With the rapid growth of rich network data available through various sources such as social media and digital archives, there is a growing interest in more powerful network visual analysis tools and methods. The rich information about the network nodes and links can be represented as multivariate graphs, in which the nodes are accompanied with attributes to represent the properties of individual nodes. An important task often encountered in multivariate network analysis is to uncover link structure with groups, e.g., to understand why a person fits a specific job or certain role in a social group well. The task usually involves complex considerations including specific requirement of node attributes and link structure, and hence a fully automatic solution is typically not satisfactory. In this work, we identify the design challenges for mining groups with complex criteria and present an interactive system, "g-Miner," that enables visual mining of groups on multivariate graph data. We demonstrate the effectiveness of our system through case study and in-depth expert interviews. This work contributes to understanding the design of systems for leveraging users' knowledge progressively with algorithmic capacity for tackling massive heterogeneous information.