Random graphical models

Causality in amongst confusion



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

  • Shalizi’s multi-factorial toy-model of intelligence in g, a Statistical Myth
  • I have a vague notion that Wigner’s random matrix model for atomic nuclei was essentially a random interaction model.
  • Random neural nets in the sense of feature maps
  • Probabilistic neural nets in the sense of posterior updating of coefficients
  • Robert May’s random trophic models (Also Jennifer Dunne’s work, presumably)
  • contagion models?
  • those N-K models?

What is a good prior over causal graphs? 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?

References

Amini, Hamed, Rama Cont, and Andreea Minca. 2013. “Resilience to Contagion in Financial Networks.” Mathematical Finance, October, n/a–. https://doi.org/10.1111/mafi.12051.
Centola, D, and Michael W Macy. 2007. “Complex Contagions and the Weakness of Long Ties.” American Journal of Sociology 113 (3): 702.
Du, Nan, Le Song, Ming Yuan, and Alex J. Smola. 2012. “Learning Networks of Heterogeneous Influence.” In Advances in Neural Information Processing Systems, 2780–88. http://papers.nips.cc/paper/4582-learning-networks-of-heterogeneous-influence.
Gauthier, Daniel J., Erik Bollt, Aaron Griffith, and Wendson A. S. Barbosa. 2021. “Next Generation Reservoir Computing.” Nature Communications 12 (1): 5564. https://doi.org/10.1038/s41467-021-25801-2.
Giryes, R., G. Sapiro, and A. M. Bronstein. 2016. “Deep Neural Networks with Random Gaussian Weights: A Universal Classification Strategy?” IEEE Transactions on Signal Processing 64 (13): 3444–57. https://doi.org/10.1109/TSP.2016.2546221.
Glasserman, Paul, and H. Peyton Young. 2016. “Contagion in Financial Networks.” Journal of Economic Literature 54 (3): 779–831. https://doi.org/10.1257/jel.20151228.
Goudarzi, Alireza, and Christof Teuscher. 2016. “Reservoir Computing: Quo Vadis?” In Proceedings of the 3rd ACM International Conference on Nanoscale Computing and Communication, 13:1–6. NANOCOM’16. New York, NY, USA: ACM. https://doi.org/10.1145/2967446.2967448.
Grzyb, B. J., E. Chinellato, G. M. Wojcik, and W. A. Kaminski. 2009. “Which Model to Use for the Liquid State Machine?” In 2009 International Joint Conference on Neural Networks, 1018–24. https://doi.org/10.1109/IJCNN.2009.5178822.
Haldane, Andrew G, and Robert M May. 2011. “Systemic Risk in Banking Ecosystems.” Nature 469: 351–55. https://doi.org/10.1038/nature09659.
Hirata, Hironori, and Robert E Ulanowicz. 1985. “Information Theoretical Analysis of the Aggregation and Hierarchical Structure of Ecological Networks.” Journal of Theoretical Biology 116 (3): 321–41. https://doi.org/10.1016/S0022-5193(85)80271-X.
Martinsson, Per-Gunnar. 2016. “Randomized Methods for Matrix Computations and Analysis of High Dimensional Data.” arXiv:1607.01649 [Math], July. http://arxiv.org/abs/1607.01649.
Roca, Carlos P, Moez Draief, and Dirk Helbing. 2011. “Percolate or Die: Multi-Percolation Decides the Struggle Between Competing Innovations.” http://www.arxiv.org/pdf/1101.0775.
Scardapane, Simone, and Dianhui Wang. 2017. “Randomness in Neural Networks: An Overview.” Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 7 (2). https://doi.org/10.1002/widm.1200.
Watts, Duncan J., and Peter Sheridan Dodds. 2007. “Influentials, Networks, and Public Opinion Formation.” Journal of Consumer Research 34 (4): 441–58. https://doi.org/10.1086/518527.
Wigner, Eugene P. 1955. “Characteristic Vectors of Bordered Matrices With Infinite Dimensions.” The Annals of Mathematics 62 (3): 548. https://doi.org/10.2307/1970079.

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