Motivated by widely observed examples in nature, society and software, where groups of already related nodes arrive together and attach to an existing network, we consider network growth via sequential attachment of linked node groups, or graphlets. We analyze the simplest case, attachment of the three node V-graphlet, where, with probability alpha, we attach a peripheral node of the graphlet, and with probability (1-alpha), we attach the central node. Our analytical results and simulations show that tuning alpha produces a wide range in degree distribution and degree assortativity, achieving assortativity values that capture a diverse set of many real-world systems. We introduce a fifteen-dimensional attribute vector derived from seven well-known network properties, which enables comprehensive comparison between any two networks. Principal Component Analysis (PCA) of this attribute vector space shows a significantly larger coverage potential of real-world network properties by a simple extension of the above model when compared against a classic model of network growth.