  Triangles are an important building block and distinguishing feature of
real-world networks, but their structure is still poorly understood. Despite
numerous reports on the abundance of triangles, there is very little
information on what these triangles look like. We initiate the study of
degree-labeled triangles -- specifically, degree homogeneity versus
heterogeneity in triangles. This yields new insight into the structure of
real-world graphs. We observe that networks coming from social and
collaborative situations are dominated by homogeneous triangles, i.e., degrees
of vertices in a triangle are quite similar to each other. On the other hand,
information networks (e.g., web graphs) are dominated by heterogeneous
triangles, i.e., the degrees in triangles are quite disparate. Surprisingly,
nodes within the top 1% of degrees participate in the vast majority of
triangles in heterogeneous graphs. We also ask the question of whether or not
current graph models reproduce the types of triangles that are observed in real
data and showed that most models fail to accurately capture these salient
features.
