## Abstract

Network control theory models how gray matter regions transition between cognitive states through associated white matter connections, where controllability quantifies the contribution of each region to driving these state transitions. Current applications predominantly adopt node-centric views and overlook the potential contribution of brain network connections. To bridge this gap, we use edge-centric network control theory (E-NCT) to assess the role of brain connectivity (i.e., edges) in governing brain dynamic processes. We applied this framework to diffusion MRI data from individuals in the Human Connectome Project. We first validate edge controllability through comparisons against null models, node controllability, and structural and functional connectomes. Notably, edge controllability predicted individual differences in phenotypic information. Using E-NCT, we estimate the brain’s energy consumption for activating specific networks. Our results reveal that the activation of a complex, whole-brain network predicting executive function (EF) is more energy efficient than the corresponding canonical network pairs. Overall, E-NCT provides an edge-centric perspective on the brain’s network control mechanism. It captures control energy patterns and brain-behavior phenotypes with a more comprehensive understanding of brain dynamics.

## 1 Introduction

Structural connectivity of the white matter facilitates communications between gray matter regions (Yeh et al., 2021). Specifically, this architecture supports the dynamics of switching between cognitive states (Sorrentino et al., 2021). Network control theory (NCT) provides a theoretical framework to quantify the energy needed for gray matter regions to switch between these cognitive states (Gu et al., 2015). The contribution of each region to a state transition is known as “controllability”. Average controllability measures a region’s ability to drive transitions to nearby states, while modal controllability measures a region’s ability to drive transitions to far-away states. NCT has furthered our understanding of the brain across various neuroimaging studies. For example, controllability develops rapidly in infancy (Sun, Jiang, et al., 2023) and continues gradually into adolescence (Tang et al., 2017) and adulthood (Gu et al., 2015). This change in controllability putatively underlies the maturation of behaviors such as executive function (EF) (Cui et al., 2020). Controllability is also altered in psychiatric (depression (Fang et al., 2021) and schizophrenia (Braun et al., 2021)) and neurologic (epilepsy (He et al., 2022) and Parkinson’s (Zarkali et al., 2020)) disorders. Finally, NCT provides a mechanistic understanding of noninvasive (neurofeedback) and invasive (transcranial magnetic stimulation) brain stimulation approaches (Bassett & Khambhati, 2017).

Nevertheless, current brain-based applications of NCT are based on node-centric frameworks, which conceptualize the brain as a graph with regions as nodes and functional and structural connections as edges. This node-centric view overlooks potentially meaningful interactions between edges (Faskowitz et al., 2022). Edge-centric frameworks instead extend the conventional concept of brain connectivity into higher dimensions, featuring interactions between the edges of a network instead of its nodes (Betzel et al., 2023). Through functional connectivity, individual-level edge time series (Sporns et al., 2021) and edge communities (Faskowitz et al., 2020, 2021; Gao et al., 2020) have produced complementary information to traditional node-centric graphs, including improved individual identifiability (Jo et al., 2021) and greater statistical power in network-level inference (Rodriguez et al., 2022). Several works from the edge-centric view have also shown promising results in detecting meaningful functional differences between study groups (Idesis et al., 2022; Jiang et al., 2022; Sun, Wang, et al., 2023; Yang et al., 2023; Zhang et al., 2023). In contrast to functional data, edge-centric communities based on structural connectivity have relied on group-averaged data due to a lack of time-series information used to construct edge-centric networks for an individual (de Reus et al., 2014). Individual-level edge-centric structural connectivity studies are not widely investigated.

In this work, we use edge network control theory (E-NCT) to study how the structural connectome facilitates system-level dynamic brain activities. First, we demonstrate an efficient way to convert the standard node-centric structural connectome to an edge-centric network via an incidence matrix. This conversion enables us to apply NCT on edge networks at the individual level. To benchmark E-NCT, we characterize edge controllability against null models, compare edge controllability with node controllability, and contrast edge controllability to structural and functional connectomes. To show the utility of E-NCT, we show that predictive models based on edge controllability perform favorably compared with standard structural connectomes in capturing individual differences in age, intelligence quotient (IQ), and EF. Finally, we calculate the energy cost necessary to change EF-related connectome patterns and compare it with the energy cost for canonical brain networks. Together, we provide a novel conceptualization of NCT for edge-centric brain networks. E-NCT can elucidate each edge’s role in supporting brain dynamics and capture individual differences in complex phenotypes.

## 2 Methods

Network control theory (NCT) aims to address the problem of how to control a complex network system through nodes (Liu et al., 2011) and edges (Nepusz & Vicsek, 2012). However, the classic application of network control theory on brain networks only focuses on measuring regional characteristics (as nodes) while overlooking the role played by connectivity (as edges) in regulating system-level activities. Therefore, edge-centric network control theory (E-NCT) may uncover distinct brain dynamics from controlling edges, further allowing the simulation of brain connectomes (Fig. 1).

