Directed Random Dot Product Graphs

In this paper we consider three models for random graphs that utilize the inner product as their fundamental object. We analyze the behavior of these models with respect to clustering, the small world property, and degree distribution. These models are motivated by the random dot product graphs developed by Kraetzl, Nickel and Scheinerman. We extend their results to fully parameterize the conditions under which clustering occurs, characterize the diameter of graphs generated by these models, and describe the behavior of the degree distribution. With the ubiquity and importance of the Internet and genetic information in medicine and biology, the study of complex networks relating to the Internet and genetics continues to be an important and vital area of study. This is especially true for networks such as the physical layer of the Internet, the link structure of the world wide web, and protein-protein and protein-gene interaction networks. Because of the size of these networks [3] and the difficulty of determining complete link information [2, 19] a significant amount of research has gone into finding models that match observed properties of these graphs in order to empirically (via simulation) and theoretically understand and predict properties of these complex networks. There are three models that, together with their variations, are the core models for these complex networks [7, 10]. The configurational model and its variants attempt to generate complex networks by specifying the degree sequence and creating edges randomly with respect to that degree sequence. On the other hand, the Barabási-Albert preferential attachment model attempts to model the process by which the network grows, specifically, it posits that vertices with high degree are more likely to increase in degree when a new vertex is added to the network. In a similar vein, the copying model [9, 18], also attempts to model the growth process of a complex networks. However, the copying model takes the more distinctly biological viewpoint of replication of existing nodes combined with mutation. All three of these types of models have had success in reproducing the hallmark features of complex networks, namely a power-law degree distribution, a diameter that grows slowly or is constant with the size of the graph, and one of several clustering properties; see [7, 10] for a collection of such results. However, there are many other aspects of complex networks that fail to be captured by these models, for example non-uniform assortativity [23] and the existence of directed cycles, among others. Thus there is considerable interest in new models for complex networks that exhibit a power-law like degree sequence, small diameter, and clustering, and are different enough from the three main model classes to exhibit other properties of complex networks that are not exhibited by the current models. One potential method to create new models is to incorporate geometry Date: June 10, 2008.