Generating complex networks through a vertex merging mechanism: Empirical and analytical analysis

One of the most well-known mechanisms contributing to the emergence of networks with a power-law degree distribution is preferential attachment. In this study, we examined a family of network evolution models based on the merging of two arbitrary vertices, which is shown to also lead to the creation of power-law distributed networks. These models simultaneously apply rules for both node addition and merging, which reflects that many real systems exhibit the processes of growth and shrink. At each iteration, when two vertices merge, the neighbors of one of the vertices become neighbors of the other, and the vertex itself is removed from the network. In addition, at each iteration, a new vertex appears that is attached to randomly selected nodes. As an enhancement, we incorporate a triadic closure mechanism into the evolution to increase the clustering coefficient, a key characteristic of real social networks. We show that in the process of evolution any initial network converges to a stationary state with a power law degree distribution, while the number of edges, the average degree, and the average clustering coefficient saturate to a certain limit values depending on the model parameters.