Statistical network modeling has focused on representing the graph as a
discrete structure, namely the adjacency matrix, and considering the
exchangeability of this array. In such cases, the Aldous-Hoover representation
theorem (Aldous, 1981;Hoover, 1979} applies and informs us that the graph is
necessarily either dense or empty. In this paper, we instead consider
representing the graph as a measure on $\mathbb{R}_+^2$. For the associated
definition of exchangeability in this continuous space, we rely on the
Kallenberg representation theorem (Kallenberg, 2005). We show that for certain
choices of such exchangeable random measures underlying our graph construction,
our network process is sparse with power-law degree distribution. In
particular, we build on the framework of completely random measures (CRMs) and
use the theory associated with such processes to derive important network
properties, such as an urn representation for our analysis and network
simulation. Our theoretical results are explored empirically and compared to
common network models. We then present a Hamiltonian Monte Carlo algorithm for
efficient exploration of the posterior distribution and demonstrate that we are
able to recover graphs ranging from dense to sparse--and perform associated
tests--based on our flexible CRM-based formulation. We explore network
properties in a range of real datasets, including Facebook social circles, a
political blogosphere, protein networks, citation networks, and world wide web
networks, including networks with hundreds of thousands of nodes and millions
of edges.

本文研究了基于测度理论的交换性研究框架，并基于一种称为完全随机测度的随机过程推导了网络结构的重要性质，提出了一种哈密顿蒙特卡罗算法，实现了网络结构的后验分布探索和图形生成，并在包括社交网络、政治博客、蛋白质网络、引用网络以及互联网网络等多个真实数据集上进行检验。

使用可交换随机测度创建稀疏图

Sparse graphs using exchangeable random measures

We present a new approach to sample from generic binary distributions, based
on an exact Hamiltonian Monte Carlo algorithm applied to a piecewise continuous
augmentation of the binary distribution of interest. An extension of this idea
to distributions over mixtures of binary and possibly-truncated Gaussian or
exponential variables allows us to sample from posteriors of linear and probit
regression models with spike-and-slab priors and truncated parameters. We
illustrate the advantages of these algorithms in several examples in which they
outperform the Metropolis or Gibbs samplers.

本文基于一种分段连续的改进二项分布的哈密顿蒙特卡洛算法， 扩展到了二项和截尾高斯或指数变量混合分布的情况，用以采样线性和 probit 回归模型中的后验概率密度，并在多个案例中展示了这些算法的优点，相比于 Metropolis 或 Gibbs 采样器具有更好的性能。