Efficient and exact sampling of simple graphs with given arbitrary degree sequence (1002.2975v1)

Abstract: Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large N, and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.

• The paper presents an efficient, polynomial-time algorithm for exact sampling of simple graphs with arbitrary given degree sequences, addressing a long-unresolved problem in network theory.
• The proposed algorithm is rejection-free, guarantees statistically independent samples, and overcomes limitations of traditional methods like MCMC or the Configuration Model.
• This advancement allows for unbiased estimation of network observables and broadens the scope for analyzing complex networks constrained solely by degree sequences.

Overview of Graph Sampling Methodology

This paper presents an efficient algorithm for sampling simple graphs that have a given arbitrary degree sequence. This advancement addresses a fundamental issue in network theory that has long been unresolved. Sampling such graphs is crucial for simulating network observables, which have numerous applications in disciplines like epidemiology, social networks, and Internet modeling. Traditional methods such as Markov-Chain Monte Carlo (MCMC) algorithms and the Configuration Model (CM) are plagued by limitations like unknown mixing times and unrestricted rejections, respectively.

Key Contributions

The proposed algorithm operates in polynomial time and guarantees the generation of statistically independent graph samples. This distinguishes it from other existing algorithms, as it is rejection-free and systematically constructs graph samples without backtracking or restarting, which are common issues in other methods. Furthermore, the algorithm produces sample weights that enhance the measurement of observable characteristics over graph ensembles, allowing for the samples to be assessed uniformly or with specific distributions.

Mathematical Foundations

The algorithm's mathematical basis is deeply rooted in the Erdős-Gallai theorem, used to ascertain whether a sequence of integers can be realized as a degree sequence of a simple graph. Several key theorems and lemmas are derived and utilized, including star-constrained graphicality and configurations that ensure the exclusion of connections to a restricted set of nodes. The introduction and application of these mathematical results pave the way for the efficient construction of the allowable set of nodes that can be linked to hub nodes, forming the basis for the sampling technique.

Algorithm Functionality and Complexity

The paper presents a rigorous explanation of the algorithm, discussing the construction of the allowed set and necessary constraints to avoid the construction of non-graphical sequences. The algorithm's complexity is determined to be at most $O(N^3)$, where $N$ is the number of nodes by leveraging recurrence relations to efficiently compute necessary calculations for graph feasibility. Empirical evaluations demonstrate that the mean and standard deviation of the logarithm of sample weights scale with network size as power laws, supporting the algorithm's efficiency.

Implications and Future Developments

The proposed algorithm advances the practice of studying network models by providing a reliable method to sample graphs from any graphical degree sequence and, by extension, any degree sequence ensemble. This allows researchers to explore network classes solely constrained by degree sequences, eliminating biases inherent in using traditional network models. The flexibility and efficacy of this algorithm suggest its potential for extension, possibly incorporating additional constraints related to graph topology or connection properties beyond degree sequences.

Conclusion

This paper contributes significantly to network theory by offering a robust solution to the graph sampling problem, supporting an unbiased estimation of network observables through precise sample weight calculation. The systematic approach and new theoretical insights underpinning the algorithm position it as a valuable tool for complex network analysis, thus broadening the scope for future research developments in this domain.