Holey graphs: very large Betti numbers are testable

2024-01-11

arXiv.org

AI Summary

This paper investigates the local testability of the $k$-th Betti number $eta_k$ in sparse graphs: given an $n$-vertex graph, can we determine with high probability—using only a constant number of adjacency queries—whether $eta_k$ achieves the maximal asymptotic scale $Omega(n^k)$? Contrary to the conventional belief that higher-order homology is inherently non-local, we establish, for the first time, constant-query testability of $eta_k$. Our method leverages graph limit theory and random local sampling, designing an $O(1)$-query algorithm that exploits the simplicial homology structure of neighborhood subgraphs. We rigorously characterize the testability threshold as $eta_k = Omega(n^k)$ and prove that the testing error remains bounded and independent of $n$. This result breaks a long-standing barrier in algebraic topology, where topological invariants were deemed difficult to verify locally, and provides the first efficient testing paradigm for homological analysis of large-scale graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Problem

• Test large Betti numbers
• Constant queries dense model
• Tolerantly test clique-freeness

Innovation

• Constant queries in dense graphs
• Tolerant testing of clique-freeness
• Combines Euler characteristic and matroid theory