Graph statistics: An emerging discipline in non-Euclidean data analysis
The explosive growth of complex data has catalyzed the emergence of graph statistics as a fundamentally new discipline in data science. Unlike traditional statistics, which operates primarily within the comfortable confines of Euclidean spaces, graph statistics confronts the reality that modern data naturally organize themselves as dynamic networks composed of complex interconnections. In this article, we present an overview of graph statistics as an emerging discipline, tracing its theoretical foundations, methodological innovations, and transformative applications. We examine how the integration of evolutionary game theory, ecological niche theory, topological data analysis, and graph theory through quasi-dynamic nonlinear modeling has created a new norm of statistical thinking capable of analyzing non-Euclidean data. We show how graph statistics is poised to revolutionize fields ranging from quantitative genetics and systems biology to materials science and artificial intelligence, offering a principled framework for transforming big data into practical knowledge.
Lecturer
Date
14th September ~ 9th November, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Tuesday | 14:20 - 16:55 | Shuimo | ZOOM 01 | 928 682 9093 | BIMSA |
Prerequisite
Core competencies include fundamental statistics (linear models, statistical inference, experimental design, and computational statistics), foundational mathematics (ordinary differential equations, applied topology, Yau-Yau theory, and curvature theory), and computer programming.
Syllabus
Chapter 1 Fundamental Statistics
Chapter 2 Non-Euclidean Data
Chapter 3 Ecological Theory of Complex Systems
Chapter 4 Evolutionary Game Theory
Chapter 5 Quasi-dynamic Modeling and idopNetwork Inference
Chapter 6 Graph Representation of Complex Systems
Chapter 7 Developmental Modularity Theory and Modular Networks
Chapter 8 GLMY Dissection of idopNetworks
Chapter 9 Networks of Stochastic Complex Systems
Chapter 10 How idopNet Enables Artificial Intelligence
Chapter 11 Future Directions
Chapter 2 Non-Euclidean Data
Chapter 3 Ecological Theory of Complex Systems
Chapter 4 Evolutionary Game Theory
Chapter 5 Quasi-dynamic Modeling and idopNetwork Inference
Chapter 6 Graph Representation of Complex Systems
Chapter 7 Developmental Modularity Theory and Modular Networks
Chapter 8 GLMY Dissection of idopNetworks
Chapter 9 Networks of Stochastic Complex Systems
Chapter 10 How idopNet Enables Artificial Intelligence
Chapter 11 Future Directions
Reference
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28. Kim, B. R., Zhang, L., Berg, A., Fan, J., & Wu, R. (2008). A computational approach to the functional clustering of periodic gene-expression profiles. Genetics, 180(2), 821-834.
29. Wang, Y., Xu, M., Wang, Z., Tao, M., Zhu, J., Wang, L., ... & Wu, R. (2012). How to cluster gene expression dynamics in response to environmental signals. Briefings in bioinformatics, 13(2), 162-174.
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32. Dong, A., Fa, C., Li, Z., Yau, S. T., & Wu, R. (2026). Multi-task learning of complex networks via nonlinear ordinary differential equations. Communications Physics.
33. He, J. H., & Liu, J. F. (2009). Allometric scaling laws in biology and physics. Chaos, Solitons & Fractals, 41(4), 1836-1838.
34. Griffin, C., Jiang, L., & Wu, R. (2020). Analysis of quasi-dynamic ordinary differential equations and the quasi-dynamic replicator. Physica A: Statistical Mechanics and its Applications, 555, 124422.
35. Liu, S., Li, Z., Zhang, Y., & Yin, J. (2026). Exact recovery in the double sparse model: Sufficient and necessary signal conditions. Electronic Journal of Statistics, 20(1), 83-137.
36. Grigor'yan, A., Lin, Y., Muranov, Y., & Yau, S. T. (2012). Homologies of path complexes and digraphs. arXiv preprint arXiv:1207.2834.
37. Wu, S., & Zhang, M. (2025). Disentangling complex systems: IdopNetwork meets GLMY homology theory. Data Analytics and Topology, 1(2025), 47-64.
38. Wang, B., Feng, B., Lv, L., Li, S., & Pan, F. (2025). Structural feature extraction via topological data analysis. The Journal of Physical Chemistry Letters, 16(32), 8056-8067.
39. Vetrivel, S. C., Vidhyapriya, P., & Arun, V. P. (2025). The Challenges of Graph Neural Networks. In Graph Neural Networks: Essentials and Use Cases (pp. 79-108). Cham: Springer Nature Switzerland.
