Why does each node in the wavelet scattering transform split into multiple paths? Why does each node in the wavelet scattering transform split into multiple paths as in this figure from https://arxiv.org/abs/1304.6763?

I understand roughly what’s happening along a single path, but I don’t understand why each node splits into three.
Thanks!
We take a wavelet transform over other wavelet transforms, which involves multiple wavelets. E.g.:

First-order uses 100 wavelets to tile the frequency domain and produce 100 coefficients
Second-order uses 16 wavelets to tile the frequency domain and produces 8 coefficients per first-order coefficient, yielding a total of 800
Third-order uses 9 wavelets ... etc

(Note second-order wouldn't actually produce 800 in implementation since xi2 &gt; xi1 wavelets are discarded, where xi == center frequency).
The idea is, first-order coefficients capture temporal modulations (AM-FM). Second-order capture the modulations of those modulations, extracting higher order structure, and the process repeats. Progressively we require less wavelets as there's less fine-grained structure to capture (also energy and non-redundancy reasons), but it's still more than one wavelet (i.e. path), as just one wavelet cannot cover entire frequency domain without losing information (unless it's very wide, but that's a bad wavelet).
See helpful lecture, and Python implementation. I also might write a detailed post on this topic in coming weeks.