Topological Data Analysis

An issue that's been raised at prior lab meetings is that correlation matrices (and higher order correlation matrices) fundamentally characterize *pairs* of interactions (or pairs of pairs of interactions, etc.)

[Topological Data Analysis (TDA)](https://en.wikipedia.org/wiki/Topological_data_analysis) may be a principled way of formally looking at triplet, quadruplet, etc. interactions. The fundamental characterization of patterns of interactions is the [persistence diagram](https://en.wikipedia.org/wiki/Topological_data_analysis#Visualization), which reflects how units interact on different scales (e.g. 1, 2, 3, 4, etc. interacting units).

We could use TDA to create persistence diagrams for each "level" at each moment, and that may provide insights into these multi-feature (beyond pair) interactions over time.