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 Table of Contents  
Year : 2022  |  Volume : 1  |  Issue : 3  |  Page : 128-135

Estimating high-order brain functional network via signed random walk for mild cognitive impairment identification

1 School of Computer Science and Technology, Shandong Jianzhu University, Jinan; School of Mathematical Sciences, Liaocheng University, Liaocheng, Shandong Province, China., China
2 School of Mathematical Sciences, Liaocheng University, Liaocheng, Shandong Province, China., China
3 Department of Radiology and BRIC, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA., USA

Date of Submission06-Jun-2022
Date of Decision05-Jul-2022
Date of Acceptance11-Jul-2022
Date of Web Publication29-Sep-2022

Correspondence Address:
Ming-Xia Liu

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Source of Support: None, Conflict of Interest: None

DOI: 10.4103/2773-2398.356522

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Brain functional network (BFN) has become an increasingly important tool to discover informative biomarkers for diagnosing neurodegenerative diseases, such as Alzheimer’s disease and its prodrome stage, namely mild cognitive impairment. Currently, the most popular BFN estimation methods include Pearson’s correlation and sparse representation. Despite their empirical success in some scenarios, such estimated BFNs only capture the low-order relationship (i.e., the direct connectivity strength between brain regions), ignoring the high-order information in the brain (e.g., the global network structure). Therefore, in this study, we proposed a novel method based on the signed random walk (SRW) to estimate high-order BFNs. Not only can SRW measure the global network structure, but it can also naturally deal with negative brain functional connectivity through the structural balance theory. To the best of our knowledge, this study was the first to use SRW in BFN estimation. Furthermore, considering the complex interaction among different brain regions, we developed a parameterized variant of SRW for improving the flexibility of the high-order BFN estimation model. To illustrate the effectiveness of the proposed method, we identified patients with mild cognitive impairment from normal controls based on the estimated high-order BFNs. Our experimental findings showed that the proposed scheme tended to achieve higher classification performance than baseline methods.

Keywords: brain disease; functional connectivity; functional magnetic resonance imaging; signed random walk

How to cite this article:
Zhang LM, Wu X, Su H, Guo TT, Liu MX. Estimating high-order brain functional network via signed random walk for mild cognitive impairment identification. Brain Netw Modulation 2022;1:128-35

How to cite this URL:
Zhang LM, Wu X, Su H, Guo TT, Liu MX. Estimating high-order brain functional network via signed random walk for mild cognitive impairment identification. Brain Netw Modulation [serial online] 2022 [cited 2023 Sep 22];1:128-35. Available from: http://www.bnmjournal.com/text.asp?2022/1/3/128/356522

Li-Mei Zhang, Xiao Wu,
Both authors contributed equally to this work.
Funding: This work was partly supported by National Natural Science Foundation of China (Nos. 61976110, 62176112, 11931008) and Natural Science Foundation of Shandong Province (No. ZR202102270451)

  Introduction Top

Brain functional network (BFN), estimated from resting-state functional magnetic resonance imaging (rs-fMRI), has been widely used in understanding the mechanism underlying brain function and revealing informative biomarkers for diagnosing neurological or mental disorders (Guye et al., 2010; Bijsterbosch et al., 2017; Song et al., 2019), such as mild cognitive impairment (MCI) (Supekar et al., 2008; Huang et al., 2010), which is an early stage of Alzheimer’s disease.

Traditional BFN-based brain disease identification generally includes three main steps (Filippi and Agosta, 2011; Wee et al., 2011). First, BFNs are estimated using, for example, Pearson’s correlation (PC) (Biswal et al., 1995) and sparse representation (SR) (Lee et al., 2011). Thereafter, features (e.g., connectivity strengths or node statistics features) are extracted from the estimated BFNs to train a diagnostic model. Finally, based on the trained model, patients with brain diseases are differentiated from normal controls (Jie et al., 2020).

Despite their empirical success in some scenarios (Wan et al., 2006; Xie et al., 2008; Zhang et al., 2016), such pipelines only extract low-order features (i.e., the correlations among brain regions) of BFNs to train the diagnostic model, but ignore high-order features (e.g., the correlations among the low-order features of brain regions) that can provide effective information for diagnosis of brain diseases (Macke et al., 2011; Chen et al., 2017; Zhou et al., 2018).