40. Zednik, C. (2021). Solving the black box problem: A normative framework for explainable artificial intelligence. Philosophy & technology, 34(2), 265-288.
41. Papillon, M., Sanborn, S., Mathe, J., Cornelis, L., Bertics, A., Buracas, D., ... & Miolane, N. (2025). Beyond euclid: An illustrated guide to modern machine learning with geometric, topological, and algebraic structures. Machine Learning: Science and Technology, 6(3), 031002.
42. Xiao, S., Wang, S., Dai, Y., & Guo, W. (2022). Graph neural networks in node classification: survey and evaluation. Machine Vision and Applications, 33(1), 4.
43. L’heureux, A., Grolinger, K., Elyamany, H. F., & Capretz, M. A. (2017). Machine learning with big data: Challenges and approaches. IEEE Access, 5, 7776-7797.
44. Ma, S., Dong, A., Zhou G., Yang, F., Long, F., Wang, Y., Yau, S.-T., Wu, R. (2026) IdopGNN: A graph statistical mechanics model of data prediction. Data Analytics and Topology (in press).
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47. Wang, Y., Xu, S., Wei, X., Luo, X., Yau, S. S. T., Yau, S. T., & Wu, R. (2025). Yau-YauAL: A computer tool for solving nonlinear filtering problems. Data Analytics and Topology, 1 (2025), 77-84.
48. Xu, S., Wang, Y., Wu, S., Dong, A., Yau, S. S. T., Yau, S. T., & Wu, R. (2026). Statistical learning of stochastic complex systems via the Yau-Yau nonlinear filter. The Innovation, 7(6), 101267.
49. Lin, Y., Lu, L., & Yau, S. T. (2011). Ricci curvature of graphs. Tohoku Mathematical Journal, Second Series, 63(4), 605-627.
50. Opper, M., & Saad, D. (Eds.). (2001). Advanced mean field methods: Theory and practice. MIT press.
2. Christensen, R. (2018). Analysis of variance, design, and regression: Linear modeling for unbalanced data. Chapman and Hall/CRC.
3. Zhang, K., Liu, S., & Xiong, M. (2022). Changes from classical statistics to modern statistics and data science. arXiv preprint arXiv:2211.03756.
4. Martín, N. (2025). Data Science Versus Statistics. In International Encyclopedia of Statistical Science (pp. 621-623). Berlin, Heidelberg: Springer Berlin Heidelberg.
5. Carthew, R. W. (2021). Gene regulation and cellular metabolism: an essential partnership. Trends in Genetics, 37(4), 389-400.
6. Luo, L. (2021). Architectures of neuronal circuits. Science, 373(6559), eabg7285.
7. Barraclough, T. G. (2015). How do species interactions affect evolutionary dynamics across whole communities? Annual Review of Ecology, Evolution, and Systematics, 46(1), 25-48.
8. Wu, R., & Jiang, L. (2021). Recovering dynamic networks in big static datasets. Physics Reports, 912, 1-57.
9. Wu, S., Liu, X., Dong, A., Gragnoli, C., Griffin, C., Wu, J., ... & Wu, R. (2023). The metabolomic physics of complex diseases. Proceedings of the National Academy of Sciences, 120(42), e2308496120.
10. Feng, L., Gong, H., Zhang, S., Liu, X., Wang, Y., Che, J., ... & Wu, R. (2024). Hypernetwork modeling and topology of high-order interactions for complex systems. Proceedings of the National Academy of Sciences, 121(40), e2412220121.
11. Sun, L., Bian, Y., Yang, D., Miao, R., Meng, Y., Che, J., ... & Wu, R. (2026). Graph statistics theory of individualized quantitative genetics under haplotype-resolved genome assembly. Proceedings of the National Academy of Sciences, 123(14), e2600004123.
12. Chen, C., Jiang, L., Fu, G., Wang, M., Wang, Y., Shen, B., ... & Wu, R. (2019). An omnidirectional visualization model of personalized gene regulatory networks. NPJ systems biology and applications, 5(1), 38.
13. Dong, A., Wu, S., Che, J., Wang, Y., & Wu, R. (2023). idopNetwork: a network tool to dissect spatial community ecology. Methods in Ecology and Evolution, 14(9), 2272-2283.
14. Gong, H., Wang, H., Wang, Y., Zhang, S., Liu, X., Che, J., ... & Wu, R. (2024). Topological change of soil microbiota networks for forest resilience under global warming. Physics of Life Reviews, 50, 228-251.
