Abstract

Evolution of random undirected graphs by the clustering attachment (CA) without node and edge deletion 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 (a local dependence measure) approximates the mean cluster size of exceedances over a sufficiently high threshold. Considering the change of the modularity at each evolution step, the extremal index of the latter random sequence indicates the consecutive large connectivity of nodes and thus, it reflects the community appearance during the network evolution. This allows to consider the community structure of the network from perspectives of the extreme value analysis. By simulation study we show that estimates of the extremal index of the modularity and the tail index of node degrees depend on the CA parameters. The latter estimates are compared both for evolution without node and edge deletion and with uniform node deletion.

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