Random graphical models

Causality in amongst confusion

October 26, 2021 — February 7, 2022

Figure 1

Reading the The “It’s really complicated and sad” theory of obesity put a question in my head about random structural models and what we can learn from them. I will be unlikely to return to and work through this right this minute, but it occurs to me that it is worth keeping around a list of models that sound like they are worth looking at

What is a good prior over causal graphs? With interactions? Good, in my current mode of thought, would mean, what classes of random causal graphs could e have that were intermediate in complexity between homogenous structure and maybe non-trivial structure. For example, monotonic, or multiplicative interactions with sparse links.

When we are concerned with sampling models over random graphs we might consider Exponential Random Graphs Model, a.k.a. ERGM models. I have some perfunctory notes on that theme under graph sampling. I wonder if that will subsume this idea or not?

I should read upon random graph theory and possible sparse random hypergraphs, e.g. Bapst and Coja-Oghlan (2016), as seen in theoretical analysis of message passing.

Figure 2

1 References

Amini, Cont, and Minca. 2013. Resilience to Contagion in Financial Networks.” Mathematical Finance.
Bapst, and Coja-Oghlan. 2016. Harnessing the Bethe Free Energy.” Random Structures & Algorithms.
Du, Song, Yuan, et al. 2012. Learning Networks of Heterogeneous Influence.” In Advances in Neural Information Processing Systems.
Gauthier, Bollt, Griffith, et al. 2021. Next Generation Reservoir Computing.” Nature Communications.
Giryes, Sapiro, and Bronstein. 2016. Deep Neural Networks with Random Gaussian Weights: A Universal Classification Strategy? IEEE Transactions on Signal Processing.
Glasserman, and Young. 2016. Contagion in Financial Networks.” Journal of Economic Literature.
Glymour. 2007. When Is a Brain Like the Planet? Philosophy of Science.
Goudarzi, and Teuscher. 2016. Reservoir Computing: Quo Vadis? In Proceedings of the 3rd ACM International Conference on Nanoscale Computing and Communication. NANOCOM’16.
Grzyb, Chinellato, Wojcik, et al. 2009. Which Model to Use for the Liquid State Machine? In 2009 International Joint Conference on Neural Networks.
Haldane, and May. 2011. Systemic Risk in Banking Ecosystems. Nature.
Hirata, and Ulanowicz. 1985. Information Theoretical Analysis of the Aggregation and Hierarchical Structure of Ecological Networks.” Journal of Theoretical Biology.
Martinsson. 2016. Randomized Methods for Matrix Computations and Analysis of High Dimensional Data.” arXiv:1607.01649 [Math].
Roca, Draief, and Helbing. 2011. Percolate or Die: Multi-Percolation Decides the Struggle Between Competing Innovations.”
Scardapane, and Wang. 2017. Randomness in Neural Networks: An Overview.” Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery.
Stadler, Fontana, and Miller. 1993. “Random Catalytic Reaction Networks.” Physica D: Nonlinear Phenomena.
Watts, and Dodds. 2007. Influentials, Networks, and Public Opinion Formation.” Journal of Consumer Research.
Wigner. 1955. Characteristic Vectors of Bordered Matrices With Infinite Dimensions.” The Annals of Mathematics.