Fast Generation of Large Scale Social Networks with Clustering
  A key challenge within the social network literature is the problem of network generation - that is, how can we create synthetic networks that match characteristics traditionally found in most real world networks? Important characteristics that are present in social networks include a power law degree distribution, small diameter and large amounts of clustering; however, most current network generators, such as the Chung Lu and Kronecker models, largely ignore the clustering present in a graph and choose to focus on preserving other network statistics, such as the power law distribution. Models such as the exponential random graph model have a transitivity parameter, but are computationally difficult to learn, making scaling to large real world networks intractable. In this work, we propose an extension to the Chung Lu ran- dom graph model, the Transitive Chung Lu (TCL) model, which incorporates the notion of a random transitive edge. That is, with some probability it will choose to connect to a node exactly two hops away, having been introduced to a 'friend of a friend'. In all other cases it will follow the standard Chung Lu model, selecting a 'random surfer' from anywhere in the graph according to the given invariant distribution. We prove TCL's expected degree distribution is equal to the degree distribution of the original graph, while being able to capture the clustering present in the network. The single parameter required by our model can be learned in seconds on graphs with millions of edges, while networks can be generated in time that is linear in the number of edges. We demonstrate the performance TCL on four real- world social networks, including an email dataset with hundreds of thousands of nodes and millions of edges, showing TCL generates graphs that match the degree distribution, clustering coefficients and hop plots of the original networks. 
Joseph J. Pfeiffer III, Timothy La Fond, Sebastian Moreno, Jennifer   Neville
cs.SI physics.soc-ph