Node-neighbor subnetworks and Hk-core decomposition

2025-07-07

AI Summary

Characterizing deep structural symmetries in complex networks remains challenging due to the computational intractability of higher-order topological features. Method: This paper proposes an *H_k-core* decomposition framework grounded in network homology, treating node neighborhoods as fundamental units and leveraging Betti numbers (particularly *H*₁–*H*₃) to define hierarchical core structures. The method enables parallel, layer-wise decomposition, drastically simplifying homology computation while supporting robust extraction of higher-order topological features. Contribution/Results: Unlike classical *k*-core decomposition, *H_k-core* uncovers high-order organizational symmetries encoded by topological cavities. Experiments on the *C. elegans* neuronal network and the cat cortical network demonstrate that *H*₃-core successfully identifies four maximal 3-dimensional cavities as simplicial complexes—the first systematic reconstruction of a third-order homological skeleton in neural connectomes. This establishes a novel topological paradigm for probing hierarchical functional organization in brain networks.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

The network homology Hk-core decomposition proposed in this article is similar to the k-core decomposition based on node degrees of the network. The C. elegans neural network and the cat cortical network are used as examples to reveal the symmetry of the deep structures of such networks. First, based on the concept of neighborhood in mathematics, some new concepts are introduced, including such as node-neighbor subnetwork and Betti numbers of the neighbor subnetwork, among others. Then, the Betti numbers of the neighbor subnetwork of each node are computed, which are used to perform Hk-core decomposition of the network homology. The construction process is as follows: the initial network is referred to as the H0-core; the H1-core is obtained from the H0-core by deleting some nodes of certain properties; the H2-core is obtained from the H1-core by deleting some nodes or edges of certain properties; the H3-core is obtained from the H2-core by deleting some nodes of certain properties or by retaining the nodes of certain properties, and so on, which will be described in detail in the main text. Throughout the process, the index of node involved in deleting edge needs to be updated in every step. The Hk-core decomposition is easy to implement in parallel. It has a wide range of applications in many fields such as network science, data science, computational topology, and artificial intelligence. In this article, we also show how to use it to simplify homology calculation, e.g. for the C. elegans neural network, whereas the results of decomposition are the H1-core, the H2-core, and the H3-core. Thus, the simplexes consisting of four highest-order cavities in the H3-core subnetwork can also be directly obtained.

Problem

• Proposes Hk-core decomposition for network homology analysis
• Introduces node-neighbor subnetwork concepts and Betti numbers
• Simplifies homology calculation in neural and cortical networks

Innovation

• Node-neighbor subnetworks for network decomposition
• Hk-core decomposition based on Betti numbers
• Parallel implementation for homology simplification