概率图子（probability graphons）与P-变量：稠密加权图极限的两种等价观点
Probability graphons and P-variables: two equivalent viewpoints for dense weighted graph limits
AI总结：
该研究引入P-变量作为稠密加权图的新连续统表示，证明其与概率图子收敛等价，迁移相关理论结果并推广至二分图、超图等组合结构，丰富图极限理论。
摘要：
我们进一步发展了稠密加权图序列的图极限理论。具体而言，我们研究了近年在图极限理论中作为加权图连续统表示出现的概率图子（probability graphons），并引入了P-变量——这类变量也出现在关于可交换无限随机阵列的Aldous-Hoover定理的研究语境中，作为加权图的另一种连续统表示。我们特别阐释了P-变量与概率图子的关联方式，类似于随机变量与概率测度的关联方式类似。我们定义了一种P-变量的度量（受稀疏图序列图极限理论中的作用收敛启发），并证明P-变量在该度量下的收敛等价于概率图子的收敛。我们利用这一等价性，将概率图子理论中的若干结果迁移到P-变量上。此外，我们证明了P-变量收敛的若干性质，由此也得到了概率图子收敛的新性质，展现了概率图子与P-变量之间关联的效力。此外，我们还展示了P-变量收敛可以如何被轻松修改并推广，以覆盖二分图、超图等其他组合结构。
英文摘要：
We develop further the graph limit theory for dense weighted graph sequences. In particular, we consider probability graphons, which have recently appeared in graph limit theory as continuum representations of weighted graphs, and we introduce P-variables, which also appear in the context of the Aldous-Hoover theorem for exchangeable infinite random arrays, as an alternative continuum representation for weighted graphs. In particular, we explain how P-variables are related to probability graphons in a similar way in which random variables are related to probability measures. We define a metric for P-variables (inspired by action convergence in the graph limit theory of sparse graph sequences) and show that convergence of P-variables in this metric is equivalent to probability graphons convergence. We exploit this equivalence to translate several results from the theory of probability graphons to P-variables. In addition, we prove several properties of P-variables convergence, thus showing new properties also for probability graphons convergence and demonstrating the power of the connection between probability graphons and P-variables. Furthermore, we show how P-variables convergence can be easily modified and generalised to cover other combinatorial structures such as bipartite graphs and hypergraphs.
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