Human Connectome Core Network

• Human Connectome Core Network is defined as the integrative backbone comprising hub regions, rich-club nodes, and recurrent motifs that support efficient communication.
• Multiple formulations such as rich-club, consensus graphs, clique percolation, and multiplex cores illustrate diverse methodological approaches in capturing core network structure.
• Its organization underpins dynamic transitions and control processes in the brain, linking specialized modules to facilitate rapid whole-brain integration.

Searching arXiv for the cited connectome core-network literature to ground the article in current records. The human connectome core network denotes the integrative backbone of the macroscale brain network: a set of regions, edges, motifs, or modules that recur as especially central for dense interconnection, cross-system communication, global integration, controllability, or robust consensus across subjects. In the literature, the term does not refer to one universally fixed mathematical object. Instead, it is instantiated as rich-club and hub architecture, consensus/reference connectomes, frequent cliques, multiplex cores, higher-order simplicial hub neighborhoods, dependency-defined tiers, and distributed clique-percolation scaffolds. Across these formulations, the common theme is that the connectome contains nonuniform, topologically privileged structure that supports efficient communication while linking otherwise specialized subsystems (Betzel, 2020).

1. Conceptual definitions and scope

At macroscale resolution, connectomics represents the brain as a graph in which nodes are parcels or regions of interest and edges are structural or functional relationships. Within this framework, the connectome core is typically identified as the “highly interconnected backbone” composed of hub regions, rich-club nodes, and often connector nodes that link modules and enable efficient integration across the brain (Betzel, 2020). A closely related formulation is the classical core-periphery picture, in which a dense core contrasts with a sparse periphery, with the defining condition that the core-core density greatly exceeds the periphery-periphery density, i.e. $\rho_1 \gg \rho_3$ (Battiston et al., 2017).

The literature also stresses that “core network” is modality- and method-dependent. In consensus approaches, the core is the set of edges that recur across many independently reconstructed connectomes and therefore form a reduced-error, low-noise reference graph (Szalkai et al., 2014). In clique-based approaches, it is a set of robust dense subgraphs or a distributed percolating clique cluster shared across subjects (Fellner et al., 2019, Tiselko et al., 2022). In control-theoretic work, the core appears as the rich-club backbone that makes certain brain-state transitions energetically cheap (Betzel et al., 2016). In multiplex work, it is the set of nodes that are collectively “rich” across structural and functional layers (Battiston et al., 2017). This suggests that the phrase names a family of related constructs rather than a single canonical subgraph.

Formalization	Defining feature	Representative source
Rich-club / hub backbone	Highly connected hubs densely interconnected with one another	(Betzel et al., 2016)
Consensus/reference connectome	Edges present across many subjects	(Szalkai et al., 2014)
Frequent clique core	Complete subgraphs present in at least 80% of connectomes	(Fellner et al., 2019)
Multiplex core	Nodes ranked by cross-layer richness $\mu_i$	(Battiston et al., 2017)
Dependency core	Highest-NDI tier of globally indispensable nodes	(Schirmer et al., 2018)
Higher-order simplicial core	Eight major hubs plus attached simplexes of all orders	(Andjelkovic et al., 2020, Tadic et al., 10 Jul 2025)

A further qualification is that some studies explicitly reject the idea of a single unique core. The synchronization study on the 998-node human connectome emphasizes a hierarchical modular structure with 12 modules at a finer level and a coarser two-module hemispheric division, arguing for locally coherent modules and inter-modular connectors rather than one privileged synchronization core (Villegas et al., 2015). Accordingly, whether the connectome core is best understood as a compact set of hubs or as a distributed multiscale scaffold remains an open modeling choice rather than a settled ontological fact.

2. Structural formulations of the core network

A major structural interpretation treats the core as a rich club: a subset of high-degree nodes more densely interconnected than expected by chance. In control-theoretic analysis of structural connectivity, this is quantified by the rich-club coefficient

$\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$

normalized against degree-preserving random networks. In that study, rich-club nodes overlap strongly with the regions most likely to appear in optimal target states, and selective destruction of rich-club organization while preserving degree significantly increases transition energy. The result indicates that the core effect depends not only on high degree but on the specific hub-to-hub wiring pattern (Betzel et al., 2016).

