Network Homology Hk-core Decomposition

• Network Homology Hk-core Decomposition is a homology-informed generalization of core decomposition that uses Betti numbers from node-neighbor subnetworks to isolate topologically critical structures.
• It integrates a local pruning scheme with a global spectral formulation via the weighted k-Hodge Laplacian, yielding homology embeddings that highlight fundamental topological components.
• Empirical studies on networks like C. elegans and cat cortex validate its effectiveness, distinguishing it from classical degree-based cores through topology-aware node and edge deletion rules.

Searching arXiv for the cited papers and closely related work on network homology, Hk-core decomposition, higher-order Laplacians, and generalized core decompositions. Network Homology Hk-core Decomposition denotes a family of decompositions that combine core-like pruning with homological descriptors of higher-order network structure. In one explicit formulation, the original simplicial network is the H0-core, and the H1-, H2-, H3-, and higher cores are obtained by iteratively deleting nodes and edges according to Betti numbers of node-neighbor subnetworks and a local characteristic number (Shi et al., 7 Jul 2025). In a complementary spectral formulation, a network is modeled as a simplicial or cubical complex, the weighted $k$-Hodge Laplacian $\mathcal L_k$ is formed, and the null space $\ker \mathcal L_k \cong \mathcal H_k$ yields a homology embedding whose factorization isolates basic topological components and generators (Chen et al., 2021). This suggests that the topic encompasses both a local homology-based pruning scheme and a global harmonic-subspace decomposition.

1. Mathematical setting and terminology

The underlying network is typically represented as a simplicial complex $K$ built from 0-simplices, 1-simplices, 2-simplices, 3-simplices, and higher simplices. If $m_k$ denotes the number of $k$-simplices, then the Euler characteristic is

$\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$

For each $k$, the chain space $C_k$ is generated by the $k$-simplices, the boundary operator $\mathcal L_k$0 is represented by a boundary matrix $\mathcal L_k$1, and if $\mathcal L_k$2, then

$\mathcal L_k$3

The Betti numbers $\mathcal L_k$4 are the dimensions of the homology groups

$\mathcal L_k$5

with $\mathcal L_k$6 counting connected components, $\mathcal L_k$7 counting independent 1-dimensional cycles, and $\mathcal L_k$8 counting 2-dimensional cavities (Shi et al., 7 Jul 2025).

A parallel formalism appears in the discrete Hodge-theoretic treatment of networks and manifolds. There, for simplicial or cubical complexes, one works with cochain spaces $\mathcal L_k$9, boundary matrices $\ker \mathcal L_k \cong \mathcal H_k$0, and the $\ker \mathcal L_k \cong \mathcal H_k$1-th homology vector space

$\ker \mathcal L_k \cong \mathcal H_k$2

which is isomorphic to the null space of the $\ker \mathcal L_k \cong \mathcal H_k$3-Laplacian $\ker \mathcal L_k \cong \mathcal H_k$4 (Chen et al., 2021).

The terminology is not fully uniform across the literature. The expression “network homology Hk-core decomposition” is used explicitly for node-neighbor-subnetwork pruning (Shi et al., 7 Jul 2025), whereas the $\ker \mathcal L_k \cong \mathcal H_k$5-Laplacian work provides “essentially all the ingredients” to define and reason about a “network homology $\ker \mathcal L_k \cong \mathcal H_k$6-core decomposition” but does not itself use that term (Chen et al., 2021). A plausible implication is that Hk-core decomposition is best understood as a homology-informed generalization of core decomposition rather than a single universally fixed construction.

2. Spectral homology embeddings and subspace factorization

In the Hodge-theoretic formulation, the weighted normalized $\ker \mathcal L_k \cong \mathcal H_k$7-Laplacian is

$\ker \mathcal L_k \cong \mathcal H_k$8

where

$\ker \mathcal L_k \cong \mathcal H_k$9

Its null space is the $K$0-th homology vector space, and if

$K$1

is an orthonormal basis of $K$2, then the $K$3-th homology embedding is defined by

$K$4

Because the null-space basis is defined only up to an orthogonal transform, the embedding is defined only up to a global unitary transform in $K$5 (Chen et al., 2021).

