2.1. Overview

As shown in Figure 1, the computational pipeline of the proposed framework consists of four main steps. In the first step, the whole‐brain tfMRI data of each subject in HCP Q1 and Q3 releases are decomposed by sparse coding (Lv et al., 2015), to extract temporal dynamics and the corresponding spatial profiles of functional network components, represented by the dictionary and sparse weighting coefficients, respectively. In the second step, a hierarchical clustering method is designed to compute the common group‐wise functional networks (CGFNs) from the network components learned in the first step. In the third step, functional connectivities and interactions among brain networks of individuals are modeled by a weighted graph. At last, the control energy of each node in the graph is calculated by the linear optimal control theory (Kim et al., 2018). In addition, energetically favorable nodes of each individual are identified and their energy consumption distributions are further assessed on the group‐wise level to characterize the energy consumption of CGFNs statistically.

Overall computational pipeline of the proposed framework. (a) The temporal dynamics and the corresponding spatial maps of brain network components are computed from the tfMRI data of each subject by the dictionary learning and sparse representation. (b) Common group‐wise functional networks (CGFNs) are obtained by a hierarchical clustering algorithm. (c) Functional connectivities and interactions among CGFNs are represented by a weighted graph model, where the nodes and edges denote the CGFNs and their spatial interactions, respectively. (d) The control energy consumption of each CGFN is calculated by the linear optimal control theory and the energy consumption characteristics are comprehensively analyzed and assessed

For the detailed description of the proposed method in this article, the following definitions and notations are used.

Xii=1,2,⋯n input matrix of tfMRI of ith subject.

Di=d1d2⋯dp dictionary.

dq(q=1,2,⋯p) atom in the dictionary.

αii=1,2,⋯n coefficient matrix.

CGFNjj=1,2,⋯N the jth common group‐wise functional network.

FNCk the kth functional network component.

FCIG=NSES functional connectivity and interaction graph with a node set NS and an edge set ES.

eij the edge between nodes CGFNi and CGFNj in FCIG.

wij the weight of the edge eij.

x∈RN the state vector of nodes in FCIG.

u∈RM control input of the dynamics of FCIG.

Eu control energy of the dynamics of FCIG.

xd∈RM the state vector of the driver nodes.

xnd∈RN−M the state vector of the nondriver nodes.

∥∙∥1 l‐1 norm.

∥∙∥F F norm.

In addition, correspondences between abbreviations used in this article and their full names are listed.

CGFNs: common group‐wise functional networks.

EMF/ELF: energetically most/least favorable.

EMFN/ELFN: EMF/ELF nodes.

EMFN‐I/ELFN‐I set: EMF/ELF node set of individuals.

J‐EMFN‐I/J‐ELFN‐I set: joint EMF/ELF node set of individuals.

EMFN‐G/EMLN‐G set: group‐wise EMF/ELF node set.

J‐EMFN‐G/J‐ELFN‐G set: group‐wise joint EMF/ELF node set.

LFD‐I/HFD‐I set: individual node set with a low/high fiber density.

LFD‐G/HFD‐G set: group‐wise node set with a low/high fiber density.

RBO: rank biased overlap.