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 self-loops 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 rank-1 inhomogeneous random graph, where all vertices are equipped with weights, and the probabilities that edges are present are moderated by asymptotically linear functions of the products of these vertex weights.
Highlights
Many real-world networks were found to have degree distributions that can be well approximated by a power-law distribution, so that the fraction of vertices of degree k scales as k−τ for some τ > 1
While our results are for triangle counts, our method extends to counting several other types of subgraphs
We have studied the number of triangles in uniform random graphs with given degrees, when the degree sequence follows a power-law distribution with degreeexponent τ ∈ (2, 3)
Summary
Many real-world networks were found to have degree distributions that can be well approximated by a power-law distribution, so that the fraction of vertices of degree k scales as k−τ for some τ > 1. The degree exponent τ of several networks was found to satisfy τ ∈ (2, 3) [26] These power-law real-world networks are often modeled by random graphs. The resulting graph has the same degree sequence as the original network, but whether other properties behave in real-world networks and uniform random graphs is an interesting question. With power-law exponent τ < 3, some edges have significantly different probabilities of being present in the erased configuration model and in the uniform graph model (see Remark 4). Rweeescpanropusoertaiosnwalittcohi√ngnm, reetshuoldtintgo count in an the number of triangles between vertices asymptotic expression for the number of triangles in a uniform random graph with power-law degree exponent τ ∈ (2, 3).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
