Toward Proper Random Graph Models for Real World Networks

Inspired by a huge amount of empirical study of real world networks such as the Internet, the Web, as well as various social and biological networks, researchers have in recent years developed several random graph models to help us to understand the most fundamental properties of these systems. Simple characteristics observed in many real world networks are 1.) a high clustering coefficient, i.e., if vertex $u$ is connected to vertex $v$, and vertex $v$ is connected to vertex $w$, then $u$ is likely connected to $w$; 2.) a power law degree distribution, i.e., the number of nodes of degree $d$ is proportional to $d^{-\alpha}$, where $\alpha$ is some constant; 3.) small mean geodesic distance, i.e., the minimum distance between two randomly chosen nodes is small. Most known random graph models with the properties above are generated by using some global graph properties (e.g., in the preferential attachment model it is assumed that each node knows the degree of any other node). However, in large real world networks the nodes usually do not possess any global information about the network. In this paper, we present new random graph models, which imitate a simple growth behavior observed in many dynamically evolving real world networks. More precisely, in our graphs we implement the fact that if a new node joins the network, then this node becomes likely connected to some old members which already share an edge in the corresponding graph. Although we do not necessarily require any global knowledge about the graph, we are able to show that in addition to a large clustering coefficient a power law degree distribution is implicitly obtained. Thus, our results provide an explanation why the behavior described above seems to be one of the essential factors for the shape of many real world networks. Additionally, by using proper parameters we can control the power law exponent in our graphs.