Chapter 3 Persistent Homology and Polarization

Persistent homology is cool and all, but how does it relate to congressional polarization? How can finding the number of loops and connected components at varyious radii reveal important features of polarization?

The key idea is that the homology helps tell something about the broad shape of the data points. For example, if data points in 2-dimensional DW-NOMINATE space are generally organized into three clusters, the barcode will show three connected components persisting for a broad range of proximity parameter values. Additionally, if the data points are generally more spread out, it will be easier to form loops, and thus 1-dimensional holes will be more persistent.

An important feature of this topological approach is that it is much more flexible than previous methods. Much of the literature has focused mainly on the ideological difference between party means, but this strategy could miss important structure in the data. For example, what if the data was most naturally clustered into 3 groups, or 4? Or what about ideological variation within a party, meaning the data points are more spread out? Or what about the presence of a few members way off at the ideological extremes, away from the rest of the members? None of these features would be well captured by standard approaches, but they would all tell us something important about the underlying “shape” of congressional ideology.