Graphons have traditionally served as limit objects for dense graph
sequences, with the cut distance serving as the metric for convergence.
However, sparse graph sequences converge to the trivial graphon under the
conventional definition of cut distance, which make this framework inadequate
for many practical applications. In this paper, we utilize the concepts of
generalized graphons and stretched cut distance to describe the convergence of
sparse graph sequences. Specifically, we consider a random graph process
generated from a generalized graphon. This random graph process converges to
the generalized graphon in stretched cut distance. We use this random graph
process to model the growing sparse graph, and prove the convergence of the
adjacency matrices' eigenvalues. We supplement our findings with experimental
validation. Our results indicate the possibility of transfer learning between
sparse graphs.

通过使用广义图同和拉伸的割距离来描述稀疏图序列的收敛性，本文探讨了基于广义图同生成的随机图过程，并证明了邻接矩阵特征值的收敛性，实验证明了稀疏图之间转移学习的可能性。

广义 Graphon 过程：在拉伸割距离中的图频率收敛

Generalized Graphon Process: Convergence of Graph Frequencies in  Stretched Cut Distance

We introduce a class of random graphs that we argue meets many of the
desiderata one would demand of a model to serve as the foundation for a
statistical analysis of real-world networks. The class of random graphs is
defined by a probabilistic symmetry: invariance of the distribution of each
graph to an arbitrary relabelings of its vertices. In particular, following
Caron and Fox, we interpret a symmetric simple point process on
$\mathbb{R}_+^2$ as the edge set of a random graph, and formalize the
probabilistic symmetry as joint exchangeability of the point process. We give a
representation theorem for the class of random graphs satisfying this symmetry
via a straightforward specialization of Kallenberg's representation theorem for
jointly exchangeable random measures on $\mathbb{R}_+^2$. The distribution of
every such random graph is characterized by three (potentially random)
components: a nonnegative real $I \in \mathbb{R}_+$, an integrable function $S:
\mathbb{R}_+ \to \mathbb{R}_+$, and a symmetric measurable function $W:
\mathbb{R}_+^2 \to [0,1]$ that satisfies several weak integrability conditions.
We call the triple $(I,S,W)$ a graphex, in analogy to graphons, which
characterize the (dense) exchangeable graphs on $\mathbb{N}$. Indeed, the model
we introduce here contains the exchangeable graphs as a special case, as well
as the "sparse exchangeable" model of Caron and Fox. We study the structure of
these random graphs, and show that they can give rise to interesting structure,
including sparse graph sequences. We give explicit equations for expectations
of certain graph statistics, as well as the limiting degree distribution. We
also show that certain families of graphexes give rise to random graphs that,
asymptotically, contain an arbitrarily large fraction of the vertices in a
single connected component.

介绍了一类随机图形，通过定义概率对称性：图形的每个排列都不影响其分布，使用简单的 Kallenberg 定理特化的表示定理得到其分布特征，可得期望与度分布等结构信息，研究表明其可以产生包含极少连接的图结构且某些构型可产生一个连通部分包含任意大分数的图随机图。