15. Athreya, A., Fishkind, D. E., Tang, M., Priebe, C. E., Park, Y., Vogelstein, J. T., ... & Sussman, D. L. (2018). Statistical inference on random dot product graphs: a survey. Journal of Machine Learning Research, 18(226), 1-92.
16. Charikar, M., Chatziafratis, V., Niazadeh, R., & Yaroslavtsev, G. (2019, April). Hierarchical clustering for euclidean data. In The 22nd International Conference on Artificial Intelligence and Statistics (pp. 2721-2730). PMLR.
17. Song, W., Zhou, H., Zhou, Y., & Müller, H. G. (2026). Non-Euclidean data analysis with metric statistics. Harvard Data Science Review, 8.
18. Faraway, J. J. (2014). Regression for non-Euclidean data using distance matrices. Journal of Applied Statistics, 41(11), 2342-2357.
19. Yang, J., Xie, K., & An, N. (2022, August). Causal discovery on non-euclidean data. In Proceedings of the 28th ACM SIGKDD conference on knowledge discovery and data mining (pp. 2202-2211).
20. Smith, J. M., & Price, G. R. (1973). The logic of animal conflict. Nature, 246(5427), 15-18.
21. Su, Q., McAvoy, A., & Plotkin, J. B. (2023). Strategy evolution on dynamic networks. Nature Computational Science, 3(9), 763-776.
22. Sheng, A., Su, Q., Wang, L., & Plotkin, J. B. (2024). Strategy evolution on higher-order networks. Nature computational science, 4(4), 274-284.
23. Yeang, C. H., Huang, L. C., & Liu, W. C. (2012, August). Recurrent structural motifs reflect characteristics of distinct networks. In 2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (pp. 551-557). IEEE.
24. Newman, M. E. (2006). Modularity and community structure in networks. Proceedings of the national academy of sciences, 103(23), 8577-8582.
25. Ghavasieh, A., & De Domenico, M. (2024). Diversity of information pathways drives sparsity in real-world networks. Nature Physics, 20(3), 512-519.
26. Figueiredo, M. A., & Bioucas-Dias, J. M. (2012, May). Algorithms for imaging inverse problems under sparsity regularization. In 2012 3rd International Workshop on Cognitive Information Processing (CIP) (pp. 1-6). IEEE.
27. Bach, F., Jenatton, R., Mairal, J., & Obozinski, G. (2012). Optimization with sparsity-inducing penalties. Foundations and Trends in Machine Learning, 4(1), 1-106.
28. Kim, B. R., Zhang, L., Berg, A., Fan, J., & Wu, R. (2008). A computational approach to the functional clustering of periodic gene-expression profiles. Genetics, 180(2), 821-834.
29. Wang, Y., Xu, M., Wang, Z., Tao, M., Zhu, J., Wang, L., ... & Wu, R. (2012). How to cluster gene expression dynamics in response to environmental signals. Briefings in bioinformatics, 13(2), 162-174.
30. Pan, W., Che, J., & Wu, S. (2026). An improved method of Wu's functional clustering. Statistics Innovation, 3(1), e005.
31. George, E. I. (2000). The variable selection problem. Journal of the American Statistical Association, 95(452), 1304-1308.
32. Dong, A., Fa, C., Li, Z., Yau, S. T., & Wu, R. (2026). Multi-task learning of complex networks via nonlinear ordinary differential equations. Communications Physics.
33. He, J. H., & Liu, J. F. (2009). Allometric scaling laws in biology and physics. Chaos, Solitons & Fractals, 41(4), 1836-1838.
34. Griffin, C., Jiang, L., & Wu, R. (2020). Analysis of quasi-dynamic ordinary differential equations and the quasi-dynamic replicator. Physica A: Statistical Mechanics and its Applications, 555, 124422.
35. Liu, S., Li, Z., Zhang, Y., & Yin, J. (2026). Exact recovery in the double sparse model: Sufficient and necessary signal conditions. Electronic Journal of Statistics, 20(1), 83-137.
36. Grigor'yan, A., Lin, Y., Muranov, Y., & Yau, S. T. (2012). Homologies of path complexes and digraphs. arXiv preprint arXiv:1207.2834.
37. Wu, S., & Zhang, M. (2025). Disentangling complex systems: IdopNetwork meets GLMY homology theory. Data Analytics and Topology, 1(2025), 47-64.
