As Large Language Models (LLMs) are used for increasingly complex cognitive tasks, a natural question is whether AI really understands. The study of understanding in LLMs is in its infancy, and the community has yet to incorporate research and insights from philosophy, psychology, and education. Here we focus on understanding algorithms, and propose a hierarchy of levels of understanding. We validate the hierarchy using a study with human subjects (undergraduate and graduate students). Following this, we apply the hierarchy to large language models (generations of GPT), revealing interesting similarities and differences with humans. We expect that our rigorous criteria for algorithm understanding will help monitor and quantify AI's progress in such cognitive domains.

The $k$-cap (or $k$-winners-take-all) process on a graph works as follows: in each iteration, a subset of $k$ vertices of the graph are identified as winners; the next round winners are the vertices that have the highest total degree from the current winners, with ties broken randomly. This natural process is a simple model of firing activity and inhibition in the brain and has been found to have desirable robustness properties as an activation function. We study its convergence on directed geometric random graphs in any constant dimension, revealing rather surprising behavior, with the support of the current active set converging to lie in a small ball and the active set itself remaining essentially random within that.