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PreprintCC0arXiv

The mathematical journey of spatial reciprocity

  • Chaoqian Wang
  • Xiang Li
  • Attila Szolnoki
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Abstract

Over the past three decades, evolutionary game theory in structured populations has witnessed explosive progress. Counterintuitive cooperative phenomena discovered in simulations have stimulated the development of rigorous mathematical theories to describe the underlying dynamics. In this paper, we review these theoretical advancements, including pair approximation and coalescent theory, which offer analytical solutions to evolutionary games on networks. We summarize the key results obtained by these theories, while also discussing their limitations, such as high computational complexity and constrained flexibility when applied to complex models. We therefore suggest that mathematical approaches provide a complementary tool for understanding cooperation in structured populations alongside simulations.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026mathematical,
  title = {{The mathematical journey of spatial reciprocity}},
  author = {Chaoqian Wang and Xiang Li and Attila Szolnoki},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35161},
  eprint = {2610.35161},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35161},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - The mathematical journey of spatial reciprocity
AU  - Chaoqian Wang
AU  - Xiang Li
AU  - Attila Szolnoki
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35161
UR  - https://arxiv.org/abs/2610.35161
AB  - Over the past three decades, evolutionary game theory in structured populations has witnessed explosive progress. Counterintuitive cooperative phenomena discovered in simulations have stimulated the development of rigorous mathematical theories to describe the underlying dynamics. In this paper, we review these theoretical advancements, including pair approximation and coalescent theory, which offer analytical solutions to evolutionary games on networks. We summarize the key results obtained by these theories, while also discussing their limitations, such as high computational complexity and constrained flexibility when applied to complex models. We therefore suggest that mathematical approaches provide a complementary tool for understanding cooperation in structured populations alongside simulations.
ER  -