Real-world networks evolve over time by adding or removing vertices and edges. In current network evolution models, vertex degrees change or increase arbitrarily. The recently introduced degree-preserving network growth (DPG) family of models preserves the vertex degree, resulting in significantly different and more diverse structures than previous models ([Nature Physics 2021,
    DOI: 10.1038/s41567-021-01417-7]). Despite its degree-preserving property, the DPG model can replicate the output of several well-known real-world network growth models. Simulations have shown that many well-studied real-world networks can be constructed from small seed graphs.

    We now begin to develop the rigorous mathematical theory underlying the network growth model of the DPG family. We prove that it is possible to reconstruct the output degree sequences of several well-known real-world network growth models via the DPG process using suitable parameterizations. We also show that the general problem of determining whether a simple graph can be obtained via the DPG process from a small seed (DPG feasibility) is, as expected, NP-complete. Clarifying whether there is a structural reason behind his DPG composability of real-world networks is an important open question.

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