### 2.1 Network control theory

- 1.
Dynamical process of brain activity

To describe the dynamics of the neural system over time, NCT employs a simplified linear dynamic network model. In the continuous setting, this model is

$x\u02d9(t)=Ax(t)+BKuK(t),$(1)where $x$ denotes the brain state at a given time, and

is the symmetric, undirected, and weighted adjacency matrix for the network. The input matrix $BK$ identifies the control points*A*in the system, where*K*= {k1,…,km} and $BK=[ek1...ekm]$, $ei$ denotes the ith canonical vector of dimension $N$ and input $uK$ denotes the input control strategy over time. In the discrete setting, this model is $x(t+1)=Ax(t)+BKuK(t)$. The variables are analogous to the continuous setting. Consistent with the literature, we define controllability (Deng et al., 2022; Gu et al., 2015) using the discrete setting and control energy using the continuous setting (Braun et al., 2021; Cui et al., 2020).*K* - 2.
Average and modal controllability of each brain region

Two metrics of controllability are defined to describe the ability to drive the network with different types of transitions as patterns of regional activity. Average controllability measures the ability to drive to nearby brain state transitions. Modal controllability measures the ability to drive to distant brain state transitions. Classic control theory provides that the controllability of the network from the set of network nodes

is equivalent to the controllability Gramian $WK$ being inverted, where $WK=\u2211\tau =0\u221eA\tau BKBKTA\tau $ for the discrete setting.*K* - (a)
Average controllability (AC)

AC is defined by the average input energy from a set of control elements and overall potential target states. The average input energy is proportional to $Trace(WK\u22121)$, the trace of the inverse of the controllability Gramian. Here, to keep the consistency with previous studies, $Trace(WK)$ is used as the measurement of average controllability to increase the accuracy of computation on small brain networks and maintain the information obtained by this measurement.

- (b)
Modal controllability (MC)

MC is defined as the ability of an element to control difficult-to-reach modes of the dynamic network system and is computed from the eigenvectors $V=[vij]$ of the adjacency matrix

. Here, we define the measurement of modal controllability as $\varphi i=\u2211j=1N(1\u2212\lambda j2(A))vij2$ of all $N$ modes $\lambda 1(A),...,\lambda N(A)$ from brain region $i$, following the definition of previous studies.*A* - 3.
Control energy to activate brain regions

To explore how the brain’s dynamic processes are constrained by the structural connectome, we model the brain system in the continuous setting and quantify the energy required to activate specific brain networks over the existing structural network topology of the brain. The baseline state

**x**(0) was set to 0 to simulate the resting state of the brain, while the target brain state**x**(T) was defined such that all regions in the desired brain network had a magnitude of 1 representing activation of the desired regions. Following the definition of the control task in the previous studies (Cui et al., 2020; Gu, Betzel, et al., 2017), where the system transitions from initial state $x(0)=x0$ to target state $x(T)=xT$ with minimum-energy input $uK(t)$ as an optimal control problem,$minu\u222b0T(xT\u2212x(t))T(xT\u2212x(t))+\rho uK(tT)uK(t))dt,s.t.\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009x(t)=Ax(t)+BK(t)uK(t),x(0)=x0,x(T)=xT,$(2)where $T$ is the control horizon, $\rho \u2208\mathbb{R}+$, $(xT\u2212x(t))$ is the distance between the state at time $t$ and the target state.

To solve the optimal control $u*$, we define the Hamiltonian as

$H(p,x,u,t)=x\Tau x+\rho u\Tau u+p\Tau (Ax+Bu).$(3)Based on the Pontryagin minimum principle (Boltyanskiy et al., 1960), if $u*$ is an optimal solution to the minimization problem with corresponding state trajectory $x*$, then there exists $p*$ such that

$\u2202H\u2202x=\u22122(xT\u2212x*)+ATp*=\u2212p\u02d9*,$(4)$\u2202H\u2202u=2\rho u*+BTp*\u2009=0$(5)With the constraints from eqs. 4 and 5, the optimization problem $u*$ can be solved (Gu, Betzel, et al., 2017). Control energy for each node $ki$ was defined as $Eki\u2009=\u222bt=0T||uki*(t)||2dt$, indicating the overall energy input required by the node to facilitate the desired state transition.

