String Topology on Networks: Loop Coproducts for Multi-Brain and Multi-Agent Connectivity

This project proposes to treat dynamic networks (such as brain-to-brain or multi-agent sensor data) as stratified simplicial objects with path spaces and define string-topological operations—like loop product, coproduct, and bracket—on their path homology or Hochschild-like models of the adjacency category. Building on Tadić et al.’s work on higher-order clique complexes and Rivera, Takeda, and Wang’s algebraic models of loop coproducts, the key novelty is moving beyond static Betti numbers to algebraic operations that quantify how loops split and merge across individuals, serving as algebraic "interaction fingerprints." The hypothesis is that the strength and algebraic type of these coproducts correlate with comprehension, engagement, or leadership roles in group communication. This approach could open a new operational topological data analysis framework for networked cognition and communication networks, with testable predictions and robust invariants under noisy edges.

References:

1. Algebraic Topology of Multi-Brain Connectivity Networks Reveals Dissimilarity in Functional Patterns during Spoken Communications. B. Tadić, M. Andjelković, B. Boshkoska, Zoran Levnajic (2016). PLoS ONE.
2. Algebraic string topology from the neighborhood of infinity. M. Rivera, Alex Takeda, Zhengfang Wang (2023).