Limitations of Chung Lu Random Graph Generation

Christopher Brissette, George Slota

Abstract: Random graph models play a central role in network analysis. The Chung-Lu model, which connects nodes based on their expected degrees, is of particular interest. It is widely used to generate null-graph models with expected degree sequences. In addition, these attachment probabilities implicitly define network measures such as modularity. Despite its popularity, practical methods for generating instances of Chung-Lu model-based graphs do relatively poor jobs in terms of accurately realizing many degree sequences. We perform a theoretical analysis of the Chung-Lu random graph model in order to understand this discrepancy. We approximate the expected output of a generated Chung-Lu random graph model with a linear system and use properties of this system to predict distribution errors. We provide bounds on the maximum proportion of nodes with a given degree that can be reliably produced by the model for both general and non-increasing distributions. We additionally provide an explicit inverse of our linear system and in cases where the inverse can provide a valid solution, we introduce a simple method for improving the accuracy of Chung-Lu graph generation. Our analysis serves as an analytic tool for determining the accuracy of Chung-Lu random graph generation as well as correcting errors under certain conditions.