### 2.2 Edge-centric network control

- 1.
Transformation from the node adjacency matrix $A$ to the incidence matrix $C$

Due to the region-based nature of brain imaging, the node-centric brain network $G$ can be directly represented by an $N\xd7N$ adjacency matrix $A$. This undirected and weighted network G can, therefore, be represented by incidence matrix $C$ (Evans & Lambiotte, 2009),

$CCT=A+D,$(6)where $A$ is the node-centric adjacency matrix, $D$ is the node degree matrix with its diagonal elements as $dii=\u2211jAij$. The $N\xd7L$ incidence matrix C can be obtained as

$Ci\alpha ={ANijif\u2009node\u2009i\u2009is\u2009incidentwith\u2009edge\u2009\alpha 0otherwise.$(7) - 2.
Transformation from the incidence matrix $C$ to the edge adjacent matrix $AE$

To project the information of network $G$ into a line graph $L(G)$, the edge-centric adjacency matrix $AE$ can be obtained by

$AE=CTC\u2212W,$(8)where $C$ is the incidence matrix and $W$ is a diagonal matrix with each element as the weight of each edge. For each element in the edge-centric adjacency matrix,

$AE\alpha \beta =\u2211iCiaCi\beta (1\u2212\delta \alpha \beta ),$(9)where C is the incidence matrix and $\delta \alpha \beta =1$ if $\alpha =\beta $.

- 3.
Calculate controllability for each edge from Section 2.1. Part 2 based on edge-centric network

Based on the edge-centric brain network, represented by edge adjacency matrix $AE$, the input vector $uL$ injects energy on the edges to drive the network from its initial state to a targeted state. Therefore, the edge-centric adjacency matrix of the structural connectome for each individual functions as the wiring diagram in the linear brain state transition. The input matrix $BL$ is in $L\xd71$ dimension, with each element indicating if the edge receives control input. Combining the controllability definition from Section 2.1. Part 2, average controllability for each edge (eAC) is defined as $Trace(WL)$, where $WL=\u2211\tau =0\u221eAE\tau BLBLTAE\tau $ and modal controllability of each edge (eMC) equals $\u2211\beta =1N(1\u2212\lambda j2(A)vE\alpha \beta 2$, where $VE=[vE\alpha \beta ]$ are the eigenvectors of edge-centric adjacency matrix $AE$ and $\lambda E$ are the corresponding eigenvalues.

- 4.
Adapt control energy to control a connectome from Section 2.1. Part 3

Control energy $EL$ is available from solving the optimal control problem

$minu\u222b0T(xT\u2212x(t))T(xT\u2212x(t))+\rho uL(tT)uL(t))dt,s.t.\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009x(t)=AEx(t)+BL(t)uL(t),x(0)=x0,x(T)=xT,$(10)and summarizing the optimal input energy $u*$ over time: $EL\u2009=\u222bt=0T||ul*(t)||2\u2009dt$.

Following the parameter setting of the previous study, we used T=1 with step size of 0.001 for the time horizon of control. Therefore, there were 1000 steps for the system to get the target state from the initial state during simulation. Here, the target states were defined as 28 inter-/intra- networks of canonical networks (i.e., visual, somatomotor, dorsal attention, ventral attention, limbic, frontoparietal, and default mode networks). Edges in the target network were assigned a value of 1, indicating activated. All other edges were assigned 0’s, indicating not activated.

### 2.3 Datasets

Two datasets were used in this study. Primary analyses were conducted on data from the Human Connectome Project Young Adult (HCP-YA; https://www.humanconnectome.org/study/hcp-young-adult) and replicated on data from the Human Connectome Project Development (HCP-D; https://www.humanconnectome.org/study/hcp-lifespan-aging), which includes only the adult part (age>18 years) of participants.

#### 2.3.1 HCP-YA

The HCP-YA imaging protocol details have been extensively documented (Glasser et al., 2013). In summary, all MRI data were obtained with a 3T Siemens Skyra using a slice-accelerated, multiband, gradient-echo, EPI sequence (72 slices acquired in the axial-oblique plane, TR = 720 ms, TE = 33.1 ms, flip angle = 52°, slice thickness = 2 mm, in-plane resolution = 2 mm × 2 mm, multiband factor = 8) and a MPRAGE (256 slices acquired in the sagittal plane, TR = 2400 ms, TE = 2.14 ms, flip angle = 8°, slice thickness = 0.7 mm, in-plane resolution = 0.7 mm × 0.7 mm).

