Raval Alpan
Keck Graduate Institute of Applied Life Sciences, 535 Watson Drive, Claremont, California 91711, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Dec;68(6 Pt 2):066119. doi: 10.1103/PhysRevE.68.066119. Epub 2003 Dec 30.
Duplication graphs are graphs that grow by duplication of existing vertices, and are important models of biological networks, including protein-protein interaction networks and gene regulatory networks. Three models of graph growth are studied: pure duplication growth, and two two-parameter models in which duplication forms one element of the growth dynamics. A power-law degree distribution is found to emerge in all three models. However, the parameter space of the latter two models is characterized by a range of parameter values for which duplication is the predominant mechanism of graph growth. For parameter values that lie in this "duplication-dominated" regime, it is shown that the degree distribution either approaches zero asymptotically, or approaches a nonzero power-law degree distribution very slowly. In either case, the approach to the true asymptotic degree distribution is characterized by a dependence of the scaling exponent on properties of the initial degree distribution. It is therefore conjectured that duplication-dominated, scale-free networks may contain identifiable remnants of their early structure. This feature is inherited from the idealized model of pure duplication growth, for which the exact finite-size degree distribution is found and its asymptotic properties studied.
复制图是通过现有顶点的复制而增长的图，是生物网络的重要模型，包括蛋白质-蛋白质相互作用网络和基因调控网络。研究了三种图增长模型：纯复制增长，以及两种双参数模型，其中复制构成增长动态的一个元素。发现幂律度分布在所有三种模型中都会出现。然而，后两种模型的参数空间的特征是存在一系列参数值，对于这些参数值，复制是图增长的主要机制。对于处于这种“复制主导”状态的参数值，结果表明度分布要么渐近地趋近于零，要么非常缓慢地趋近于非零幂律度分布。在任何一种情况下，趋近于真实渐近度分布的过程都具有缩放指数对初始度分布属性的依赖性。因此推测，复制主导的无标度网络可能包含其早期结构的可识别残余。这一特征继承自纯复制增长的理想化模型，对于该模型，找到了精确的有限尺寸度分布并研究了其渐近性质。
