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Degree switching and partitioning for enumerating graphs to arbitrary orders of accuracy

12 November 2015
D. Burstein
Jonathan Rubin
ArXiv (abs)PDFHTML
Abstract

We provide a novel method for constructing asymptotics (to arbitrary accuracy) for the number of directed graphs that realize a fixed bidegree sequence d=a×bd = a \times bd=a×b with maximum degree dmax=O(S12−τ)d_{max}=O(S^{\frac{1}{2}-\tau})dmax​=O(S21​−τ) for an arbitrarily small positive number τ\tauτ, where SSS is the number edges specified by ddd. Our approach is based on two key steps, graph partitioning and degree preserving switches. The former idea allows us to relate enumeration results for given sequences to those for sequences that are especially easy to handle, while the latter facilitates expansions based on numbers of shared neighbors of pairs of nodes. While we focus primarily on directed graphs allowing loops, our results can be extended to other cases, including bipartite graphs, as well as directed and undirected graphs without loops. In addition, we can relax the constraint that dmax=O(S12−τ)d_{max} = O(S^{\frac{1}{2}-\tau})dmax​=O(S21​−τ) and replace it with amaxbmax=O(S1−τ)a_{max} b_{max} = O(S^{1-\tau})amax​bmax​=O(S1−τ). where amaxa_{max}amax​ and bmaxb_{max}bmax​ are the maximum values for aaa and bbb respectively. The previous best results, from Greenhill et al., only allow for dmax=o(S13)d_{max} = o(S^{\frac{1}{3}})dmax​=o(S31​) or alternatively amaxbmax=o(S23)a_{max} b_{max} = o(S^{\frac{2}{3}})amax​bmax​=o(S32​). Since in many real world networks, dmaxd_{max}dmax​ scales larger than o(S13)o(S^{\frac{1}{3}})o(S31​), we expect that this work will be helpful for various applications.

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