To address this issue, some high-order BFN (HOBFN) estimation methods have been proposed recently (Plis et al., 2014; Zhang et al., 2016; Guo et al., 2017). For example, Zhang et al. (2016) used two sequential correlations, also known as Pearson correlation’s correlation (PC2), to describe high-order relations among brain regions. Additionally, a recent study (Chen et al., 2016) proposed to construct HOBFNs by considering the time-varying properties of brain regions (in terms of rs-fMRI time series). Despite their effectiveness, these methods failed to explicitly capture the global feature of BFNs, since two nodes, even without local overlap in their neighborhood, could still be members of the same community from a global perspective.

In this paper, we propose a novel model to estimate HOBFNs based on the signed random walk (SRW). It can not only capture the global structure of a network (Yang, 2005; Tong et al., 2006) but also naturally deal with negative edges of BFNs through the structural balance theory (Cartwright and Harary, 1956; Leskovec et al., 2010). Herein, we simply regard the social network as an analogy to BFN for explaining the structural balance theory. The members in a social network correspond to nodes (in analogy with brain regions in BFN), and the friend (enemy) relationship corresponds to the positive (negative) edges (in analogy with the functional connection in BFN). Furthermore, in balanced networks, as shown in [Figure 1], a random surfer with + sign starts at node s (we define the sign of the surfer as positive at the beginning of random walking), then walks across a negative edge to node v, and its sign becomes negative. Finally, the sign of the surfer walks across a positive edge to node t remains negative. Therefore, the node s and node t have a positive edge (i.e., friend relationship). Otherwise, they have a negative edge (i.e., enemy relationship). Based on the advantages of the SRW, it is used to obtain HOBFNs by encoding the global structure of BFN.
Figure 1: The walker can change his sign when walking on negative edges.
Note: A random walker with a sign starts at node s, passes through node v, and finally reaches node t. If the sign random walker walks to other nodes and the sign does not change, then the relationship between the nodes is positive, otherwise it is negative. For example, the relationship between nodes s and t is negative (A, B), the relationship between nodes s and t is positive (C, D).

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However, due to the complexity of relationships among different functional connectivity (FC) in the brain (Li et al., 2017; Jie et al., 2020), we assume that some of the relationships between FC do not conform to the structural balance theory (Cartwright and Harary, 1956; Leskovec et al., 2010), just as the friend of my friend could be my enemy in a real social network (Jung et al., 2016). To deal with this problem, we modify the SRW model by introducing two parameters, γ and β (called SRW-P). Specifically, a two-step learning strategy is adopted. First, we construct initial BFNs using SR according to the sparsity prior of the brain (Lee et al., 2011; Li et al., 2019). Thereafter, SRW-P constructs HOBFNs, which can consider the global structure and complex edge relationships based on the initial BFNs. To verify the effectiveness of our proposed method, we apply the SRW and SRW-P to construct HOBFNs for identifying patients with MCI from normal controls. We select SR and PC2 as baseline methods in this work. Experimental results show that our methods outperform the baseline methods.

The rest of this report is organized as follows. In Section II, we review two methods used in constructing HOBFN. In Section III, we propose our HOBFN learning strategy, including the motivation and model. In Section IV, we evaluate our proposed method with applications for MCI identification. In Section V, we discuss our results and several aspects that affect the final performance. Finally, we conclude this paper in Section VI.

  Materials and methods Top

In this section, we first describe the data preparation (rs-fMRI acquisition and preprocessing), then introduce the methods that closely related to this study, and finally propose our new scheme for estimating HOBFNs.

Data acquisition and preprocessing

For MCI identification, we used the same preprocessed method, as previously reported in Zhou et al. (2018). Specifically, 137 participants, including 68 MCIs and 69 normal controls, are selected from the ADNI dataset, by excluding the participants with framewise displacement (>0.5) exceeding 2.5 minutes. They are scanned using 3.0 T Philips scanners. The acquired rs-fMRI data were processed using the SPM8 toolbox (http://www.fil.ion.ucl.ac.uk/spm/). For each participant, the scanning time was 7 minutes, resulting in 140 volumes. To ensure signal stabilization, the first three volumes of each participant were discarded. The remaining volumes were corrected to remove the effect of head movement, and participants with head movement > 2 mm or 2° were excluded. All participants were registered into the Montreal Neurological Institute space, and the Gaussian kernel with full width at half maximum of 6 × 6 × 6 mm3 was employed for spatial smoothing Thereafter, to further reduce the influences of nuisance signals, regression of ventricle, white matter signals, and Friston 24-parameter model were performed (Burkhardt et al., 2022). Additionally, according to the AAL template (Tzourio-Mazoyer et al., 2002), the rs-fMRI was divided into 116 regions. For each region, its mean rs-fMRI time series was band-pass filtered from 0.015 Hz to 0.15 Hz. Finally, all the mean rs-fMRI time series were used to form a data matrix χЄR137×161

