School of Architecture, Southeast University, 2 Sipailou, Nanjing, 210096, China.
Key Laboratory of Urban and Architectural Heritage Conservation (Southeast University), Ministry of Education, Nanjing, China.
Sci Rep. 2021 Jan 8;11(1):17. doi: 10.1038/s41598-020-79507-4.
The Erdős-Rényi (ER) random graph G(n, p) analytically characterizes the behaviors in complex networks. However, attempts to fit real-world observations need more sophisticated structures (e.g., multilayer networks), rules (e.g., Achlioptas processes), and projections onto geometric, social, or geographic spaces. The p-adic number system offers a natural representation of hierarchical organization of complex networks. The p-adic random graph interprets n as the cardinality of a set of p-adic numbers. Constructing a vast space of hierarchical structures is equivalent for combining number sequences. Although the giant component is vital in dynamic evolution of networks, the structure of multiple big components is also essential. Fitting the sizes of the few largest components to empirical data was rarely demonstrated. The p-adic ultrametric enables the ER model to simulate multiple big components from the observations of genetic interaction networks, social networks, and epidemics. Community structures lead to multimodal distributions of the big component sizes in networks, which have important implications in intervention of spreading processes.
埃尔德什-雷尼（Erdős-Rényi）随机图 G(n,p) 分析了复杂网络中的行为。然而，要拟合实际观测结果，需要更复杂的结构（例如多层网络）、规则（例如 Achlioptas 过程），以及在几何、社会或地理空间上的投影。p-进数系统为复杂网络的层次结构提供了自然的表示形式。p-进随机图将 n 解释为 p-进数集合的基数。构建层次结构的广阔空间相当于组合数序列。尽管巨型组件在网络的动态演化中至关重要，但多个大组件的结构也很重要。将最大组件的大小拟合到经验数据的情况很少见。p-进超度量使 ER 模型能够从遗传相互作用网络、社交网络和传染病的观测中模拟多个大组件。社区结构导致网络中大组件大小的多峰分布，这对传播过程的干预具有重要意义。
