Degree-preserving graph dynamics -- a versatile process to construct random networks

TL;DR

This paper develops a rigorous mathematical framework for degree-preserving graph dynamics, demonstrating its ability to replicate real-world network structures and analyzing the complexity of network construction via this process.

Contribution

It introduces a formal mathematical theory for degree-preserving graph dynamics and proves the NP-completeness of the DPG feasibility problem.

Findings

Degree sequences of real-world networks can be reconstructed using DPG models.

Deciding DPG feasibility is NP-complete.

DPG models can generate diverse network structures from small seeds.

Abstract

Real-world networks evolve over time via additions or removals of vertices and edges. In current network evolution models, vertex degree varies or grows arbitrarily. A recently introduced degree-preserving network growth (DPG) family of models preserves vertex degree, resulting in structures significantly different from and more diverse than previous models ([Nature Physics 2021, DOI: 10.1038/s41567-021-01417-7]). Despite its degree preserving property, the DPG model is able to replicate the output of several well-known real-world network growth models. Simulations showed that many well-studied real-world networks can be constructed from small seed graphs. Here we start the development of a rigorous mathematical theory underlying the DPG family of network growth models. We prove that the degree sequence of the output of some of the well-known, real-world network growth models can be reconstructed via the DPG process, using proper parametrization. We also show that the general problem of deciding whether a simple graph can be obtained via the DPG process from a small seed (DPG feasibility) is, as expected, NP-complete. It is an important open problem to uncover whether there is a structural reason behind the DPG-constructibility of real-world networks.