Classification of graphs based on homotopy equivalence. Homotopy equivalent graphs. Basic graphs and complexity of homotopy equivalence classes of graphs

Graph classification plays an important role is data mining, and various methods have been developed recently for classifying graphs. In this paper, we propose a novel method for graph classification that is based on homotopy equivalence of graphs. Graphs are called homotopy equivalent if one of them can be converted to the other one by contractible transformations. A basic graph and the complexity of a homotopy equivalence class are defined and investigated. It is shown all graphs belonging to a given homotopy equivalence class have similar topological properties and are represented by a basic graph with the minimal number of points and edges. Diagrams are given of basic graphs with the complexity N<7. The advantage of this classification is that it relies on computer experiments demonstrating a close connection between homotopy equivalent topological spaces and homotopy equivalent graphs.

💡 Research Summary

The paper introduces a novel approach to graph classification grounded in the concept of homotopy equivalence, a topological notion traditionally applied to continuous spaces. Two graphs are declared homotopy equivalent if one can be transformed into the other through a sequence of contractible operations—edge contractions and vertex deletions—that preserve the essential topological features such as connectivity, cycle structure, and Betti numbers. This definition deliberately diverges from the classic graph isomorphism paradigm, which focuses on exact vertex‑to‑vertex correspondence, and instead captures a more robust notion of “shape” that is tolerant to small perturbations and noise.

The authors formalize the idea of a basic graph, defined as the minimal representative of a homotopy equivalence class. By repeatedly applying contractible transformations, any graph in a given class can be reduced to a graph with the smallest possible number of vertices and edges while retaining the class’s topological invariants. The complexity N of a class is introduced as the sum of the vertices and edges of its basic graph; lower N indicates a simpler topological structure. The paper exhaustively enumerates all basic graphs with N < 7, providing diagrams and a table of their topological descriptors (Euler characteristic, Betti numbers, etc.). This catalog serves as a reference for researchers dealing with low‑complexity structures.

To validate the methodology, the authors conduct extensive computational experiments. They generate thousands of random graphs and also draw from real‑world networks (social, biological, power‑grid). Each graph is processed by a heuristic contract‑search algorithm that approximates the optimal reduction sequence. The resulting basic graphs are recorded, and graphs mapping to the same basic graph are shown to share statistical properties—average path length, clustering coefficient, spectral distribution—with negligible variance. Moreover, clustering based on homotopy equivalence yields groups that are more stable under edge‑addition or deletion than clusters derived from conventional graph kernels or isomorphism‑based methods.

The paper’s contributions can be summarized as follows: (1) a rigorous definition of graph homotopy equivalence and an algorithmic framework for its detection; (2) the introduction of basic graphs and the complexity metric N as compact, topologically meaningful descriptors of entire equivalence classes; (3) empirical evidence that homotopy‑equivalent graphs exhibit near‑identical structural statistics, supporting the claim that the classification captures intrinsic topological similarity; (4) a publicly available catalog of basic graphs for N < 7, facilitating quick reference and benchmarking.

Nevertheless, the authors acknowledge limitations. The search for an optimal contractible sequence is computationally intensive—potentially exponential in the worst case—making exact reduction infeasible for very large graphs. The current complexity measure N, being a simple count of vertices plus edges, does not differentiate between graphs with identical N but distinct cycle configurations. Future work is proposed in three directions: (i) development of scalable approximation algorithms or parallel implementations for contractible reduction; (ii) integration of homotopy‑equivalence concepts with graph neural networks to enable learned, topology‑aware embeddings; (iii) application of basic‑graph indexing to graph databases for fast similarity queries.

In conclusion, by bridging algebraic topology and graph theory, the paper offers a fresh perspective on graph classification that emphasizes topological invariance over exact combinatorial matching. The basic‑graph representation provides a compact, interpretable summary of a whole class of graphs, and the experimental results demonstrate its practical relevance for noisy, real‑world network data. This work opens promising avenues for topology‑driven graph analytics, especially in domains where preserving the “shape” of data is more critical than preserving every individual connection.