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

We provide a novel method for constructing asymptotics (to arbitrary accuracy) for the number of directed graphs that realize a fixed bidegree sequence $d = a \times b$ with maximum degree $d_{max}=O(S^{\frac{1}{2}-\tau})$ for an arbitrarily small positive number $\tau$, where $S$ is the number edges specified by $d$. 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 $d_{max} = O(S^{\frac{1}{2}-\tau})$ and replace it with $a_{max} b_{max} = O(S^{1-\tau})$. where $a_{max}$ and $b_{max}$ are the maximum values for $a$ and $b$ respectively. The previous best results, from Greenhill et al., only allow for $d_{max} = o(S^{\frac{1}{3}})$ or alternatively $a_{max} b_{max} = o(S^{\frac{2}{3}})$. Since in many real world networks, $d_{max}$ scales larger than $o(S^{\frac{1}{3}})$, we expect that this work will be helpful for various applications.

💡 Research Summary

This paper introduces a novel methodological framework for asymptotically approximating, to arbitrarily high accuracy, the number of directed graphs (allowing loops) that realize a given bidegree sequence. The key advancement is the substantial relaxation of constraints on the maximum degree compared to prior state-of-the-art results. While previous work by Greenhill et al. required the maximum degree to be o(S^{1/3}) (or alternatively, the product of maximum in- and out-degree to be o(S^{2/3})), this new method is valid for sequences where the maximum degree is O(S^{1/2-τ}) for an arbitrarily small τ > 0, where S is the total number of edges. Since many real-world networks exhibit degree distributions where the maximum degree scales larger than o(S^{1/3}), this extension is of significant practical relevance.

The core technical innovation rests on two intertwined ideas: graph partitioning and degree-preserving switches. Graph partitioning involves decomposing the adjacency matrix into a submatrix containing two specific rows (corresponding to two nodes i and j) and the complementary submatrix containing the remaining rows. This allows the enumeration problem for the original degree sequence d to be related to enumeration problems for smaller, derived “residual” degree sequences. Specifically, the number of graphs kG_dk can be expressed as a sum over k, where k is the number of common neighbors shared by nodes i and j, of terms involving the number of graphs realizing these residual sequences.

Degree-preserving switches are edge-replacement operations that eliminate a common neighbor of two nodes without altering any node’s degree. Leveraging this concept, the authors prove that for sparse graphs (under the stated maximum degree constraint), the dominant term in the partition-based expansion corresponds to the case where nodes i and j have zero common neighbors (k=0). This leads to a fundamental asymptotic relation: for two degree sequences d and d* that differ only by 1 in the in-degree of a single node i, the ratio of graph counts kG_{d*}k / kG_dk is asymptotically equal to a_i / a_j, where a_i and a_j are the in-degrees of nodes i and j in d.

Building upon this foundational ratio estimate, the paper develops a recursive strategy to achieve arbitrarily high-order approximations. Theorems 3 and 4 form the engine of this refinement process. They show that if an approximation for the ratio kG_{d1}k / kG_{d2}k is known with an error of O(S^{-γ}), then a new approximation with an error of O(S^{-γ-2τ}) can be constructed. Theorem 4 specifically establishes that this improvement is possible without additional assumptions once γ ≥ 1/2. Theorem 5 then explains how to translate these precise ratio estimates into an approximation for the absolute count kG_{d}k itself. This is done by constructing a path from a “reference” degree sequence d0, for which the count can be computed exactly using a combinatorial formula (Lemma 1), to the target sequence d, such that consecutive sequences along the path differ by a taxicab norm of 2. Multiplying the estimated ratios for each step yields the desired approximation for kG_{d}k / kG_{d0}k, and hence for kG_{d}k.

Sections 5 and 6 provide detailed justifications, showing that the assumptions required for the recursive process hold for error terms up to O(S^{-1/2}) (Theorem 12). Consequently, one can iteratively apply Theorem 3 to reduce errors to the O(S^{-1/2}) level and then repeatedly apply Theorem 4 to push the error to any desired order of accuracy, such as O(S^{-10τ}).

The paper also discusses significant generalizations. Appendix A shows how to relax the maximum degree condition further to a_max * b_max = O(S^{1-τ}). Appendix B outlines extensions of the method to other graph classes, including bipartite graphs, undirected graphs, and directed graphs without loops, as well as to graphs where edges between certain node pairs are prohibited.

In summary, this work provides a powerful and flexible framework for the asymptotic enumeration of graphs with prescribed degree sequences, breaking previous barriers on allowable degree concentrations and offering a pathway to approximations of arbitrary precision relevant for the analysis of real-world networks.