Coarse graining

Persistent Homolgy

What’s that?

Petri, G., Expert, P., Turkheimer, F., Carhart-Harris, R., Nutt, D., Hellyer, P. J., & Vaccarino, F. (2014). Homological scaffolds of brain functional networks. in Journal of The Royal Society Interface, 11(101), 20140873. DOI

Talks about a fun-sounding “persistent homology” idea, which sounds a little like some kind of topological measure theory to my analytics-biased perspective:

Persistent homology is a recent technique in computational topology developed for shape recognition and the analysis of high dimensional datasets [36,37]. It has been used in very diverse fields, ranging from biology [38,39] and sensor network coverage [40] to cosmology [41]. Similar approaches to brain data [42,43], collaboration data [44] and network structure [45] also exist. The central idea is the construction of a sequence of successive approximations of the original dataset seen as a topological space X. This sequence of topological spaces \(X_0, X_1, \dots{}, X_N = X\) is such that \(X_i \subseteq X_j\) whenever \(i < j\) and is called the filtration.

Canon

• Bar-Yam, Y. (2003). Dynamics Of Complex Systems. Westview Press.
• Castiglione, P., & Falcioni, M. (2008). Chaos and Coarse Graining in Statistical Mechanics. Cambridge, UK ; New York: Cambridge University Press.
• NESCI’s multiscale methods page