Recently, there arise a large number of graphs with massive sizes and complex structures in many new applications, such as biological networks, social networks, and the Web, demanding powerful data mining methods. Due to inherent noise or data diversity, it is crucial to address the issue of approximation, if one wants to mine patterns that are potentially interesting with tolerable variations. In this paper, we investigate the problem of mining frequent approximate patterns from a massive network and propose a method called gApprox. gApprox not only finds approximate network patterns, which is the key for many knowledge discovery applications on structural data, but also enriches the library of graph mining methodologies by introducing several novel techniques such as: (1) a complete and redundancy-free strategy to explore the new pattern space faced by gApprox; and (2) transform "frequent in an approximate sense" into an anti-monotonic constraint so that it can be pushed deep into the mining process. Systematic empirical studies on both real and synthetic data sets show that frequent approximate patterns mined from the worm protein-protein interaction network are biologically interesting and gApprox is both effective and efficient.