相关随机增长图
Correlated randomly growing graphs
摘要：
我们引入了一个新的相关随机增长图模型，并研究了检测相关性和估计相关结构方面的基本问题。该模型很简单，可以从任何随机增长图的模型开始，例如均匀连接（UA）或优先连接（PA）。给定这样的模型，一对图$(G_1, G_2)$在两个阶段中生长：直到时间$t_{\star}$，它们一起生长（即，$G_1 = G_2$），之后它们根据基本的增长模型独立增长。我们发现，每当种子图有一个影响，在底层的图形增长模型-这已经显示为PA和UA树，并被证明持有广泛-然后相关性可以检测到在这个模型中，即使图形一起生长，只是一个单一的时间步。我们还给出了一个一般的充分条件（适用于PA和UA树），在此条件下，检测是可能的概率去$1$作为$t_{\star} \to \infty$。最后，我们表明PA和UA树的相关性，$t_{\star}$测量的量，可以估计与消失的相对误差为$t_{\star} \to \infty$。
英文摘要：
We introduce a new model of correlated randomly growing graphs and study the fundamental questions of detecting correlation and estimating aspects of the correlated structure. The model is simple and starts with any model of randomly growing graphs, such as uniform attachment (UA) or preferential attachment (PA). Given such a model, a pair of graphs $(G_1, G_2)$ is grown in two stages: until time $t_{\star}$ they are grown together (i.e., $G_1 = G_2$), after which they grow independently according to the underlying growth model. We show that whenever the seed graph has an influence in the underlying graph growth model---this has been shown for PA and UA trees and is conjectured to hold broadly---then correlation can be detected in this model, even if the graphs are grown together for just a single time step. We also give a general sufficient condition (which holds for PA and UA trees) under which detection is possible with probability going to $1$ as $t_{\star} \to \infty$. Finally, we show for PA and UA trees that the amount of correlation, measured by $t_{\star}$, can be estimated with vanishing relative error as $t_{\star} \to \infty$.
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