This construction generalizes the standard graph Laplacian embedding used in spectral clustering. For $K$6, $K$7 encodes connected components; for $K$8, $K$9 encodes loops; for $m_k$0, cavities; and in general each harmonic $m_k$1-cochain corresponds to a homology class. The supports of these cochains identify representative cocycles for the corresponding holes (Chen et al., 2021).

The central structural result concerns connected sums. If

$m_k$2

then under assumptions that the complexes approximate the manifolds with isomorphic homology, that connected sum does not create or destroy $m_k$3-homology, and that only a small fraction of $m_k$4-simplices are created or destroyed in gluing, the homology decomposes as

$m_k$5

and the corresponding embedding behaves as a perturbation of the block-diagonal disjoint-union embedding. The resulting geometry is a union of linear subspaces, each associated with a distinct topological component. The paper therefore describes a “subspace clustering structure in the homology embedding” (Chen et al., 2021).

To extract these subspaces algorithmically, the homology basis $m_k$6 is factorized by blind source separation. The proposed “Subspace identification” procedure constructs the complex, computes $m_k$7 and $m_k$8, forms $m_k$9, computes the null-space basis $k$0, and then applies ICA without sphering to obtain

$k$1

The columns of $k$2 are treated as an independent homology basis, and each $k$3 can be interpreted as a “core” $k$4-homology mode or basic generator (Chen et al., 2021). This does not define Hk-cores in the node-peeling sense, but it furnishes a spectral notion of decomposition into topological cores.

3. Node-neighbor subnetworks and the H0/H1/H2/H3 hierarchy

The node-neighbor formulation begins with a local subnetwork for each node $k$5, obtained from its neighbors and the simplices supported on that neighbor set. For each such node-neighbor subnetwork, the Betti numbers are computed exactly as for the full simplicial network, yielding a local tuple

$k$6

The paper also introduces a local characteristic number. If the simplex counts of the deleted local piece associated with node $k$7 are $k$8, then

$k$9

The triple

$\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$0

is the new node index, where $\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$1 is the number of neighbors of node $\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$2 in the original graph (Shi et al., 7 Jul 2025).

The decomposition itself is recursive. The H0-core is the original network. The H1-core is obtained by deleting nodes whose neighbor subnetworks have Betti numbers

$\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$3

A key result states that if a node-neighbor subnetwork has Betti numbers $\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$4, then deleting node $\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$5 preserves the topology of the original network when $\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$6, whereas if $\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$7, deleting node $\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$8 changes the Betti numbers and decreases the global $\chi = m_0 - m_1 + m_2 - m_3 + \cdots .$9 by $k$0 (Shi et al., 7 Jul 2025).

The H2-core is then obtained from the H1-core by deleting nodes with

$k$1

and, if $k$2 is still nonzero, deleting edges in subnetworks with

$k$3

that are attached to branches without 1-cycles. The H3-core is obtained from the H2-core by deleting nodes with

$k$4

and then deleting edges that connect to $k$5-branches in a way that reduces $k$6 while retaining the higher-order structure. The construction continues inductively: the Hk-core is intended to retain the subnetwork that still carries homology in dimensions $k$7 after lower-dimensional contributions have been removed (Shi et al., 7 Jul 2025).

Two local propositions explain why these deletions are topology-aware. First, if a neighbor subnetwork has a central node, then its Betti numbers are $k$8. Second, deleting edges connected to nodes whose neighbor subnetworks have branches without $k$9 reduces $C_k$0. The method therefore peels off locally contractible pieces, then lower-order branches, then branches supporting progressively higher-dimensional holes (Shi et al., 7 Jul 2025).