Another structural tradition defines the core as a consensus graph. The Budapest Reference Connectome Server v2.0 constructs a graph from 96 Human Connectome Project MRI datasets, each yielding a 1015-vertex cortex graph. An edge $\{u,v\}$ is included if it appears in at least $k$ source graphs; the default minimum edge confidence is 14. Edge weight is

$w(\{u,v\})=\frac{n}{L},$

where $n$ is the number of fibers connecting the regions and $L$ is the average fiber length. The default reference graph has 1015 vertices and 8507 edges, and the server provides CSV and GraphML outputs with FreeSurfer annotations and a 3D rotating brain visualization. In this usage, the core is the common edge set repeatedly observed across subjects (Szalkai et al., 2014).

A denser variant of structural core extraction uses frequent complete subgraphs. In 414 diffusion-MRI-derived connectomes with 463 nodes each, complete subgraphs present in at least 80% of subjects are taken as robust structural cores. The study reports 812 frequent complete subgraphs overall, with the largest frequent cliques having 7 vertices. Two such 7-vertex cliques are in the left hemisphere, and among the 48 different 6-vertex complete subgraphs present in at least 80% of connectomes, only 6 are in the right hemisphere whereas 42 are in the left hemisphere. Several maximal complete subgraphs are present in all 414 subjects, including caudate-pallidum-putamen-thalamus clusters, hippocampus-putamen-thalamus clusters, putamen-thalamus-insula clusters, and a right superior frontal clique (Fellner et al., 2019).

A more distributed structural formulation comes from $k$-clique percolation. In that work, the core is a percolating clique cluster formed by overlapping high-order cliques. Human structural connectomes exhibit clique percolation up to about $k=7$ in the data, whereas the corresponding random-graph threshold

$\mu_i$0

would predict only much lower-order percolation around $\mu_i$1–$\mu_i$2 at comparable density. Common structural connections sustain this overlap-rich scaffold, whereas individual-specific edges contribute more to variability in local community interaction (Tiselko et al., 2022).

3. Multiplex, higher-order, and dependency-based cores

When structural and functional connectivity are treated jointly, the core can be defined by multiplex richness. For a multiplex $\mu_i$3, node $\mu_i$4 has layer-specific degree $\mu_i$5 and multiplex richness

$\mu_i$6

Nodes are ranked by $\mu_i$7, and the multiplex core is the prefix ending where $\mu_i$8 reaches its maximum. Applied to a two-layer structural-functional cortex network derived from 171 healthy subjects in the NKI Rockland dataset, this procedure shows that the multiplex core is not identical to the structural core. At $\mu_i$9, the overlap between structure-only and function-only cores is low, with core similarity $\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$0. Posterior medial and parietal regions remain central, but sensorimotor regions such as the precentral and postcentral gyri become important multiplex-core nodes, and frontal regions such as medial prefrontal cortex do not survive the multiplex criterion (Battiston et al., 2017).

Higher-order approaches replace pairwise graphs with simplicial complexes. In one line of work, the core network is the subgraph centered on eight global hubs—Left and Right Putamen, Caudate, Thalamus-Proper, and Hippocampus—identified by topological dimension

$\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$1

the number of simplexes of all orders containing node $\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$2. The resulting Fc-network and Mc-network contain the leading hubs, all nodes directly attached to them, and all edges among those nodes. At threshold $\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$3, the study reports 948 edges common to both sex-specific core networks, 204 edges unique to the male network, and 419 unique to the female network; the corresponding node counts are 517 for the female core network and 418 for the male core network. As threshold increases, the maximum simplex order decreases from $\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$4 at $\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$5 to $\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$6 at $\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$7, and both sexes retain a residual 6-clique structure at high threshold (Andjelkovic et al., 2020).

A related consensus-simplicial formulation defines the core as the common F/M-connectome, the overlap between 100-subject female and 100-subject male consensus networks derived from the Budapest Connectome Server 3.0. At 1000K launched fibers, the common network practically coincides with the male network, has simplices up to $\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$8—equivalently simplexes of order 14 in the paper’s wording—and is organized into six anatomical communities. The female network contains additional structured edges, including a robust F-excess1195 set of 1195 edges that remains absent from the male connectome even at 1000K (Tadic et al., 2019).