38. Wang, B., Feng, B., Lv, L., Li, S., & Pan, F. (2025). Structural feature extraction via topological data analysis. The Journal of Physical Chemistry Letters, 16(32), 8056-8067.
39. Vetrivel, S. C., Vidhyapriya, P., & Arun, V. P. (2025). The Challenges of Graph Neural Networks. In Graph Neural Networks: Essentials and Use Cases (pp. 79-108). Cham: Springer Nature Switzerland.
40. Zednik, C. (2021). Solving the black box problem: A normative framework for explainable artificial intelligence. Philosophy & technology, 34(2), 265-288.
41. Papillon, M., Sanborn, S., Mathe, J., Cornelis, L., Bertics, A., Buracas, D., ... & Miolane, N. (2025). Beyond euclid: An illustrated guide to modern machine learning with geometric, topological, and algebraic structures. Machine Learning: Science and Technology, 6(3), 031002.
42. Xiao, S., Wang, S., Dai, Y., & Guo, W. (2022). Graph neural networks in node classification: survey and evaluation. Machine Vision and Applications, 33(1), 4.
43. L’heureux, A., Grolinger, K., Elyamany, H. F., & Capretz, M. A. (2017). Machine learning with big data: Challenges and approaches. IEEE Access, 5, 7776-7797.
44. Ma, S., Dong, A., Zhou G., Yang, F., Long, F., Wang, Y., Yau, S.-T., Wu, R. (2026) IdopGNN: A graph statistical mechanics model of data prediction. Data Analytics and Topology (in press).
45. Yau, S. T., & Yau, S. S. T. (2000). Real time solution of nonlinear filtering problem without memory I. Mathematical Research Letters, 7(6), 671-693.
46. Yau, S. T., & Yau, S. S. T. (2008). Real time solution of the nonlinear filtering problem without memory II. SIAM Journal on Control and Optimization, 47(1), 163-195.
47. Wang, Y., Xu, S., Wei, X., Luo, X., Yau, S. S. T., Yau, S. T., & Wu, R. (2025). Yau-YauAL: A computer tool for solving nonlinear filtering problems. Data Analytics and Topology, 1 (2025), 77-84.
48. Xu, S., Wang, Y., Wu, S., Dong, A., Yau, S. S. T., Yau, S. T., & Wu, R. (2026). Statistical learning of stochastic complex systems via the Yau-Yau nonlinear filter. The Innovation, 7(6), 101267.
49. Lin, Y., Lu, L., & Yau, S. T. (2011). Ricci curvature of graphs. Tohoku Mathematical Journal, Second Series, 63(4), 605-627.
50. Opper, M., & Saad, D. (Eds.). (2001). Advanced mean field methods: Theory and practice. MIT press.
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
Chinese
, English
Lecturer Intro
Rongling Wu, received a Ph.D. in Quantitative Genetics from the University of Washington (Seattle) in 1995. He was a Distinguished Professor of Statistics and Public Health Sciences at Pennsylvania State University, and Director of the Center for Statistical Genetics. He is currently the Zeng Siming Chair Professor of Yau Mathematical Sciences Center, Tsinghua University. He is also a researcher at Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, and also serves as editor-in-chief, associate editor, special editor and editorial board member of several journals in the fields of genetics, bioinformatics and computational biology. He was selected as a fellow of the American Association for the Advancement of Science and the American Statistical Association, and won the Distinguished Researcher Award of the American Institute of Applied Mathematics and Statistics (SAMSI), the University of Florida Research Fund Professor Award, the Pennsylvania State University Distinguished University Professor Award, and the Floyd Science Innovation Award. Research interests include: developing interdisciplinary statistical methods to reveal the genetic control mechanisms of complex traits and human complex diseases. The proposed functional mapping method can effectively discover the genetic rules of trait development and describe the key patterns of gene effects changing over time and space. Combining functional mapping with evolutionary game theory, scale theory, and prey-predator theory, a series of computational methods have been developed to construct multi-level, multi-space, and multi-scale genotype-phenotype relationships from molecules to phenotypes The three-dimensional network provides analysis tools for systems biology, systems medicine, and systems pharmacology research. Published more than 400 SCI papers in important international journals such as Nature Reviews Genetics, Nature Communications, PNAS, Journal of the American Statistical Association, Annals of Applied Statistics, Physics of Life Reviews, Physics Reports, Briefings in Bioinformatics, Cell Reports, Evolution, etc. The research results have been cited or highlighted by important journals such as Science and Cell.