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Degree-preserving network growth


Real-world networks evolve over time through the addition or removal of nodes and edges. In current network-evolution models, the degree of each node varies or grows arbitrarily, yet there are many networks for which a different description is required. In some networks, node degree saturates, such as the number of active contacts of a person, and in some it is fixed, such as the valence of an atom in a molecule. Here we introduce a family of network growth processes that preserve node degree, resulting in structures substantially different from those reported previously. We demonstrate that, despite it being an NP (non-deterministic polynomial time)-hard problem in general, the exact structure of most real-world networks can be generated from degree-preserving growth. We show that this process can create scale-free networks with arbitrary exponents, however, without preferential attachment. We present applications to epidemics control via network immunization, to viral marketing, to knowledge dissemination and to the design of molecular isomers with desired properties.

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Fig. 1: Degree-preserving growth.
Fig. 2: Real-world networks from DPG process.
Fig. 3: DPG models.
Fig. 4: Scale-free DPG networks.
Fig. 5: DPG in network design.

Data availability

Source data are provided with this paper. The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The codes used for simulation and analysis are available from the corresponding author upon reasonable request.


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We thank D. Soltész, I. Miklós, N. Rupprecht and J. Baker for useful discussions. The project was supported in part by the United States National Science Foundation through grant IIS-1724297 (Z.T.) and by the National Research, Development and Innovation Office of Hungary through NKFIH grants K 116769, KH 126853, K 132696 and SNN 135643 (P.L.E. and T.R.M.).

Author information

Authors and Affiliations



S.R.K. performed mathematical modelling, contributed proofs, provided analysis tools, designed and ran experiments, analysed data and generated the figures. S.C. ran experiments. T.R.M. and P.L.E. performed mathematical modelling and provided proofs. Z.T. proposed the main idea, performed mathematical modelling, contributed proofs and was the lead writer of the manuscript. All authors contributed to proofreading the paper.

Corresponding author

Correspondence to Zoltan Toroczkai.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks Thilo Gross and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary text, derivations, proofs, Figs. 1–11 and Tables 1 and 2.

Source data

Source Data Fig. 2

Statistical source data.

Source Data Fig. 3

Statistical source data.

Source Data Fig. 4

Statistical source data.

Source Data Fig. 5

Statistical source data.

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Kharel, S.R., Mezei, T.R., Chung, S. et al. Degree-preserving network growth. Nat. Phys. 18, 100–106 (2022).

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