On the Uniform Sampling of the Configuration Model with Centrality Constraints

TL;DR

This paper introduces a k-edge swapping MCMC algorithm for uniformly sampling simple graphs with fixed Onion Decomposition and degree sequences, improving the realism of null models for complex networks.

Contribution

It develops an efficient k-edge swap method for constrained graph sampling, addressing connectivity issues and enhancing the structural realism of null models.

Findings

The 2-edge swap algorithm is non-connected for small graphs.

Numerical experiments show 2-edge swaps are effective in practical cases.

The proposed model better reproduces meso-scale structures of real networks.

Abstract

The Onion Decomposition has recently been shown to provide principled models of complex graphs that better reproduce the sparse networks found in nature, but at the cost of complicated connection rules. We propose a k-edge swapping MCMC algorithm to efficiently obtain a uniform sample from the ensemble of simple graphs with a fixed Onion Decomposition and degree sequence. We prove the non-connectivity of the 2-edge swap algorithm for some small graphs, but then provide numerical experiments to show that this non-connectivity is not a problem for 2-edge swap in many practical cases, and likely irrelevant when using k-edge swaps with k>2. We finish by comparing our null model to other well-known models in the literature, and show that keeping constraints on the meso-scale structures of the Onion Decomposition greatly increases both the structural and functional realism of random graph null model.