This H0/H1/H2/H3 hierarchy is not degree-based. It is driven by the local homology of node-neighbor subnetworks and by repeated updates of the node indices after every node or edge deletion. Throughout the process, “the index of node involved in deleting edge needs to be updated in every step,” and the method is described as easy to implement in parallel (Shi et al., 7 Jul 2025).

4. Relation to classical and generalized core decompositions

The classical $C_k$1-core of a graph is the maximal induced subgraph in which every vertex has degree at least $C_k$2. It is obtained by iteratively pruning vertices of degree $C_k$3, and the resulting family of cores is nested. A later random-network formalization preserves arbitrary $C_k$4-core structure through a Hard-core Random Network model and treats the $C_k$5-core hierarchy as a filtration of the graph (Hébert-Dufresne et al., 2013).

Several generalizations broaden the notion of “core” without using homology. The Generalized $C_k$6-core ($C_k$7-core) is produced by a $C_k$8-leaf removal algorithm in which a vertex of degree strictly less than $C_k$9, together with its first neighbors and all edges incident to those neighbors, is progressively removed. This yields a nested hierarchy

$k$0

interpreted as a robustness hierarchy under “weak-plus-neighbors” pruning (Azimi-Tafreshi et al., 2018).

Distance-generalized core decomposition defines the $k$1-core as the maximal induced subgraph in which every vertex has at least $k$2 other vertices at distance $k$3 within the subgraph. For $k$4, $k$5-core is exactly the classical $k$6-core. The $k$7-core preserves uniqueness, nesting, and a unique core index per vertex, but replaces local degree by $k$8-hop support; later work scales this computation to large graphs through a local $k$9-degree updating technique, bitmap implementation, sampling, and parallelization (Bonchi et al., 2019, Dai et al., 2020).

A higher-order clique-based generalization is nucleus decomposition. For $\mathcal L_k$00, a $\mathcal L_k$01-$\mathcal L_k$02 nucleus is a maximal $\mathcal L_k$03-connected set of $\mathcal L_k$04-cliques each having at least $\mathcal L_k$05 incident $\mathcal L_k$06-cliques; $\mathcal L_k$07 recovers $\mathcal L_k$08-core, $\mathcal L_k$09 recovers $\mathcal L_k$10-truss, and $\mathcal L_k$11 yields triangle-based higher-order nuclei (Sariyuce et al., 2017). Hypergraphs admit a related $\mathcal L_k$12-core, the maximal subhypergraph in which each vertex has at least $\mathcal L_k$13 hypergraph degrees and each hyperedge contains at least $\mathcal L_k$14 vertices (Lee et al., 2023).

Against this background, Hk-core decomposition is distinguished by its retention rule. Classical $\mathcal L_k$15-core uses node degree, $\mathcal L_k$16-core uses $\mathcal L_k$17-hop degree, $\mathcal L_k$18-core uses a pruning dynamics that also removes neighbors, nucleus decomposition uses higher-order clique support, and hypergraph $\mathcal L_k$19-core uses simultaneous vertex-degree and hyperedge-size constraints. By contrast, Hk-core decomposition uses Betti numbers of node-neighbor subnetworks and local characteristic numbers to decide which nodes or edges are homologically redundant and which belong to the carrier of higher-dimensional cavities (Shi et al., 7 Jul 2025).

5. Empirical decompositions and computational use

The method is illustrated in detail on the C. elegans neural network. The original H0-core has simplex counts

$\mathcal L_k$20

boundary ranks

$\mathcal L_k$21

and Betti numbers

$\mathcal L_k$22

The maximum nonzero Betti order is $\mathcal L_k$23 (Shi et al., 7 Jul 2025).

