This section outlines the data acquisition, network construction, and analytical methodologies employed in this study. Cognitive test data and sMRI data were obtained from 29 male children, who were divided into a control group (CG) and a gifted group (GG). Two individual-level brain networks–MSN and MIND–were constructed for subsequent comparative analysis. Group-level comparisons were performed between CG and GG average networks, focusing on hemispheric connectivity and nodal topological features across different connection densities. At the individual level, Spearman correlation analysis was conducted to assess the relationships between global network metrics (including seven topological and five hemispheric metrics) and cognitive performance scores.

DataParticipants

In this study, we use the dataset consisting of sMRI and cognitive tests from 29 healthy right-handed male children with no history of psychiatric or neurological disorders. This dataset was previously used for group-level SCN analysis in our earlier work (Solé-Casals et al., 2019). The raw (anonymized) MRI data are available in the OpenNeuro repository at https://openneuro.org/datasets/ds001988.

Participants were divided into two groups: the control group (CG, 14 subjects) and the gifted group (GG, 15 subjects). Participant details are presented in Table 1. The table indicates no notable age differences between groups but reveals significant differences in full-scale IQ scores.

Table 1 Participant informationsMRI Data

Participants underwent MRI scanning using a 3T scanner. High-resolution T1-weighted images were acquired with the MPRAGE 3D protocol (TR = 2300ms; TE = 3ms; TI = 900ms; FOV = \(244\times 244 mm^\); 1mm isotropic voxel). In this study, we adopted the sMRI preprocessing method outlined in prior research (Solé-Casals et al., 2019). FreeSurfer v5.3 was used for preprocessing to estimate cortical thickness (CT) from a three-dimensional cortical surface model based on intensity and continuity information (Fischl et al., 2000). Cortical reconstructions were independently reviewed by two experienced researchers to ensure adherence to quality control criteria. Each brain was parcellated into 308 regions (\(R=308\); approximately \(500\ \textrm^2\) each) using the standard FreeSurfer fsaverage template and a backtracking algorithm (Romero-Garcia et al., 2012; Solé-Casals et al., 2019), which further subdivides the regions defined in the Desikan-Killiany atlas (Desikan et al., 2006). Among them, 152 regions belong to the left hemisphere (\(N_L=152\)) and 156 to the right hemisphere (\(N_R=156\)). The surface-based (non-linear) registration by the FreeSurfer command mri_surf2surf was then applied to warp the parcellation from the standard template to each individual’s native MPRAGE space. This method is recommended for superior cortical landmark alignment and for avoiding the introduction of age-associated bias (Ghosh et al., 2010).

In this work, we chose five morphometric features for both MIND network and MSN construction to maintain consistency for comparison: surface area (SA), gray matter volume (GMV), cortical thickness (CT), mean curvature (MC), Gaussian curvature (GC). Specifically, while regional-level morphometric features were employed for MSN estimation, vertex-level morphometric features of each region were utilized for MIND estimation in each subject.

Methods

In this study, we construct and analyze two types of individual-level brain networks–the morphometric similarity network (MSN) and the morphometric inverse divergence (MIND) network–for comparative analysis at both group and individual levels. In the group analyses, we compare the average brain networks of CG and GG groups across MSN and MIND networks, including intra-/inter-hemispheric distinctions, effects of varying connection densities, and nodal topological features. In the individual cognitive analysis, we assess the correlation between the global topological features and the scores of cognitive indices using Spearman correlation methods.

Fig. 1

MSN construction and processing for group average MSNs. Five morphometric features (tSA, tGMV, aCT, iMC, iGC) are extracted from each brain region, resulting in \(308\times 5\) features per sample. These features are normalized using the z-score method. MSNs (\(308\times 308\)) are then computed by calculating the PCCs between the normalized features of all region pairs. Finally, group average MSNs are generated for both CG and GG

Network Construction

In this study, we construct two types of individual-level brain networks for comparative analysis: the MSN (Li et al., 2017; Seidlitz et al., 2018) and the MIND network (Sebenius et al., 2023). Both network types are analyzed as weighted networks, which more accurately capture brain features compared to binary networks (Qi et al., 2015). To ensure network connectivity while controlling for sparsity, we employ a minimum spanning tree (MST)-based thresholding approach (Van Wijk et al., 2010) across a range of connection densities p. This method guarantees that all thresholded networks remain fully connected while preserving the strongest connections.

