# Degree Relations of Triangles in Real-world Networks and Graph Models

### Abstract

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 investigate whether current graph models reproduce the types of triangles that are observed in real data and observe that most models fail to accurately capture these salient features

Publication
In CIKM’12: Proceedings of the 21st ACM International Conference on Information and Knowledge Management
Date
Tags
Citation
N. Durak, A. Pinar, T. G. Kolda, C. Seshadhri. Degree Relations of Triangles in Real-world Networks and Graph Models. In CIKM’12: Proceedings of the 21st ACM International Conference on Information and Knowledge Management, Maui, Hawaii (2012-10-29 to 2012-11-02), ACM, pp. 1712-1716, 2013. https://doi.org/10.1145/2396761.2398503

### Keywords

graph models, social networks, triangles in graphs

Short paper: 28% acceptance rate.

### BibTeX

@inproceedings{DuPiKoSe12,
author = {Durak, Nurcan and Pinar, Ali and Kolda, Tamara G. and Seshadhri, C.},
title = {Degree Relations of Triangles in Real-world Networks and Graph Models},
booktitle = {CIKM'12: Proceedings of the 21st ACM International Conference on Information and Knowledge Management},
venue = {Maui, Hawaii},
eventdate = {2012-10-29/2012-11-02},
publisher = {ACM},
pages = {1712--1716},
year = {2012},
doi = {10.1145/2396761.2398503},
}