Effects of triad formations stimulated by intermediaries on network topology

Triad formation of vertices is considered a significant mechanism in the emergence of highly clustered structures in real networks. However, the net effect of triad formations on network topology has yet to be understood completely, since triad formations are usually studied with additional effects including the attachment of new vertices to prevent a saturation of the number of edges, where almost all vertices are directly linked to each other. In this paper, we focus on the net effects of triad formations stimulated by randomly chosen intermediaries on network topologies such as local clustering and evolution of degrees. We show that the local clustering of vertices with degree k is divided into an essential term ∼1/k which can be widely seen in real networks and additional terms depending on the initial network topology. Also, we derive an equation which measures the influence of local structures of networks on the time evolution of vertex degrees, according to which triad formations lead to the so called “rich get richer” phenomenon in the evolution of degrees. Local events like a triad formation stimulated by pre-existing vertices leads not only to highly clustered structures but to a typical power-law form in the degree distribution with a power-law exponent of about 2.