The MSN (Seidlitz et al., 2018) transformed each individual’s set of multimodal MRI features into a morphometric similarity matrix of pairwise inter-regional correlations of morphometric feature vectors. In this study, as depicted in Fig. 1, a set of \(\eta\) (\(\eta =5\)) morphometric features (\(308\times 5\) for each sample) derived from any T1-weighted MRI scans: total SA (tSA), total GMV (tGMV), average CT (aCT), integrated rectified MC (iMC), integrated rectified GC (iGC), was employed to construct MSNs. It has been demonstrated that the MSNs based on these five features are similar to MSNs utilizing a broader array of features (Seidlitz et al., 2018), with tSA, tGMV, aCT, and iGC identified as the most discriminative features (Zhang et al., 2021). In accordance with prior studies (Li et al., 2017; Seidlitz et al., 2018), each feature vector is standardized by the z-score values prior to the correlation calculation. The morphometric similarity between each possible pair of regions was estimated by the Pearson’s correlation coefficient (PCC) between their normalized morphometric feature vectors, resulting in a \(308\times 308\) MSN for each sample. As commonly acknowledged, a PCC value close to -1 denotes anti-correlation between the pair of features, while a PCC value close to 1 denotes strong correlation between the pair of features (Heinsfeld et al., 2018). Hence, the diagonal elements of MSN equal to 1. However, we uniformly assign NaN values to the diagonal elements of MSNs.

The MIND (Sebenius et al., 2023) estimates the similarity between cortical areas at the individual level by the symmetric KL divergence (Jeffreys, 1973) between their multivariate distributions of MRI vertex features. In this study, the MIND network’s input comprises mesh representations of cortical surfaces derived from T1-weighted MRI scans. Each surface is delineated by a set of 290,487 vertices. To maintain consistency with MSN methodology for comparison, each vertex is characterized by five structural MRI features: SA, GMV, CT, MC and GC. In Sebenius et al. (2023), as depicted in Fig. 2, z-score was employed to standardize each feature across all vertices in the brain prior to parcellating the data into vertex-level distributions. Thus, each surface can be described by a set of vertices \(\varvec_i \in \mathcal \), where \(\varvec_i\) represents the vector of \(\eta\) (\(\eta =5\)) structural features of the ith vertices in the surface. \(\varvec_i\) in a surface can be grouped into \(R (R=308)\) regions according to the parcellation, such that \(\mathcal =\_1\},\ldots ,\_R\}\}\). Given regions a and b, vertices \(\mathcal _a\) and \(\mathcal _b\), with true multivariate distributions \(P_a\) and \(P_b\), the KL divergence between \(P_a\) and \(P_b\) is defined as:

$$\begin d_(P_a\parallel P_b)=\int _^}P_a(\varvec)\log )})}} d\varvec \ge 0 \end$$

(1)

The k-nearest (\(k=1\)) neighbor divergence approximation (Pérez-Cruz, 2008) was used to estimate \(d_(P_a\parallel P_b)\):

$$\begin \hat_(P_a\parallel P_b)=-\frac\sum \limits _^\log _i)}_i)}}+\log _b\parallel }_a \parallel -1}} \end$$

(2)

Here, \(r_k(\varvec_i)\) and \(s_k(\varvec_i)\) represent the Euclidean distances of \(\varvec_i\) to the k-th most similar vertex of \(\varvec_i\) in \(\mathcal _a\) (with the sample \(\varvec_i\) removed) and \(\mathcal _b\), respectively. A symmetric measure of KL divergence is then defined as:

$$\begin \begin \hat(P_a,P_b)=\max (\hat_(P_a\parallel P_b),0) +\max _(P_b\parallel P_a),0)} \end \end$$

(3)

KL divergence is always greater than or equal to zero, and it equals zero only when the two distributions are identical. Finally, the KL divergence for regions a and b was transformed by Eq. 4 to estimate the inter-areal MIND similarity. The values of MIND are bounded between 0 and 1, with higher values denoting a stronger degree of similarity.