Diffusion MRI was sampled using a multishell diffusion scheme with a maximum b-values of
3000 s/mm^{2}, in-plane resolution of 1.5 mm, and slice thickness of 1.5 mm. The
diffusion data were reconstructed using generalized q-sampling imaging (Yeh et al., 2010) with a diffusion sampling length ratio of 1.25. The
tensor metrics were calculated and analyzed using the resource allocation (TG-CIS200026) at
Extreme Science and Engineering Discovery Environment (XSEDE) resources (Towns et al., 2014). Whole-brain fiber tracking was conducted with
DSI-studio with quantitative anisotropy (QA) as the termination threshold. QA values were
computed in each voxel in their native space for each subject. These QA values are used to
warp the brain to a template QA volume in Montreal Neurological Institute (MNI) space using
the statistical parametric mapping (SPM) nonlinear registration algorithm. Once in MNI space,
spin density functions were again reconstructed with a mean diffusion distance of 1.25 mm
using three fiber orientations per voxel. Fiber tracking was performed in DSI studio with an
angular cutoff of 60°, step size of 1.0 mm, minimum length of 30 mm, spin density
function smoothing of 0.0, maximum length of 300 mm, and a QA threshold determined by DWI
signal. Deterministic fiber tracking using a modified FACT algorithm was performed until
10,000,000 streamlines were reconstructed for each individual. Here, we used AAL2 atlas (Rolls et al., 2015) in MNI space with 120 nodes to construct
individual structural connectome. The pairwise connectivity strength was calculated as the
average QA value of each fiber connecting the two end regions. Edges with a value less than
0.001 were interpreted as not being connected and having a value not equal to 0 due to
numerical instability in the fiber tracking algorithm, and, therefore, set to 0. This process
resulted in a 120 x 120 matrix for each participant.

The preprocessing of resting-state functional MRI data from HCP-YA followed the steps from previous studies (Dadashkarimi et al., 2023; Greene et al., 2018). The HCP minimal preprocessing pipeline was used for artifact removal, motion correction, and registration. All subsequent preprocessing was performed in BioImage Suite (https://bioimagesuiteweb.github.io/) and included standard preprocessing procedures, including removal of motion-related components of the signal; regression of mean time courses in white matter, cerebrospinal fluid, and gray matter; removal of the linear trend; and low-pass filtering. Like the structural connectome construction, the AAL2 atlas was applied to the preprocessed fMRI data to create a mean time series for each brain region. Functional connectomes were then generated by calculating the Pearson correlation between each pair of node-wise time series and then taking the Fisher transform.

We restricted our analyses to those subjects who participated in all fMRI and dMRI scans, whose mean frame-to-frame displacement during fMRI was less than 0.1 mm, and for whom IQ measures and executive function task (i.e., List Sorting, Card Sorting, and Flanker task) scores were available (n = 515; 241 males; ages 22–36+ years).

#### 2.3.2 HCP-D

HCP-D imaging protocol details are documented by Somerville
et al. (2018). Here, we only included 149 subjects over age 18 years for replication
and validation. Demographic information is summarized in Table S1. All HCP-D brain imaging is
obtained from 3T Siemens Prisma scanners. Diffusion imaging samples 185 directions on 2 shells
of b = 1500 and 3000 s/mm^{2}, along with 28 b = 0 s/mm^{2} images. Construction of structural connectomes with the diffusion data used the same pipeline
as above in HCP-YA. The diffusion data were reconstructed using generalized q-sampling imaging
with quantitative anisotropy (QA) as the termination threshold. The AAL2 atlas with 120 nodes
was used to construct an individual structural connectome with averaged QA values from fibers
tracked between two regions, resulting in a 120 x 120 adjacency matrix for each
participant.

### 2.4 Null model

To validate specificity of our controllability results, we constructed 5000 null models for the group-averaged structural connectome by randomly rewiring connections from the original network with weight, degree, and strength distribution (Rubinov & Sporns, 2010). We calculated the corresponding controllability measures (i.e., control energy, average controllability) for each rewired connectome. The number of times that empirical controllability is higher/lower than the permuted controllability provides a nonparametric p-value. When appropriate, multiple comparisons were corrected with false discovery rate (FDR).

### 2.5 Connectome-based prediction model

We used connectome-based predictive modeling (CPM) to test edge controllability’s ability to predict individual phenotypes. Age, fluid intelligence (IQ), and three EF tasks were predicted. We also predicted a composite score of the three EF tasks based on a principle component analysis (PCA) of the tasks. The literature routinely uses these phenotypes for benchmarking predictive models (Butler et al., 2014; Litwińczuk et al., 2022; Schumacher et al., 2019). Independently, node controllability (including both AC and MC in a single model), structural connectivity, eAC, and eMC were the input features for CPM. We compared prediction performance from edge controllability to node controllability and structural connectivity.

