Efficient Construction of 2K+ Graphs

I. INTRODUCTION Researchers often need to generate synthetic graphs with key properties resembling those of networks of interest. This is for example the case when the full topology is unavailable or impractical to measure or use. In our work, we are interested in generating graphs that resemble online social networks in terms of their joint degree distribution (JDM) and their degree-dependent average clustering coefficient (¯ c(k)). The above two graph properties have been chosen based on the systematic framework of dK-series [2], which characterizes the properties of a graph using a series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. The joint degree distribution (JDM) and the degree-dependent average clustering coefficient (¯ c(k)) provide a sweet spot between accurate representation of online social networks and practical constraints in estimation and construction. One one hand, the joint degree distribution (2K-distribution) alone is insufficient for modeling online social networks, which exhibit high clustering. On the other hand, the 3K-distribution captures all information about subgraphs of three nodes, but there are currently no efficient algorithms for estimating 3K parameters or constructing graphs with a target 3K. We design two algorithms for constructing graphs with (1) an exact 2K distribution and (2) an exact 2K and an approximate ¯ c(k). Our approach is the first that achieves exact 2K, high clustering, and speed, whereas previous methods could get only two out of three.