On the Evolution of Finite-Sized Complex Networks with Constrained Link Addition

Scale-free characteristics, where degree distribution of a network follows the power-law distribution, are observed in most of the existing real-world complex networks. Barabasi and Albert first studied the evolution of random complex networks and observed that complex networks with node growth via preferential attachment can evolve to be scale-free. However, some complex networks such as neural networks inside the human brain, employees of an organization, and networks of closed social groups can be considered finite-sized complex networks which are relatively static with respect to the number of nodes where only the number of edges grow with time. This paper studies the gradual evolution of such finite-sized complex networks. It can be observed from our study that a finite sized complex network, with average path optimal edge growth, evolves as the following: a regular network -> a small-world network -> a scale-free network -> a scale-free network with the truncated degree distribution -> a fully connected network with unconstrained link addition. Therefore, it can be concluded that in finite-sized complex networks edge growth can result in transitional scale-free networks.