# Spectral graph theory

Linear signals on graphs

November 24, 2014 — October 27, 2021

Placeholder for the application of classic linear systems theory applied to networks in the form of spectral graph theory, which I split off from the networks notebook, since that started to feel like algebraic graph methods, mostly. As used in graph nueral networks.

I am interested in how filters can be defined on graphs; apparently they can?

Defferrard, Bresson, and Vandergheynst (2016) is credited with introducing localized Cheybshev filtering defined in terms of graph Laplacian to the neural nets. David I. Shuman, Vandergheynst, and Frossard (2011) seems to introduce this in a signal processing setting. (see overview in D. I. Shuman et al. (2013)). Isufi et al. (2017) defines an analogue of standard linear filters. There is a helpful visual comparison of both methods in Andreas Loukas’ blog post.

Many overlaps, different phrasing: matrix factorisation.

## 2 Tooling

Laplacians.jl (Julia):

Laplacians is a package containing graph algorithms, with an emphasis on tasks related to spectral and algebraic graph theory. It contains (and will contain more) code for solving systems of linear equations in graph Laplacians, low stretch spanning trees, sparsifiation, clustering, local clustering, and optimization on graphs.

All graphs are represented by sparse adjacency matrices. This is both for speed, and because our main concerns are algebraic tasks. It does not handle dynamic graphs. It would be very slow to implement dynamic graphs this way.

Much more tooling under graph neural networks.

## 3 References

Belkin, and Niyogi. 2003. Neural Computation.
Butler, and Chung. 2013. In Handbook of Linear Algebra.
Chung, and Graham. 1997. Spectral Graph Theory.
Defferrard, Bresson, and Vandergheynst. 2016. In Advances In Neural Information Processing Systems.
Diaconis, and Stroock. 1991. The Annals of Applied Probability.
Ding, He, and Simon. 2005. In Proceedings of the 2005 SIAM International Conference on Data Mining. Proceedings.
Hogben, ed. 2014. Handbook of Linear Algebra. Discrete Mathematics and Its Applications.
Isufi, Loukas, Simonetto, et al. 2017. IEEE Transactions on Signal Processing.
Kipf, and Welling. 2016. In arXiv:1609.02907 [Cs, Stat].
Luxburg. 2007. “A Tutorial on Spectral Clustering.”
Mahoney. 2016. arXiv Preprint arXiv:1608.04845.
Newman. 2006. Proceedings of the National Academy of Sciences.
Shuman, D. I., Narang, Frossard, et al. 2013. IEEE Signal Processing Magazine.
Shuman, David I., Vandergheynst, and Frossard. 2011. 2011 International Conference on Distributed Computing in Sensor Systems and Workshops (DCOSS).
Solé-Ribalta, De Domenico, Kouvaris, et al. 2013. Physical Review E.
Spielman, and Teng. 2008a. arXiv:0808.4134 [Cs].
———. 2008b. arXiv:0809.3232 [Cs].
Xia, Sun, Yu, et al. 2021. IEEE Transactions on Artificial Intelligence.