We present a Bayesian formulation of weighted stochastic block models that
can be used to infer the large-scale modular structure of weighted networks,
including their hierarchical organization. Our method is nonparametric, and
thus does not require the prior knowledge of the number of groups or other
dimensions of the model, which are instead inferred from data. We give a
comprehensive treatment of different kinds of edge weights (i.e. continuous or
discrete, signed or unsigned, bounded or unbounded), as well as arbitrary
weight transformations, and describe an unsupervised model selection approach
to choose the best network description. We illustrate the application of our
method to a variety of empirical weighted networks, such as global migrations,
voting patterns in congress, and neural connections in the human brain.

本文提出了一种基于贝叶斯方法的加权随机块模型，可以用于推断加权网络的大规模模块结构，方法为无参数方法，使用数据来推断模型中的群组数量和其他维度，并提供了不同种类的边权（如连续或离散、有符号或无符号、有界或无界等）以及任意权重转换的综合处理。作者还描述了无监督模型选择方法来选择最佳网络描述，并且将该方法应用于各种实际加权网络，例如全球移民、国会中的投票模式和人类大脑中的神经连接。

非参数加权随机块模型

Nonparametric weighted stochastic block models

Discovering and characterizing the large-scale topological features in
empirical networks are crucial steps in understanding how complex systems
function. However, most existing methods used to obtain the modular structure
of networks suffer from serious problems, such as being oblivious to the
statistical evidence supporting the discovered patterns, which results in the
inability to separate actual structure from noise. In addition to this, one
also observes a resolution limit on the size of communities, where smaller but
well-defined clusters are not detectable when the network becomes large. This
phenomenon occurs not only for the very popular approach of modularity
optimization, which lacks built-in statistical validation, but also for more
principled methods based on statistical inference and model selection, which do
incorporate statistical validation in a formally correct way. Here we construct
a nested generative model that, through a complete description of the entire
network hierarchy at multiple scales, is capable of avoiding this limitation,
and enables the detection of modular structure at levels far beyond those
possible with current approaches. Even with this increased resolution, the
method is based on the principle of parsimony, and is capable of separating
signal from noise, and thus will not lead to the identification of spurious
modules even on sparse networks. Furthermore, it fully generalizes other
approaches in that it is not restricted to purely assortative mixing patterns,
directed or undirected graphs, and ad hoc hierarchical structures such as
binary trees. Despite its general character, the approach is tractable, and can
be combined with advanced techniques of community detection to yield an
efficient algorithm that scales well for very large networks.

通过具备多重尺度的完全描述网络层次结构的嵌套概率模型，可避免现有方法的局限性，在更高的分辨率下检测模块化结构并确保不产生伪模块，且具有一般性和可扩展性，适用于大型网络的高效算法