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Triangles are an important building block and distinguishing feature ofreal-world networks, but their structure is still poorly understood. Despitenumerous reports on the abundance of triangles, there is very littleinformation on what these triangles look like. We initiate the study ofdegree-labeled triangles -- specifically, degree homogeneity versusheterogeneity in triangles. This yields new insight into the structure ofreal-world graphs. We observe that networks coming from social andcollaborative situations are dominated by homogeneous triangles, i.e., degreesof vertices in a triangle are quite similar to each other. On the other hand,information networks (e.g., web graphs) are dominated by heterogeneoustriangles, i.e., the degrees in triangles are quite disparate. Surprisingly,nodes within the top 1% of degrees participate in the vast majority oftriangles in heterogeneous graphs. We also ask the question of whether or notcurrent graph models reproduce the types of triangles that are observed in realdata and showed that most models fail to accurately capture these salientfeatures.

Degree Relations of Triangles in Real-world Networks and Models