Clusters of Exceedances for Evolving Random Graphs

Evolution of random undirected graphs by the clustering attachment (CA) and with uniform node deletion is investigated. The CA causes clusters of consecutive exceedances of the modularity over a sufficiently high threshold. The modularity is a measure that allows us to divide graphs into communities. It shows the connectivity of nodes in the community. An extremal index approximates the mean cluster size and thus, it reflects a local dependence. It is shown by simulation study that estimates of the extremal index of the modularity and tail index of node degrees depend on the CA parameters.