Differential calculus on graphon space and statistical applications of graph limit theory
- In this thesis, we build on the beautiful work on dense graph limit theory in two directions. In the first, we develop a framework for differential calculus on the space of graph limits. In the process, we discover a structure theory for differentiable graphon parameters and find that homomorphism densities can be used to expand such parameters in Taylor series. The methods developed are novel, robust and can be generalized. In the second, we use dense graph limit theory to provide a new framework for the study of stability of graph partitioning methods. By formulating statistical consistency as a continuity result on the graphon space, we obtain robust consistency results independent of needing to assume a specific form of the data generating mechanism. We derive the consistency of commonly used clustering algorithms such as clustering based on local graph statistics as well as spectral clustering using the normalized Laplacian. In the final chapter, we indicate how this work can lead to the discovery of new necessary mathematical abstractions to serve as foundations for modern data analysis in areas such as network science or machine learning.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Diao, Peter Zhiyi
|Stanford University, Department of Mathematics.
|Statement of responsibility
|Peter Zhiyi Diao.
|Submitted to the Department of Mathematics.
|Thesis (Ph.D.)--Stanford University, 2016.
- © 2016 by Peter Diao
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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