We used CPM with a feature selection threshold at p = 0.01 (Shen et al., 2017). Tenfold cross-validation was performed. The dataset was randomly divided into 10 subgroups, among which 9 subgroups were used as training and the left 1 as a testing group. Model training involves feature selection of relative brain features within the training group using Pearson’s correlation. The brain features (e.g., nAC/nMC for each node or eAC/eMC/edge strength of SC for each edge) significantly correlated (p-value < 0.01) with the phenotype measure (e.g., EF, IQ, or age) were retained. Next, the selected features for each individual were summarized into a single number. Linear regression was then used to model this summary score and the phenotype in the training group. Finally, this model was applied to the testing group. This process was repeated iteratively, with each subgroup being the testing group once, generating a predicted result for each individual in the dataset. We predicted each phenotype independently (EF, IQ, and age). To compare the prediction performance across different brain features, we performed 1000 repeats of 10-fold cross-validation in the same train-test group across the different brain features. p-Values were determined by the proportion of predictions for edge controllability that were better than that of other features among the 1000 repetitions.

## 3 Results

### 3.1 Edge controllability distribution across whole brain

The whole-brain distribution of edge controllability is shown in Figure 2a. Edges with the highest average controllability were observed within
the parietal lobe and between the parietal and occipital lobes. Edges with the highest modal
controllability are observed within and between the temporal lobe, cerebellum, and subcortical
regions. We define eAC_{whole-brain} and eMC_{whole-brain} as the sum of eAC or
eMC across all edges. Across individuals, eAC_{whole-brain} and
eMC_{whole-brain} are negatively correlated (r = -0.81, p<0.001),
suggesting that individuals with better controllability on nearby-state transitions tend to
display less capability to control distant brain state transitions. There were no sex
differences in eAC_{whole-brain} (t = -0.86, p = 0.38) and
eMC_{whole-brain} (t = 0.71, p = 0.47). For each edge, eAC is negatively
associated with eMC across all individuals (r = -0.93, p<0.001), suggesting an
edge can only control nearby or distant state transitions. These results are consistent with
previous node controllability results (Gu et al.,
2015).

To investigate the specificity of our results, both measures of edge controllability were
significantly different than the null models on the whole-brain level (Fig. S1). eAC_{whole-brain} in
the real network was significantly lower than in the null models (p<0.001). These
results may be caused by the decrease of rich clubs and increase in local connections during
the rewiring step in creating null models (Rubinov &
Sporns, 2010; Van Den Heuvel and Sporns, 2011).
This topology leads to the null models that are more capable to drive short-distance system
transitions on the energy landscape (i.e., increased eAC_{whole-brain}).
eMC_{whole-brain} in the real networks was significantly higher than in the null
models ( p<0.001), suggesting that the structural connectome’s rewiring leads to
less efficient, long-distance state transitions. After correcting for multiple comparisons
using FDR, 4259 edges had an average controllability significantly lower than the null model.
In total, 4517 edges had a modal controllability significantly higher than the null models.
This corresponds to 59.6% and 63.3% of edges exhibiting significantly different controllability
than the null models. The distribution of significant edges is shown on network level in Figure S2, highlighting the large number
of edges in the cerebellar and limbic networks.

We also investigated edge controllability for nine canonical networks by summing eAC and eMC over all edges in a canonical network. A critical advantage of edge controllability is that controllability can be separated into within-network (e.g., the controllability of edges in the default mode network—DMN) and between-network controllability (e.g., the controllability of edges between the DMN and the frontoparietal network—FPN). Edges with highest eAC located within the limbic network and between limbic and somatosensory, and default mode networks. Edges with highest eMC lived within the cerebellum, and between ventral-attention and subcortical networks (Fig. 2b).

Separating controllability into within-network (e.g., the controllability of edges in the DMN) and between-network components via E-NCT allowed us to observe these reversed patterns.

### 3.2 Consistency and distinctiveness of edge controllability

To provide face validity of edge controllability, we tested the consistency between
controllability based on node-centric and edge-centric networks. First, we summed the edge
controllability of every edge linked to one node and divided it by the number of edges,
resulting in the mean edge controllability for each node. We labeled these
eAC_{node-mean} and eMC_{node-mean}.On the group level, node controllability
and node-wise mean edge controllability are strongly correlated (Pearson’s correlation;
node AC and eAC_{node-mean}: r = 0.94, p<0.001; node MC and
eMC_{node-mean}: r = 0.98, p<0.001; Fig. 3a).

Moreover, eAC_{node_mean} was positively correlated with the node strength (r
= 0.91) and eMC_{node_mean} was negatively correlated with the node strength (r
= -0.99) in the structural connectome (nSC); eAC_{node_mean} was correlated with
the positive node strength (r = 0.20) and negative node strength (r = -0.43), and
eMC_{node_mean} was correlated with the positive node strength (r = -0.29) and
negative node strength (r = 0.52) in the functional connectome (nFC). These results
suggest that edge controllability measures information similar to conventional brain network
measurements for nodes when summarized at the node level.

