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

Real-world networks evolve over time via additions or removals of nodes and edges. In current network evolution models, node degree varies or grows arbitrarily. A recently introduced degree-preserving network growth (DPG) family of models preserves node degree, resulting in structures significantly different from and more diverse than previous models (Nature Physics 2021, DOI:10.1038/s41567-021-01417-7). Here we present a rigorous mathematical theory underlying the DPG family of network growth models. We prove that the general problem of deciding whether a simple graph can be obtained via the DPG process from a small "kernel" graph (DPG feasibility) is NP-complete, in contrast with the surprising numerical observation that most real-world networks are actually easily constructible by this process; a dichotomy that still needs to be understood. We demonstrate how some of the well-known network models can be constructed via the DPG process, using proper parametrization.