Persistent A-homotopy of Complex Networks
Abstract
We study how loop classes in a network change when cycles up to a prescribed length are declared contractible. The resulting sequence of quotient groups contains the fundamental group of the clique complex at length three and the graph A-fundamental group at length four. Its mod-two abelianisation has a barcode whose death scales are the lengths in a minimum cycle basis. This separates the choice of a network filtration from the choice of local loop relations and provides a computable connection between persistent A-homotopy and persistent homology. We illustrate the construction on lattices, Watts--Strogatz networks and nine benchmark networks. Filling the plaquettes of a periodic square lattice recovers the fundamental group of the torus. In the Watts--Strogatz simulations, the normalised largest death scale falls below the normalised mean path length over the sampled range, while most clique-homology classes survive the square relations at small rewiring probabilities. The power grid has fewer surviving classes but a longer tail of death scales than five degree-preserving randomisations. We also examine connectivity graphs of maximal cliques in ego networks. Explicit certificates establish non-trivial A-classes even though the original ego clique complexes are contractible. The group construction and its linear summary answer related but distinct questions about which loop relations a network model permit.
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BibTeX
@misc{rajkovic2026persistent,
title = {{Persistent A-homotopy of Complex Networks}},
author = {Milan Rajkovic},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.09967},
eprint = {2611.09967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.09967},
}
RIS
TY - UNPB
TI - Persistent A-homotopy of Complex Networks
AU - Milan Rajkovic
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.09967
UR - https://arxiv.org/abs/2611.09967
AB - We study how loop classes in a network change when cycles up to a prescribed length are declared contractible. The resulting sequence of quotient groups contains the fundamental group of the clique complex at length three and the graph A-fundamental group at length four. Its mod-two abelianisation has a barcode whose death scales are the lengths in a minimum cycle basis. This separates the choice of a network filtration from the choice of local loop relations and provides a computable connection between persistent A-homotopy and persistent homology. We illustrate the construction on lattices, Watts--Strogatz networks and nine benchmark networks. Filling the plaquettes of a periodic square lattice recovers the fundamental group of the torus. In the Watts--Strogatz simulations, the normalised largest death scale falls below the normalised mean path length over the sampled range, while most clique-homology classes survive the square relations at small rewiring probabilities. The power grid has fewer surviving classes but a longer tail of death scales than five degree-preserving randomisations. We also examine connectivity graphs of maximal cliques in ego networks. Explicit certificates establish non-trivial A-classes even though the original ego clique complexes are contractible. The group construction and its linear summary answer related but distinct questions about which loop relations a network model permit.
ER -