However, edge controllability revealed distinctive information on the edge level compared with edge features from structural and functional connectomes. For structural connectomes, on the edge level, edge strength showed weak correlations with eAC (r = -0.06) or eMC (r = 0.05). Similarly, for functional connectomes, edge strength showed no significant correlations with eAC (r = -0.005) or eMC (r = 0.02) (Fig. S3). Further, eAC (r = 0.03) and eMC (r = -0.03) were not strongly correlated with the Euclidean distance between nodes. In contrast, SC (r = -0.20) and FC (r = -0.45) showed typical correlations with the Euclidean distance between nodes (Fig. S4). Overall, we showed that edge controllability remained consistent with conventional nodal measurements on the node level but allowed information to be investigated at a finer grain on the edge level (Fig. 3b).

### 3.3 Edge controllability better predicts behavior

Numerically, eMC models exhibited the strongest correlation between observed and predicted phenotypes (Fig. 4a; Table S2). Among all predictable tasks, edge controllability outperformed node controllability, particularly in the Flanker test and the executive function principal component, for which reliable predictions could not be generated based on node controllability (Fig. 4b). Additionally, for the fluid intelligence scores (p = 0.001) and list sorting (p = 0.003) scores, the prediction with edge modal controllability was significantly higher than those of structural connectivity. Card sorting scores cannot be predicted from the brain features investigated here. Additional prediction results are shown in Table S2, including prediction from functional connectivity and controlling feature dimensionality. Node controllability has 120 features, whereas all other data have 7140 features. CPM models were rerun limiting the number of features to 120. The prediction results were replicated with kernel ridge regression as shown in Table S3.

### 3.4 Energy-efficient activation of the EF-related functional network

Previous NCT work has shown that the energy cost of transition to the activation state for a particular network relates to behavioral and developmental patterns. For example, the energetic cost to activate the FPN decreases over adolescence, allowing for improved EF as one becomes an adult (Cui et al., 2020). We extend this approach beyond node activation to understand the energy cost of changing the strength of functional connectivity.

We calculated the energy cost necessary to change connectivity within and between canonical brain networks using E-NCT. Similar to previous control energy studies, edges in the target network (i.e., 1 of the 28 between/within networks) were assigned a value of 1. Edges in every other network were assigned a value of 0. Normalized network control energy for each edge-level canonical network, defined as the control energy to change the target network divided by the network size (i.e., number of edges in the network; Fig. 5a), was significantly higher than those of null models except edges within the dorsal attention network (Fig. S5). Consistent with node-centric results (Cui et al., 2020), changing the edges within the FPN requires the highest amount of control energy. However, some edges to the FPN, like edges between the FPN and limbic networks, were recruited with substantially less energy to change. Together, these results suggest that the high cost to activate FPN nodes is driven by within FPN connections rather than connections between other networks. The network pairs that included the limbic network were the easiest to change.

We also investigate the cost of changing connectome patterns in complex, whole-brain networks from predictive modeling. First, we used resting-state functional connectomes to predict EF using CPM with 10-fold cross-validation and a feature selection threshold of p<0.01. We observed significant prediction performance (r = 0.11, p = 0.018). Next, we calculated the required energy to change the connectivity of this predictive network as defined by the positive network from CPM. Similar to the above, the magnitude of the edges in the positive network from CPM (Fig. 5b) was assigned 1, and all other edges were set to 0. Surprisingly, the energy cost to change the EF predictive network is lower than any within or between network connectivity. This suggests that the energy to change complex, predictive networks—which putatively better reflect the underlying behavior—might be lower than that to change canonical networks. Next, we further broke down the energy cost to change the EF network by canonical networks on the population level. Edges within the FPN mostly explain the EF network’s energy costs and not connections between the FPN and other networks (Fig. 5c). Further, for each individual, differences in contributions of canonical network control energy for the EF network can be associated with their fluid intelligence performance. The energy contributions of FP-VS (r = -0.095, p = 0.028, FDR-corrected) and VA-LM (r = -0.087, p = 0.045, FDR-corrected) were negatively correlated with individual fluid intelligence scores, while those of DM-DM were positively correlated with intelligence (r = 0.10, p = 0.024, FDR-corrected; Fig. S6). This result suggested that—in addition to the total energy consumption—the personal strategy of using control energy to change brain networks might be related to individual cognition performance.

### 3.5 Replication of E-NCT in external data

Finally, we replicated our E-NCT results in a secondary dataset using young adults (>18 years old; n = 149) from the HCP-D dataset. As shown in Figure S7, edge controllability between the two datasets was highly correlated (eAC: r = 0.93; eMC: r = 0.93), indicating high consistency between which edges exhibited large edge controllability across datasets. Additionally, associations between edge controllability and node controllability, structural connectivity, and functional connectivity also hold in the replication from the HCP-D dataset (Table S4).

