This paper studies how close random graphs are typically to their
expectations. We interpret this question through the concentration of the
adjacency and Laplacian matrices in the spectral norm. We study inhomogeneous
Erd\"os-R\'enyi random graphs on $n$ vertices, where edges form independently
and possibly with different probabilities $p_{ij}$. Sparse random graphs whose
expected degrees are $o(\log n)$ fail to concentrate; the obstruction is caused
by vertices with abnormally high and low degrees. We show that concentration
can be restored if we regularize the degrees of such vertices, and one can do
this in various ways. As an example, let us reweight or remove enough edges to
make all degrees bounded above by $O(d)$ where $d=\max np_{ij}$. Then we show
that the resulting adjacency matrix $A'$ concentrates with the optimal rate:
$\|A' - \mathbb{E} A\| = O(\sqrt{d})$. Similarly, if we make all degrees
bounded below by $d$ by adding weight $d/n$ to all edges, then the resulting
Laplacian concentrates with the optimal rate: $\|L(A') - L(\mathbb{E} A')\| =
O(1/\sqrt{d})$. Our approach is based on Grothendieck-Pietsch factorization,
using which we construct a new decomposition of random graphs. We illustrate
the concentration results with an application to the community detection
problem in the analysis of networks.

本文通过研究谱范数中邻接矩阵和拉普拉斯矩阵的浓度来探索随机图与其期望值之间的典型接近程度，其中包括不同概率的独立形成的具有 n 个顶点的不均匀 Erdos-Renyi 随机图，对于稀疏随机图，其期望度数小于 o（logn），需要使这种度数正则化，本文通过一些方法，例如重量重排或删除足够的边等操作来实现，演示了在社区检测问题中，集中结果的应用。