Zigzag Persistence of Large Language Models Representations

We analyze internal representations of large language models with zigzag persistent homology, treating depth as a discrete time axis for point clouds of last-token embeddings. At each layer we build a k-nearest-neighbors clique complex, connect adjacent layers via intersections, and summarize the resulting diagrams with effective persistence images. From these we derive two descriptors: Births’ Relative Frequency (at what rate new p-dimensional features appear) and Inter-Layer Persistence (how long they survive across depth). On the SST movie reviews dataset and three open-source models (Llama-3.1, OSS-20B, Phi-4), we consistently observe three evolving phases: early rapid changes, a middle regime of stable organization, and a final reorganization before output. Using the stability signal (inter-layer persistence) to guide where to remove contiguous blocks of layers, we find that pruning within high-persistence regions maintains 5-shot MMLU performance (with the same trend visible even for the more pruning-sensitive OSS-20B). This suggests that zigzag-based summaries capture meaningful, system-level dynamics and can inform lightweight pruning.