$$\begin MIND(a,b)=\frac(P_a,P_b)} \end$$

(4)

Fig. 2

MIND network construction and processing for group average MIND networks. Five features (SA, GMV, CT, MC, GC) of each vertex normalized by z-score are used to compute the similarity between cortical regions at the individual level based on the symmetric KL divergence, through this, we can get the MIND networks (\(308\times 308\)) for each sample. Finally, the group average MIND networks are generated for both groups CG and GG

Based on the fundamental premise that the brain operates as an integrated system where no neural elements should be completely isolated (Fornito et al., 2016), we employ a minimum spanning tree (MST) based thresholding method (Van Wijk et al., 2010; Tewarie et al., 2015) to sparsify brain networks at varying connection densities p while ensuring the graphs remain node connected. This approach, widely used in brain network analysis (Alexander-Bloch et al., 2010; Solé-Casals et al., 2019; Han et al., 2024), utilizes Kruskal’s algorithm to construct an MST that connects all R nodes with \(R-1\) edges, effectively maximizing the sum of edge weights in weighted networks. The thresholding procedure starts with the MST as a backbone, to which additional edges are added to achieve the desired density p, defined as the ratio of actual edges to the maximum possible edges \((R-1)R/2\). To ensure methodological consistency across different network types, when computing MST-based graphs for MSNs, negative values are set to zero, aligning with MIND networks where all values are inherently non-negative. Here, we denote \(\mathcal _p\) as the MST-based thresholding network with connection density p derived from brain network \(\mathcal \).

For a symmetric brain network \(\mathcal \), the MST-based thresholding procedure is implemented as follows:

1.

MST construction:

Compute distance matrix: \(\mathcal _ = \frac_}\) (for \(i \ne j\)), \(\mathcal _ = 0\)

Build MST using Kruskal’s algorithm: \(E_} = \text (\mathcal )\)

2.

Edge selection:

Sort edges in descending order: \(E_} = \text (\_ \mid i < j\}, \text )\)

Calculate target number of edges: \(k_} = \left\lfloor p \cdot \frac \right\rfloor\)

Determine additional edges to add: \(k_} = k_} - |E_}|\)

Construct selected edge set: \(E_} = E_} \cup \ k_} \text E_} \setminus E_}\}\)

3.

Output matrix construction:

Group Analyses for MSNs and MIND Networks

We conducted comprehensive group-level comparisons between the CG and GG groups using average brain networks derived from both MSN and MIND frameworks (Figs. 1, 2). Our analysis encompassed two key dimensions: hemispheric connectivity patterns and nodal topological features, both evaluated across different connection densities.

Particularly, we use the (mean) signed Euclidean distance (SED) to evaluate the difference between the two groups across each vector metric. The SED directly quantifies differences at corresponding nodes/edges, with positive values indicating higher means in GG compared to CG, and negative values indicating the reverse.

To investigate hemispheric connectivity differences, we first rearrange the brain networks accordingly. For an MST-based network \(\mathcal _p\), we extract three hemispheric connection types: left intra-hemispheric (LL), right intra-hemispheric (RR), and inter-hemispheric (LR). These are formally defined as:

$$\begin \begin \varvec_^p = \left\^_p |i< j; i,j \in N_L \right\} , \\ \varvec_^p = \left\^_p | i < j; i,j \in N_R \right\} , \\ \varvec_^p = \left\^_p | i \in N_L; j \in N_R \right\} . \end \end$$

(5)

where:

\(N_\) and \(N_\) are the node sets in left and right hemisphere, respectively.

\(\mathcal ^_p\) is the edge between node i and node j in the network \(\mathcal _p\).

\(\varvec_^p\), \(\varvec_^p\), and \(\varvec_^p\) denote the vectors containing the connections for left intra-hemispheric, right intra-hemispheric, and inter-hemispheric edges, respectively.

(1) Effects of varying connection densities.

Given that all nodes in an MST-based network \(\mathcal _p\) are connected, the minimum connection density p of MST-based network \(R=308\) nodes is \((R-1)/[R(R-1)/2] = 0.0065\). In this study, we examine 16 connection densities as follows: \(p \in \\).

