We count the number of ways to build paths, stars, cycles, and complete
graphs as a sequence of vertices and edges, where each edge follows both of its
endpoints. The problem was considered 50 years ago by Stanley but the explicit
sequences corresponding to graph families seem to have been little studied. A
cost-based variant is introduced and applications are considered.

本篇论文通过顶点和边构建路径、星形图、环和完全图的序列的方式来探讨问题，并引入一种基于成本的变体，并研究了其应用。

构造数：如何构建图形？

Construction numbers: How to build a graph?

We develop a definition of Ricci curvature on directed hypergraphs and
explore the consequences of that definition. The definition generalizes
Ollivier's definition for graphs. It involves a carefully designed optimal
transport problem between sets of vertices.

本文提出了有向超图的 Ricci 曲率定义，并探讨了这个定义的后果，它推广了图的 Ollivier 定义，并涉及到一个精心设计的顶点集之间的最优输运问题。

定向超图的 Ollivier Ricci 曲率

Ollivier Ricci Curvature of Directed Hypergraphs

One property of networks that has received comparatively little attention is
hierarchy, i.e., the property of having vertices that cluster together in
groups, which then join to form groups of groups, and so forth, up through all
levels of organization in the network. Here, we give a precise definition of
hierarchical structure, give a generic model for generating arbitrary
hierarchical structure in a random graph, and describe a statistically
principled way to learn the set of hierarchical features that most plausibly
explain a particular real-world network. By applying this approach to two
example networks, we demonstrate its advantages for the interpretation of
network data, the annotation of graphs with edge, vertex and community
properties, and the generation of generic null models for further hypothesis
testing.

研究网络中的等级结构和生成任意等级结构的一般模型，通过两个示例网络演示了该方法在解释网络数据、注释图与边、顶点及社区属性以及生成更多假设测试的一般空模型方面的优势。

网络层次结构的结构推断

Structural Inference of Hierarchies in Networks

We consider methods for quantifying the similarity of vertices in networks.
We propose a measure of similarity based on the concept that two vertices are
similar if their immediate neighbors in the network are themselves similar.
This leads to a self-consistent matrix formulation of similarity that can be
evaluated iteratively using only a knowledge of the adjacency matrix of the
network. We test our similarity measure on computer-generated networks for
which the expected results are known, and on a number of real-world networks.

本文提出了一种基于顶点相邻度量相似性的方法，通过邻接矩阵迭代计算得到自洽矩阵，测试结果表明该方法可用于计算人工生成和真实网络中的节点相似性。