In the H1 step, the decomposition deletes nodes whose neighbor subnetworks have a central point and then nodes with $\mathcal L_k$24. The resulting H1-core has

$\mathcal L_k$25

and preserves the global Betti vector

$\mathcal L_k$26

In the H2 step, after node deletions and deletion of 17 edges connected to isolated branches in node-neighbor subnetworks, the resulting H2-core has

$\mathcal L_k$27

and

$\mathcal L_k$28

In the H3 step, the network reduces to 16 nodes,

$\mathcal L_k$29

with

$\mathcal L_k$30

and

$\mathcal L_k$31

Thus the entire 3-dimensional homology of the original network is concentrated in a small H3-core, and “the simplexes consisting of four highest-order cavities in the H3-core subnetwork can also be directly obtained” (Shi et al., 7 Jul 2025).

A second case study is the cat cortical network. Its H0-core has

$\mathcal L_k$32

A first retention-based attempt that keeps 16 nodes with $\mathcal L_k$33 changes the global $\mathcal L_k$34 from 2 to 1. The paper then adopts a deletion-and-update strategy: after deleting 17 nodes with central-point neighborhoods and 33 nodes with $\mathcal L_k$35, the resulting H3-core has 15 nodes,

$\mathcal L_k$36

with

$\mathcal L_k$37

and

$\mathcal L_k$38

This preserves the two independent 3-dimensional cavities of the original network (Shi et al., 7 Jul 2025).

The spectral line of work supports a different set of applications. The homology embedding derived from $\mathcal L_k$39 has been applied to the shortest homologous loop detection problem, which is NP-hard in general, and the spectral loop detection algorithm is reported to scale better than existing methods and to be effective on diverse data such as point clouds and images (Chen et al., 2021). This suggests a practical division of labor: node-neighbor Hk-core decomposition localizes higher-order cavities, whereas null-space factorization yields harmonic generators and homology modes.

6. Interpretation, limitations, and open issues

A common misconception is to treat Hk-core decomposition as a direct synonym for graph $\mathcal L_k$40-core. The literature does not support that identification. Classical $\mathcal L_k$41-core is degree-threshold peeling on the 1-skeleton; Hk-core decomposition uses local Betti numbers, Euler-characteristic-derived node indices, and node-neighbor simplicial subnetworks (Shi et al., 7 Jul 2025). Another misconception is to equate Hk-core decomposition with persistent homology. The node-neighbor method operates at a single combinatorial scale and is presented as a way to simplify homology calculation; it is not a substitute for persistence but a complementary tool to structurally localize and simplify cavity search (Shi et al., 7 Jul 2025).

The spectral interpretation also has explicit assumptions. Its perturbation theorem requires that the simplicial or cubical complex approximate the underlying manifold with isomorphic homology, that connected sum not create or destroy $\mathcal L_k$42-homology, and that only a small fraction of $\mathcal L_k$43-cells be created or destroyed in gluing. The ICA-based identification of independent harmonic modes is data-driven and heuristic; the paper states that formal guarantees that each component aligns exactly with a single homology generator are not fully proven, though the behavior is supported empirically and by perturbation analysis (Chen et al., 2021).

The node-neighbor formulation has its own sensitivities. The cat cortical example shows that a naive retention strategy can change the target Betti number, whereas deletion with repeated local updates preserves the intended homology (Shi et al., 7 Jul 2025). This indicates that the exact deletion rules and update order are structural components of the method rather than incidental implementation details. A plausible implication is that Hk-core decomposition is best viewed as an algorithmic homology-preserving reduction scheme, not merely as a static invariant.

The broader research landscape suggests several convergences. Generalized cores indexed by distance, clique support, or hyperedge size already provide multi-scale and higher-order decompositions (Bonchi et al., 2019, Sariyuce et al., 2017, Lee et al., 2023). Hodge-theoretic embeddings furnish harmonic bases for $\mathcal L_k$44 (Chen et al., 2021). Hk-core decomposition occupies the intersection of these lines by using topological rather than purely combinatorial support to define the “core” of a network. In that sense, it reframes core decomposition from a degree-centric concept into a homology-centric one.