Examining Higher Order Transformations for Scale-free Small World Graphs

The degree distribution of scale-free Small World networks follows a power law. For random graph generators, its exponent is constrained by the construction mechanism, whereas in real-world data, different slopes can be observed. However, the degree distribution alone does not reveal much of the local structure of these graphs. Therefore, we propose a graph transformation we call ”higher order” transformation, which encodes the number of common neighbours two vertices share in its edge weights. Studying the degree distribution of secondand third order graphs and comparing it to natural language cooccurrence data, we find that the higher order transformation reveals differences that cannot be detected by only looking at traditional measures on the original graph.