To address scale disparities between MSN and MIND networks, we first normalize their average MST-based networks using Min-Max scaling across both groups to preserve relative magnitudes:

$$\begin \left[ \hat}^C_p, \hat}^G_p\right] = \text _\left( \left[ \mathcal ^C_p, \mathcal ^G_p\right] \right) . \end$$

(6)

where:

\(\hat}^C_p\) and \(\hat}^G_p\) represent the normalized networks for the CG and GG average at connection density p, respectively.

MinMax scaling normalizes the values in the range [0, 1].

For each hemispheric connection type (LL, RR, LR) and density p, we calculate the mean SED (\(\bar_p\)) between the normalized CG and GG average networks, separately for MSN and MIND networks:

$$\begin \bar_^p = }\left( \mu (\hat}_^) - \mu (\hat}_^) \right) \cdot \frac}_^ - \hat}_^ \right\| _2}}_^|} \end$$

(7)

where:

\(\hat}_^\) and \(\hat}_^\) are the left intra-hemispheric connections from the normalized GG (\(\hat}^G_p\)) and CG (\(\hat}^C_p\)) MST-based networks with connection density p following the Eq. 5.

\(\mu (\cdot )\) denotes the mean value.

\(\left\| \cdot \right\| _2\) is the Euclidean norm.

\(|\cdot |\) represents the length of the vector.

\(}(\cdot )\) is the sign function preserving directionality of group differences.

Analogous calculations yield \(\bar_^p\) (right intra-hemispheric) and \(\bar_^p\) (inter-hemispheric) mean SEDs for both MSN and MIND.

(2) Hemispheric connection strength comparison.

For a given MST-based network \(\mathcal _p\), we compute the strengths for each hemispheric connection type (LL, RR, LR):

$$\begin S_^p = \frac_^p}}_^p|}, \\ S_^p = \frac_^p}}_^p|}, \\ S_^p = \frac_^p}}_^p|}. \end} \end$$

(8)

where \(|\varvec_^p|\), \(|\varvec_^p|\) and \(|\varvec_^p|\) represent the number of connections in the corresponding connection types.

For group-level comparison, we compute these strength measures of CG and GG at connection density \(p=0.1\) for both MSN and MIND.

Eight nodal topological features (including node degree \(\varvec_\), node strength \(\varvec_\), eigenvector centrality \(\varvec_\), participation coefficient \(\varvec_\), node (betweenness) centrality \(\varvec_\), local efficiency \(\varvec_\), clustering coefficient \(\varvec_\), node versatility \(\varvec_\)), are adopted to characterize the nodal topological organization of brain networks. Each nodal topological feature is a vector with a length equivalent to the number of regions (\(R=308\) in this study).

1.

The node degree \(\varvec_(i)\) refers to the number of connections (or edges) that node i has to other nodes in a graph.

2.

The node strength \(\varvec_(i)\) is the sum of weights of links connected to node i.

3.

The eigenvector centrality (Newman et al., 2010) of node i, \(\varvec_(i)\), is the i-th element in the eigenvector corresponding to the largest eigenvalue of the adjacency matrix.

4.

The participation coefficient \(\varvec_(i)\) (Guimera & Nunes Amaral, 2005) measures how the connections of node i are distributed across various modules by representing the proportion of its connectivity allocated to each module: \(\varvec_(i)= 1 - \sum _^ (\kappa _ / \kappa _i)^2\), where \(N_M\) is the number of modules in the network, \(\kappa _\) is the total weight of connections between node i and all nodes in module s, and \(\kappa _i\) is the strength of node i.

5.

The node (betweenness) centrality \(\varvec_(i)\) (Brandes, 2001) quantifies the proportion of shortest paths between all node pairs in the network that pass through a given index node i.

6.

The local efficiency \(\varvec_(i)\) (Rubinov & Sporns, 2010) quantifies the global efficiency within the neighborhood of node i and is closely related to the clustering coefficient.

7.

The (weighted) clustering coefficient \(\varvec_\) (Onnela et al., 2005) is defined as the average “intensity” (geometric mean) of all triangles associated with each node.

8.

The node versatility \(\varvec_\) (Shinn et al., 2017) assesses the consistency with which a node in a modular decomposition is linked to a particular module.