## 4 Discussion

Research in network science has advanced in descriptions of the structural and dynamic properties of networks, on both nodes (Liu et al., 2011) and edges (Nepusz & Vicsek, 2012; Pang et al., 2017). Integrating network control theory (NCT) and edge-centric brain networks, we use edge-centric NCT (E-NCT) to calculate the controllability for each edge (i.e., edge controllability) in the connectome. We converted the standard node-centric connectome into an edge-centric network using an incidence matrix to create edge-centric networks from structural data for each individual. This step enabled the application of E-NCT for each individual. To demonstrate the framework’s effectiveness, the analysis of edge controllability on null models provided evidence that edge controllability could reflect the topological characteristics of structural connectomes. Furthermore, the validity of edge controllability was tested by mapping the edges back to each node and comparing it with node controllability. Additionally, edge controllability predicted individual differences in phenotypic information. Finally, simulating control energy to change edges that predict EF revealed a distinct energy consumption pattern from canonical networks. By considering the interactions between edges, E-NCT naturally distributes a node’s controllability among its edges, distinguishing the contribution of network-to-network collaborations and giving a more comprehensive view of brain dynamics.

When summing the edge controllability of each edge connected to a node, we found a high consistency with traditional node controllability. This result suggests that E-NCT allows us to break down how each edge contributes to a node’s controllability. The consistency between edge and node controllability also suggests that traditional node controllability can be recovered from edge controllability. Computationally, one could explicitly calculate only edge controllability and recover node controllability rather than calculate both independently. This association between edge and node controllability provides an intuitive interpretation. Node controllability is proportional to summing edge controllability for edges belonging to that node—analogous to node degree and strength. Nevertheless, this result is expected (Novelli & Razi, 2022) because the same structural connectome is used to estimate both controllability measures. Overall, edge-centric methods originate from the same information source but at a finer resolution than node-centric methods.

Furthermore, moving from node to edge controllability benefitted downstream analyses. Edge controllability showed an improved prediction of phenotypic information over node controllability. One explanation is that brain features on the edge level include considerably greater information than those on the node level, with complexity in the order of $O(n2)$ for edges compared with $O(n)$ for nodes. This interpretation aligns with other works showing that connectomes achieve better predictions than regional approaches (Jo et al., 2021; Sun, Wang, et al., 2023; Zhang et al., 2023). Additionally, as node controllability appears to be proportional to the sum of edge controllability, this summing averages out effects, reducing information for prediction. Analyzing controllability at the edge level retains this information, which could be necessary to better understand cognition or psychiatric and neurological disorders in future applications of E-NCT.

A further example of this benefit is seen in the energy consumption simulations, where the energy cost of complex, whole-brain task-predictive networks can be simulated and broken down by canonical network pairings. This advance is essential as many behavioral phenotypes manifest through the interaction between multiple networks and regions (Mišić & Sporns, 2016). As such, connectome-based predictive models better predict and, putatively, better reflect the underlying functional anatomy of cognition (Avery et al., 2020; Sui et al., 2020) and clinical disorders (Garrison et al., 2023; Greene & Constable, 2023; Vogel et al., 2023) than models involving only functional specific regions. Our preliminary results suggest that these whole-brain task-predictive networks are easier to change than canonical brain networks and may better reflect the energetic costs of behaviors such as EF. Additionally, for the EF network, changing the connectivity state for edges within the FPN required the most energy, consistent with control energy studies with a node-centric approach (Cui et al., 2020). This high control energy cost was not observed in connections between the FPN and other networks. These findings are consistent with the FPN being a flexible hub (Adeli et al., 2019) that changes its connectivity to other networks to switch between different tasks—a component of EF. Calculating control energy for within- and between-network edges independently exhibits how the FPN can have high internal activation costs while being highly adaptive in its connectivity to other networks.

NCT promises to explain and predict how one can exogenously steer the brain away from pathological states and toward healthy states (Muldoon et al., 2016). Although node-centric NCT is currently used in brain stimulation studies (Hahn et al., 2023), growing noninvasive and invasive brain stimulation approaches have been moving from targeting a single region to multiple regions and their connections. For example, connectome-based neurofeedback has shown promise in targeting the connectivity of edges (Scheinost et al., 2020). The field of deep brain stimulation also is shifting from targeting anatomical regions toward activation of white matter pathways (Choi et al., 2021). For example, full activation of white matter pathways predicted better treatment response to deep brain stimulation of the subcallosal cingulate for treatment-resistant depression (Song et al., 2023). Highly controllable edges may be better white matter pathways to activate. Additionally, E-NCT could potentially serve as a computational model to guide transcranial electrical stimulation (Woods et al., 2016), such as transcranial direct current stimulation (tDCS) and transcranial alternating current stimulation (tACS). Highly controllable edges could be targeted by placing anodes and cathodes at the two ends of those edges.

