Text categorization is an essential task in Web content analysis. Considering the ever-evolving Web data and new emerging categories, instead of the laborious supervised setting, in this paper, we focus on the minimally-supervised setting that aims to categorize documents effectively, with a couple of seed documents annotated per category. We recognize that texts collected from the Web are often structure-rich, i.e., accompanied by various metadata. One can easily organize the corpus into a text-rich network, joining raw text documents with document attributes, high-quality phrases, label surface names as nodes, and their associations as edges. Such a network provides a holistic view of the corpus' heterogeneous data sources and enables a joint optimization for network-based analysis and deep textual model training. We therefore propose a novel framework for minimally supervised categorization by learning from the text-rich network. Specifically, we jointly train two modules with different inductive biases - a text analysis module for text understanding and a network learning module for class-discriminative, scalable network learning. Each module generates pseudo training labels from the unlabeled document set, and both modules mutually enhance each other by co-training using pooled pseudo labels. We test our model on two real-world datasets. On the challenging e-commerce product categorization dataset with 683 categories, our experiments show that given only three seed documents per category, our framework can achieve an accuracy of about 92%, significantly outperforming all compared methods; our accuracy is only less than 2% away from the supervised BERT model trained on about 50K labeled documents.