Counting triangles in powerlaw uniform random graphs
Abstract
We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent $\tau\in(2,3)$. We also analyze the local clustering coefficient $c(k)$, the probability that two random neighbors of a vertex of degree $k$ are connected. We find that the number of triangles, as well as the local clustering coefficient, scale similarly as in the erased configuration model, where all selfloops and multiple edges of the configuration model are removed. Interestingly, uniform random graphs contain more triangles than erased configuration models with the same degree sequence. The number of triangles in uniform random graphs is closely related to that in a version of the rank1 inhomogeneous random graph, where all vertices are equipped with weights, and the edge probabilities are moderated by asymptotically linear functions of the products of these vertex weights.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 DOI:
 10.48550/arXiv.1812.04289
 arXiv:
 arXiv:1812.04289
 Bibcode:
 2018arXiv181204289G
 Keywords:

 Mathematics  Probability