[Paper Review] Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph

Anita Liebenau, Nick Wormald|arXiv (Cornell University)|Feb 27, 2017

Limits and Structures in Graph Theory4 citations

TL;DR

This paper proves the Binomial Approximation Conjecture for random graphs, establishing that the degree sequence of a random graph in both the G(n,m) and G(n,p) models is asymptotically well-approximated by independent binomial random variables conditioned on the sum. The key result is an asymptotic formula for the number of d-regular graphs for all d, resolving a long-standing conjecture from 1990 and enabling precise analysis of degree distribution properties such as the median degree.

ABSTRACT

In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. This confirms a conjecture made in 1997. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of $d$-regular graphs for all $d$, as $n o\infty$.

Motivation & Objective

• To resolve the 1990 Binomial Approximation Conjecture on the asymptotic enumeration of graphs by degree sequence across all densities.
• To establish a rigorous connection between the degree sequence of random graphs and independent binomial random variables, extending beyond sparse and dense regimes.
• To provide a unified model for the degree sequence of G(n,m) and G(n,p) that enables estimation of complex functionals, such as the median degree.
• To develop a new analytical framework based on fixed-point analysis of operators to estimate ratios of point probabilities in the degree sequence space.
• To extend the applicability of binomial approximation models to broader classes of random graphs, including bipartite and directed graphs.

Proposed method

• Derive an asymptotic formula for the number of graphs with a given degree sequence using a novel approach to estimating ratios of point probabilities in the degree sequence space.
• Introduce and analyze fixed points of associated operators to control error terms in the approximation of degree sequence probabilities.
• Use a perturbation method to compare the true graph degree probabilities with those of independent binomial variables, showing convergence under the binomial model.
• Apply a coupling argument via auxiliary graphs and size-biased sampling to bound the total variation distance between the true degree sequence and the binomial model.
• Employ error term analysis with parameters μ₁, ε, and δ to control deviations in the approximation, especially in intermediate density regimes.
• Leverage results from previous works on sparse and dense graphs as base cases, extending them to the full range of degrees via operator-based interpolation.

Experimental results

Research questions

• RQ1 Does the binomial approximation model accurately describe the degree sequence of a random graph across all densities, including intermediate ones?
• RQ2 What is the asymptotic number of d-regular graphs for d in the range c√n ≤ d = o(n/log n), as conjectured in 1990?
• RQ3 Can the degree sequence of G(n,m) be well-approximated by a sequence of independent binomial variables conditioned on summing to 2m?
• RQ4 How can the distribution of order statistics, such as the median degree, be analyzed using the binomial approximation model?
• RQ5 What techniques allow for the extension of asymptotic enumeration results to broader classes of random graphs, such as bipartite or directed graphs?

Key findings

• The Binomial Approximation Conjecture is proven, confirming that the degree sequence of a random graph is asymptotically equivalent to a sequence of independent binomial variables under appropriate conditioning.
• An asymptotic formula is established for the number of d-regular graphs on n vertices for all d satisfying 1 ≤ d ≤ n−2, with the formula given by |(n−1 choose d)^n * (n choose 2 choose m) / (n(n−1)/2 choose 2m) * e^{1/4}|, where m = dn/2.
• The approximation error in the binomial model is shown to be O(μ₁ε⁴ + 1/n²), with μ₁ and ε controlling the deviation from independence.
• The method enables precise estimation of the distribution of the median degree and other order statistics in random graphs, previously inaccessible with prior techniques.
• The framework is generalizable to other random graph models, including random bipartite graphs, loopless directed graphs, and hypergraphs.
• The proof technique, based on fixed-point analysis and operator ratios, provides a robust foundation for future asymptotic enumeration in combinatorics.