A different criterion defines the core by global network dependency rather than density or degree. The Network Dependency Index framework removes each node, recomputes the information matrix on the residual graph, and measures how much network information loss results. Natural-log-transformed NDI values are fit by a Gaussian Mixture Model; with $\phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)},$9 Gaussians, the method yields four tiers, including one for nodes with NDI $\{u,v\}$0. Across age groups, NDI rankings are highly stable, with average Spearman correlation $\{u,v\}$1, and Tier 1 contains key relay nuclei and cortical regions including thalamus, caudate, pallidum, putamen, insula, and posterior cingulate/parahippocampal-related cortex (Schirmer et al., 2018).

4. Dynamical and control significance

One of the clearest functional interpretations of the connectome core comes from network control theory. Structural dynamics are modeled as the linear time-invariant system

$\{u,v\}$2

with node energy

$\{u,v\}$3

A proxy for transition difficulty is

$\{u,v\}$4

In this framework, control energy is tightly related to weighted degree $\{u,v\}$5, and optimal target states disproportionately recruit high-strength hub regions. Communicability,

$\{u,v\}$6

predicts compensatory energy increases when control of another region is suppressed. The main implication is that the rich club acts as a control backbone that makes hub-rich target states especially low-energy and supports compensation through direct and indirect routes (Betzel et al., 2016).

A broader network-neuroscience synthesis places this result within the segregation-integration tradeoff. Hubs, rich clubs, connector nodes, short path lengths, and high efficiency constitute the integrative scaffold that links specialized modules. In this view, the connectome core explains how the brain can combine modular segregation with rapid whole-brain communication and dynamic reconfiguration across cognitive demands (Betzel, 2020).

Not all dynamical studies converge on a single compact core. In noisy Kuramoto simulations on a 998-node human connectome, synchronization unfolds through a broad Griffiths-like intermediate regime generated by hierarchical modularity. Modules become coherent at different thresholds, the global order parameter fluctuates strongly for roughly $\{u,v\}$7, and metastable local attractors can be escaped by intermediate noise. This supports a multi-scale picture in which local modules and inter-modular connectors, rather than one unique synchronization core, organize the dynamic repertoire (Villegas et al., 2015).

Other dynamical models recover core-like structure in different terms. Near the critical point of an Ising model on the human connectome, Belief Propagation with no external input produces a DMN-like posterior activation map, while Susceptibility Propagation yields correlation modules resembling resting-state networks; both results are stronger on the empirical connectome than on random nulls (Peraza-Goicolea et al., 2019). In the partition-stability framework, consensus structural communities are defined by how well they trap a random walk over Markov time, and the agreement matrix $\{u,v\}$8 correlates with resting-state functional connectivity with a peak of about $\{u,v\}$9 around $k$0, indicating that functionally relevant structural backbone communities exist across a range of dynamical scales (Betzel et al., 2013).

5. Variability across modality, subjects, lifespan, and parcellation

A recurring finding is that the connectome core depends on which connectivity layer is analyzed. Structural cores, functional cores, and multiplex cores only partially overlap. In the two-layer multiplex study, posterior medial and parietal cortices remain prominent, but sensorimotor cortex becomes newly central, whereas some frontal regions emphasized in structural-only analyses are relegated to the periphery (Battiston et al., 2017). This suggests that no single modality yields a complete account of the core network.

Across subjects, consensus methods identify a reproducible backbone without assuming identical individual graphs. The Budapest Reference Connectome Server explicitly begins from the observation that individual connectomes are pairwise distinct and therefore that “we cannot talk about an abstract ‘graph of the brain’.” The consensus edge formalism treats repeatedly observed edges as more reliable, and the frequent-clique formalism applies the same logic to dense motifs rather than single edges (Szalkai et al., 2014, Fellner et al., 2019).

Sex differences have been reported at several levels. In the frequent complete subgraph analysis, 224 complete subgraphs are significantly more frequent in females and 812 are significantly more frequent in males, using a $k$1 test with $k$2 and Holm–Bonferroni correction (Fellner et al., 2019). In the simplicial hub-centered analysis, the same eight major hubs appear in both sexes, but edge identities, edge weights, entropy minima, and simplex organization differ (Andjelkovic et al., 2020). In the common F/M-connectome analysis, the common backbone practically coincides with the male network at each fiber count, whereas the female connectome contains additional structured edges that enlarge simplex size and introduce new cycles (Tadic et al., 2019).

