 Feb 23.
We consider triad dynamics as it was recently considered by Antal [Phys. Rev. E 72, 036121 (2005)] as an approach to social balance. Here we generalize the topology from all-to-all to the regular one of a two-dimensional triangular lattice. The driving force in this dynamics is the reduction of frustrated triads in order to reach a balanced state. The dynamics is parametrized by a so-called propensity parameter p that determines the tendency of negative links to become positive. As a function of p we find a phase transition between different kinds of absorbing states. The phases differ by the existence of an infinitely connected (percolated) cluster of negative links that forms whenever p<or=p(c). Moreover, for p<or=p(c), the time to reach the absorbing state grows powerlike with the system size L, while it increases logarithmically with L for p>p(c). From a finite-size scaling analysis we numerically determine the static critical exponents beta and nu(perpendicular) together with gamma, tau, sigma, and the dynamical critical exponents nu(parallel) and delta. The exponents satisfy the hyperscaling relations. We also determine the fractal dimension d(f) that satisfies a hyperscaling relation as well. The transition of triad dynamics between different absorbing states belongs to a universality class with different critical exponents. We generalize the triad dynamics to four-cycle dynamics on a square lattice. In this case, again there is a transition between different absorbing states, going along with the formation of an infinite cluster of negative links, but the usual scaling and hyperscaling relations are violated.
我们像安塔尔最近在[《物理评论E》72, 036121 (2005)]中所考虑的那样，将三元组动力学视为一种实现社会平衡的方法。在这里，我们将拓扑结构从全对全推广到二维三角形晶格的规则拓扑结构。这种动力学中的驱动力是减少受挫三元组以达到平衡状态。该动力学由一个所谓的倾向参数p来参数化，p决定了负链接变为正链接的趋势。作为p的函数，我们发现了不同吸收态之间的相变。这些相的区别在于当p≤p(c)时会形成一个无限连通（渗流）的负链接簇。此外，对于p≤p(c)，达到吸收态的时间随系统大小L呈幂律增长，而对于p > p(c)，它随L呈对数增长。通过有限尺寸标度分析，我们数值确定了静态临界指数β和ν⊥以及γ、τ、σ，还有动态临界指数ν∥和δ。这些指数满足超标度关系。我们还确定了满足超标度关系的分形维数d(f)。三元组动力学在不同吸收态之间的转变属于具有不同临界指数的普适类。我们将三元组动力学推广到正方形晶格上的四循环动力学。在这种情况下，同样存在不同吸收态之间的转变，伴随着负链接无限簇的形成，但通常的标度和超标度关系被违反。
