Counting Triangles in Power-Law Uniform Random Graphs

Uniform generation of random graphs with power-law degree sequences

We give a linear-time algorithm that approximately uniformly generates a randomsimple graph with a power-law degree sequence whose exponent is at least 2.8811.While sampling graphs with power-law degree sequence of exponent at least 3 is fairlyeasy, and many samplers work efficiently in this case, the problem becomes dramaticallymore difficult when the exponent drops below 3; ou...

Triangles in random graphs

Triangles in Random Graphs

OF THE DISSERTATION Triangles in Random Graphs by Robert DeMarco Dissertation Director: Jeff Kahn We prove four separate results. These results will appear or have appeared in various papers (see [10], [11], [12], [13]). For a gentler introduction to these results, the reader is directed to the first chapter of this thesis. Let G = Gn,p. With ξk = ξ n,p k the number of copies of Kk in G, p ≥ n−...

Counting Triangles in Massive Graphs with MapReduce

Graphs and networks are used to model interactions in a variety of contexts. There is a growing need to quickly assess the characteristics of a graph in order to understand its underlying structure. Some of the most useful metrics are triangle-based and give a measure of the connectedness of mutual friends. This is often summarized in terms of clustering coefficients, which measure the likeliho...

Eigenvalues of Random Power Law Graphs

Many graphs arising in various information networks exhibit the “power law” behavior – the number of vertices of degree k is proportional to k−β for some positive β. We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1 + o(1)) √ m where m is the maximum degree. Moreover, the k largest eigenvalues of a random power law graph with exponent β have power l...