Extended power laws and inhomogeneous connections are structural patterns often found in empirical networks. Mechanisms based on the formation of triads are able to explain the power law behavior of the degree distribution of such networks. The proposed model introduces a two-step mechanism of attachment and triad formation that illustrates how preferential linkage plays an important role in shaping the inhomogeneity of connections and the division of the network into groups of nodes (i.e., the growth of community structures). In particular, we identify conditions under which the scaling exponent of the power law correlates to a widely-used modularity measure of non-overlapping communities. Our analytical results characterize the asymptotic behavior of both the scaling exponent and the modularity, as a function of the strength with which nodes with similar characteristics tend to link to each other.