For the calculation of \(\varvec_\) and \(\varvec_\), which require module assignments, we employed a robust multi-resolution consensus community detection approach based on the Louvain algorithm (Blondel et al., 2008). This procedure was conducted as follows:

(1)

The Louvain algorithm was applied across 22 resolution parameters \(\gamma =0.4\sim 2.5\) (step size 0.1). For each \(\gamma\), \(L=100\) independent iterations were performed to account for algorithmic stochasticity.

(2)

For each \(\gamma\), a consensus matrix \(\mathcal (\gamma )\) was constructed, where its element \(\mathcal _(\gamma )\) represents the probability that nodes i and j belong to the same community across the 100 iterations: \(\mathcal _(\gamma ) = \frac \sum _^ \mathbb \left( c_i^(\gamma ) = c_j^(\gamma )\right)\), where \(L=100\), \(c_i^(\gamma )\) is the community assignment of node i in the \(\ell\)-th run at resolution \(\gamma\), and \(\mathbb (\cdot )\) is the indicator function;

(3)

To compute \(\varvec_\), a final stable community partition was obtained by applying the Louvain algorithm (with \(\gamma =1.0\)) to the average consensus matrix \(\bar} = \frac \sum _ \mathcal (\gamma )\). Here, \(\Gamma\) denotes the set of resolution parameters, with \(|\Gamma | = 22\).

(4)

Node versatility \(\varvec_(i)\) was calculated by first computing a metric at each \(\gamma\): \(\varvec_i(\gamma )=\sum _j\sin (\pi \mathcal _(\gamma ))/\sum _j \mathcal _(\gamma )\). The final \(\varvec_(i)\) is the average across all \(\gamma\) values: \(\varvec_(i) = \frac \sum _ \varvec_i(\gamma )\).

To analyze nodal topological properties in group networks, we employ the Signed Euclidean Distance (SED) to quantify discrepancies between CG and GG average networks. This comparison is performed for each nodal topological feature at various connection densities p in both MSN and MIND networks. Prior to SED computation, each nodal features from both groups of each connection density p are normalized to the [0, 1] range using Min-Max normalization:

$$\begin \left[ \hat}^C_\Phi (p), \hat}^G_\Phi (p)\right] = \text _ [0,1] \end}\left( \left[ \varvec^C_\Phi (p), \varvec^G_\Phi (p)\right] \right) , \end$$

(9)

where \(\Phi \in \left\\) denotes the nodal feature type. Here, \(\varvec^C_\Phi\) and \(\varvec^G_\Phi\) represent raw features of CG and GG average networks at connection density p, respectively, while \(\hat}^C_\Phi\) and \(\hat}^G_\Phi\) are their normalized counterparts. The SED is defined as:

$$\begin D^u_\Phi (p) = }\left( \mu \left( \hat}^G_\Phi (p)\right) - \mu \left( \hat}^C_\Phi (p)\right) \right) \cdot \left\| \hat}^G_\Phi (p) - \hat}^C_\Phi (p) \right\| _2 \end$$

(10)

The term \(D^u_\Phi (p)\) quantifies the SED for feature \(\Phi\) in brain network u (\(u\in \left\\)).

To further examine hemispheric differences, we compute the mean SED between CG and GG using hemispheric subsets of each nodal feature vector:

$$\begin \bar^_\Phi (p) = }\left( \mu \left( \hat}^_\Phi (p)\right) - \mu \left( \hat}^_\Phi (p)\right) \right) \cdot \frac}^_\Phi (p) - \hat}^_\Phi (p) \right\| _2}, \end$$

(11)

where \(\hat}^_\Phi (p)\) and \(\hat}^_\Phi (p)\) denote the first \(N_L\) elements of \(\hat}^_\Phi (p)\) and \(\hat}^_\Phi (p)\), respectively, corresponding to the left hemisphere. A similar definition applies to \(\bar^_\Phi (p)\) for the right hemisphere.

Individual Cognitive Analysis for MSNs and MIND Networks

This section examines how global brain network metrics relate to cognitive performance for both MSN and MIND. Seven typical global topological features (assortativity \(f_a\), transitivity \(f_t\), global efficiency \(f_\), characteristic path length \(f_\), mean participation coefficient \(f_\), mean (global) weighted clustering coefficient \(f_\), mean versatility \(f_\)) and five hemispheric metrics (left-to-right intra-hemispheric ratio \(R_\), intra-to-inter-hemispheric ratio \(R_\), left intra-hemispheric strength \(S_\), right intra-hemispheric strength \(S_\) and inter-hemispheric strength \(S_\)) are analyzed.

