4th Online NETWORKS Seminar Talk

Abstract

Two key structural features observed in many complex networks are the large variability in the degrees and the presence of triangles and communities. However, for a long time, researchers struggled to design models that would combine these two key features. In 2010 Krioukov et al. introduced such model which has become a well-established approach to model complex networks. In this talk we will consider the version of the model were points are sprinkled, uniformly at random, on the hyperbolic disc of radius R, and then points are connected if their hyperbolic distance is less than R. Since its introduction, many properties of this model, such as degree distribution, coefficient, size of the largest component and scaling of clique sizes, have been analyzed.

 

This talk will focus on the clustering of the model. While results exist for the global clustering coefficient, this statistic only gives a global average indication of the triangle structure of the network. To obtain more detailed insights, the local clustering function is more useful. This function gives for any degree value k, the average fraction of triangles in which nodes of degree k participate. Due to its dependence on the degree, the local clustering function gives a more detailed profile of the clustering structure in the network, compared to the global clustering coefficient. I will present new results for the local clustering function in the hyperbolic random graph. These include both pointwise converges, as well as the asymptotic behavior as the degree value k tends to infinity. Our results also establish an interesting phase-transition in the scaling of the local clustering function with the degree, which depends on the exponent of the power-law degree distribution.