Lifespan work suggests both stability and gradual rebalancing. The NDI framework yields highly consistent central nodes across age groups and a stable Tier 1 relay core (Schirmer et al., 2018). By contrast, generative modeling indicates that the balance of wiring rules changes with age: the best-fitting model combines geometric constraints with homophilic attachment,

$k$3

and in a lifespan sample aged 7–85 years, the geometric penalty weakens with age while $k$4 does not change significantly. The model reproduces clustering, modularity, characteristic path length, global efficiency, and long-distance hub connectivity, implying that core organization may arise from an evolving tradeoff between wiring economy and topological affinity (Betzel et al., 2015).

Parcellation choice also changes inferred core architecture. A structural connectome analysis that explicitly includes the brainstem reports that the brainstem ranks highest in degree centrality, eigenvector centrality, and betweenness centrality in a 104-structure extended connectome, and remains highly ranked after normalization by node volume. Removing the brainstem and related structures produces an 84-node restricted connectome in which rankings shift and no single structure dominates in the same way. The authors conclude that omitting the brainstem can substantially distort inferred core structure (Salhi et al., 2022).

6. Methodological limitations, debates, and current extensions

Several methodological cautions recur throughout the literature. First, all macroscale cores are coarse-grained constructs. Frequent complete subgraphs and simplicial cores are defined on ROI-level graphs; a complete subgraph among ROIs does not imply literal all-to-all neuronal connectivity (Fellner et al., 2019). Consensus reference connectomes likewise do not claim to recover an exact universal brain graph, only a low-noise, robust graph representation built from repeated observations (Szalkai et al., 2014). This distinction is central to preventing overinterpretation.

Second, tractography and preprocessing constrain what counts as core structure. Consensus and clique methods are motivated precisely by the long and error-prone pipeline of acquisition, segmentation, parcellation, tractography, and graph construction (Fellner et al., 2019). Cross-hemispheric clique detection may be especially limited by fiber-crossing problems in the corpus callosum (Fellner et al., 2019). In consensus-server work, the choice of minimum edge confidence, minimum edge weight, and median versus mean aggregation directly changes the resulting reference core (Szalkai et al., 2014).

Third, the computational definition of core can alter both interpretation and downstream dynamics. In modular coarse-graining of 104-node structural connectomes, reduction to about 10–14 modules preserves substantial global structure and critical behavior only when inter-module weights are summed and not renormalized. Under that prescription, global efficiency correlates strongly between original and coarse-grained networks with about $k$5, the Wilson–Cowan transition value correlates with about $k$6, and the Ising critical temperature with about $k$7; alternative coarse-graining choices can largely destroy these relations (Kora et al., 2023). A plausible implication is that “core network” is meaningful only relative to a specified reduction map.

Recent machine-learning work turns the core-periphery principle into an inductive bias for functional connectome classification. The CP-SSM architecture imposes a CP mask with core node rate $k$8, uses Mamba as a selective state-space model with linear complexity, and introduces a CP-guided Mixture-of-Experts: $k$9 On ABIDE, CP-SSM achieves AUROC $w(\{u,v\})=\frac{n}{L},$0 and ACC $w(\{u,v\})=\frac{n}{L},$1; on ADNI, AUROC $w(\{u,v\})=\frac{n}{L},$2 and ACC $w(\{u,v\})=\frac{n}{L},$3. The model is explicitly motivated by the view that dense core interactions support efficient information transmission, although in this case the core is imposed by design rather than recovered by community detection (Chen et al., 18 Mar 2025).

Current work also extends higher-order core analysis into synchronization dynamics. A 418-node human connectome core network centered on the same eight hubs and their attached simplicial structure exhibits partial synchrony and multiscale oscillations of the Kuramoto order parameter below the master-stability threshold. Spectral graph analysis and eigenvector localization identify three major synchronized clusters; topology largely determines cluster membership, while empirical weights tighten hub-cluster synchronization, shrink the instability interval, and reduce the multifractal spectrum (Tadic et al., 10 Jul 2025). Taken together, these developments indicate that the human connectome core network is increasingly treated not merely as a static hub set, but as a mesoscale object linking topology, higher-order organization, dynamics, control, and computation.