1.

Assortativity (Newman, 2002) \(f_a\) is defined as the correlation coefficient for the degrees of neighboring nodes, which refers to the tendency of nodes in a network to link with other similar nodes.

2.

Transitivity \(f_t\) quantifies the ratio of closed to all possible triplets in a network, providing a global measure of segregation by capturing the extent of locally clustered connectivity (Rubinov & Sporns, 2010).

3.

A network’s global efficiency (Latora & Marchiori, 2001) \(f_\) is the reciprocal of the harmonic mean of its path lengths.

4.

The characteristic path length (Watts et al., 1998) \(f_\) is the average shortest path length between all possible pairs of nodes in a network. Moreover, the characteristic path length of a network is strongly positively correlated with the network’s mean strength.

5.

The mean participation coefficient can measure the global integration of a network (Solé-Casals et al., 2019), which is denoted as \(f_=\frac \sum \limits _ \varvec_(i)\), where \(R=308\), \(\varvec_\) represents the participation coefficient vector of a network.

6.

The mean clustering coefficient is denoted as \(f_=\frac \sum \limits _ \varvec_(i)\), where \(\varvec_\) represents the clustering coefficient vector of a network.

7.

The mean versatility can measure the global integration of a network (Solé-Casals et al., 2019), which is denoted as \(f_=\frac \sum \limits _ \varvec_(i)\), where \(\varvec_\) represents the versatility vector of a network.

8.

The left intra-hemispheric strength \(S_\), right intra-hemispheric strength \(S_\) and inter-hemispheric strength \(S_\) are defined in Eq. 8.

9.

The left-to-right intra-hemispheric ratio is calculated as \(R_=\frac}}\).

10.

The intra-to-inter-hemispheric ratio is denoted as \(R_=\frac +S_}}\).

We assess the relationship between cognitive performance and global metrics of individual brain networks using Spearman correlation coefficients (SCCs). SCC is a non-parametric measure of rank correlation, evaluating the statistical dependence between the rankings of two variables. It determines how well the relationship between two variables can be described using a monotonic function. The SCC, denoted as \(\rho\), ranges from -1 to 1: \(\rho =1\) signifies a perfect positive monotonic relationship; \(\rho =-1\) signifies a perfect negative monotonic relationship; \(\rho =0\) indicates no monotonic relationship.

In this work, given the small sample size (\(N = 29\)) and the exploratory nature of the study, we interpret \(\rho> 0.2\) as indicating a potentially meaningful monotonic association, corresponding approximately to a small-to-moderate effect, rather than as a formal threshold for statistical significance.

For this analysis, we define \(\rho ^p_(\varvec^p,\varvec_)\) to represent the SCC between a global metric \(\varvec^p\) of networks with connection density p and IQ scores \(\varvec_\), where \(u \in \left\\) denotes different brain networks, \(\varvec \in \_a, \varvec_t, \varvec_, \varvec_, \varvec_,\) \(\varvec_, \varvec_, \varvec_, \varvec_, \varvec_, \varvec_, \varvec_ \}\) represents various global metric vectors of all subjects.

To further assess group differences across each global metric, we apply the Mann-Whitney U test, a non-parametric test used to determine whether there is a significant difference between the distributions of two independent groups. This test is particularly useful when the assumptions of normality are not met. The Mann-Whitney U test is defined as follows:

$$\begin P_p^}=U(\varvec_p^G, \varvec_p^C) \end$$

(12)

where \(\varvec_p^G\) and \(\varvec_p^C\) represent the metric vectors for the GG and CG groups, respectively, at density p. The resulting \(P_p^}\) is the probability of the difference between the two groups for each global metric, with lower values (typically \(P_p^} \le 0.05\)) indicating statistical significance. Specifically, we computed these measures for the GG and CG groups at a connection density of \(p = 0.1\).

Furthermore, to account for multiple comparisons and control the false discovery rate (FDR), we