Our work has several strengths. Previous work has investigated edge-centric functional
networks using their temporal characteristics (Sporns et al.,
2021) to study the network features such as community structures (Chumin et al., 2022; Idesis et al.,
2022) and modularity (Gu, Yang, et al., 2017) on
the individual level. However, edge-centric structural connectomes have primarily been studied
at the group-averaged level (de Reus et al., 2014) rather
than at the individual level. This limitation arises from the difficulty in constructing an
edge-centric connectome for each individual. First, we efficiently construct edge-centric
connectomes for each individual utilizing the incidence matrix transform between a graph
(node-centric network) and a line graph (edge-centric network). Second, eMC and eAC were unique
in the edge strengths of both structural and functional connectomes. However, when summing edge
strength over all edges connected to a node (i.e., the node level), eAC_{node_mean} and
eMC_{node_mean} were strongly correlated with equivalent measures from structural and
functional connectomes. Other real-world networks, including the US telecommunication network
and most World Wide Web networks, also show significant differences when using an edge-centric
view (Nepusz & Vicsek, 2012), mainly due to the
highly hierarchical structure of the networks (Battiston et al.,
2021). Third, we compared edge controllability with null models, preserving the
network’s degree distribution, and observed that the edge controllability differed
significantly from that of null models. Additionally, the effect of rewiring during null-model
reconstruction is hierarchically distinct for each edge. The higher the controllability in the
network, the more it is influenced by rewiring when creating the null networks. Fourth, we
reproduced results in a second dataset of young adults (participants in HCP-D >18 years
old).

The current study also has several limitations. First, edge-centric approaches square the number of elements compared with node-centric approaches. As such, E-NCT requires comparatively more memory and computational resources than node-centric NCT. The AAL2 atlas was chosen over other parcellations due to the lower number of nodes and associated reduced computational time. Computational complexity for E-NCT grows at $O(n2)$. Currently, calculating eAC and eMC takes ~150 s, while calculating control energy takes ~1.2 hours for each subject. Similar results were observed when using the Dosenbach 160 node atlas (Dosenbach et al., 2010), suggesting E-NCT is not atlas dependent. Second, it may be difficult to generalize E-NCT resting-state fMRI data. Functional connectivity includes positive and negative correlations between regions depending on preprocessing choices (Murphy & Fox, 2017; Murphy et al., 2009). However, “negative” edge weights are difficult to conceptualize in NCT. Future work may focus on silencing brain activity to activate a “negative” brain network (Tu et al., 2021), parallel to stimulating brain regions to activate a positive one. Investigating E-NCT with resting-state fMRI and associations with various processing choices (e.g., global signal regression) is future work. Third, to fully model the information flow within structural networks, it is necessary to know the axonal directionality or theoretically define the direction of edges in the network. Diffusion MRI used in most human studies can quantify the strength of the structural connection but fail to provide information about the direction of information flow. Future research should incorporate directional information into NCT. Fourth, we formulate E-NCT with different settings (discrete or continuous) based on the measures. We use the discrete setting for eAC and eMC and the continuous setting for control energy. This approach was chosen to be consistent with the papers that introduced these measures in neuroimaging. Future work is needed to compare E-NCT under the two settings.

## 5 Conclusion

In conclusion, we proposed an E-NCT to explore the capability of each edge of the structural connectome in supporting brain state transitions. Our findings show that edge controllability provides a complementary measure of NCT. By considering the interactions between edges, E-NCT naturally distributes a node’s controllability among its edges, distinguishing between within- and between-network effects and giving a more comprehensive view of brain dynamics. In sum, E-NCT provides an edge-centric perspective on the brain’s network control mechanism.

## Ethics

All participants from whom data were used in this manuscript provided written informed consent (and consent to publish) according to the declaration of Helsinki.

## Data and Code Availability

The Human Connectome Project (HCP) dataset is publicly available at https://db.humanconnectome.org/ with the acceptance of HCP Open Access Data Use Terms. The diffusion data were processed with DSI studio https://dsi-studio.labsolver.org/ and functional data were processed with https://bioimagesuiteweb.github.io/. The original edge-centric network control theory code is publicly available at https://github.com/huiliii/edge_control.

## Author Contributions

H.S.: conceptualization, methodology, investigation, visualization, and writing—original draft; J.D. and L.T.: methodology, writing—review & editing; M.R. and R.R.: writing—review & editing; D.S.: conceptualization, funding acquisition, supervision and writing—review & editing.

## Declaration of Competing Interest

The authors have no competing interests.

## Acknowledgement

National Institutes of Health grant R01MH121095 (DS)

## Supplementary Materials

Supplementary material for this article is available with the online version here: https://doi.org/10.1162/imag_a_00191.

## References

*44*,