Related work

Low-order BFN estimation

Partial correlation is a method that can remove the confounding effect from other regions of interest (ROI), as compared with the full correlation methods, such as PC (Zhou et al., 2018). However, it is generally calculated based on the estimation of inverse covariance matrix, which may result in an ill-posed problem (Bartlett, 1947). To avoid this problem, SR based partial correlation estimation, a typical method that can produce a more reliable BFN, is proposed, and its model is expressed as follows:

where xiЄ Rm, i=1,...,n is the rs-fMRI time series associated with the ith ROI, Wij is the estimated FC (network edge weight) between the ith and jth ROIs, m is the number of temporal image volumes, and λ is a regularization parameter that can control the balance between two terms in Equation 1. For simplicity, we further express Equation 1 into a matrix form as follows:

where X=[x1, x2...,xn] is the rs-fMRI data matrix, W is the adjacency matrix of the estimated BFN, ∥.∥1 is the l1 matrix norm that can encode the sparsity prior, and the constraint Wii=0 is used to remove the xi from X for avoiding the problem of trivial solutions.

HOBFN estimation

To model high-order relationships among brain regions, PC2, a simple and straightforward method, has been proposed, and its model is given as follows:

where Hij is the high-order relationship between the ith and jth ROIs. C Є is the ith column of adjacency matrix C estimated by Pearson’s correlation. Ci is the mean of the entries in vector Ci. Under the assumption that the Ci was centralized and normalized as shown in Equation 3, the PC2 can be simplified into the following matrix form:

In fact, we can continue the correlation operation to construct higher-order BFNs, but a recent study (Guo et al., 2021) has found that the higher order correlations generally fail to result in more discriminative BFN. Additionally, it is unclear that whether the PC2-based HOBFN can capture the global structure of networks that, as mentioned earlier, may play an import role in cognitive function of the brain.


Studies have found that the global structure of the brain network plays an important role in the classification of brain diseases (Busatto et al., 2003; Filippi and Agosta, 2011; Jie et al., 2014a, b). Note that SRW does not only obtain the global structure information of the network but also encodes negative edges in a signed network (e.g., BFN) based on the structural balance theory. According to Jung et al. (2016), the recursive equations of SRW model is defined as:

where Nt+ is the set of nodes that have a positive relationship with thetthnode, |Nt+|(|Nt- is the sum of the edge weights between the tth and other nodes in Nt+ (Nt+), |Nv|=|Nv+| + |Nv-| is the edge weight (absolute value of edge weight) between nodes v and t. rt+ is the probability that the positive surfer at node t after SRW moved from the seed node s. Otherwise, rt- is the probability that the negative surfer at node t after SRW moved from the seed node s. The relationship between node sand node t is positive (or negative), when rtd=rt+-rt- is greater (or less) than 0.

Proposed method

Most of the existing BFN estimation methods, such as SR and PC2, ignore the global structure of brain network. But such a global structure can potentially provide high-level brain functional interactions or supplementary information in the diagnosis of brain diseases (Macke et al., 2011; Chen et al., 2017; Zhou et al., 2018; Sun et al., 2021). Therefore, as a successful technique for capturing the global structure of a network, SRW is introduced in this study for estimating HOBFNs. Additionally, for better catching the complex relationship between nodes in BFNs, we use two parameters, γand β to improve the flexibility of SRW (called SRW-P) by assuming that some relationships do not always match the structural balance theory.

Model and algorithm

In this study, two main steps were used to build a BFN estimation model.

Step 1. Estimating initial BFNs based on traditional methods. SR was used to obtain the initial BFNs due to its simplicity and popularity.

Step 2. Estimating HOBFNs through SRW and SRW-P. SRW has been introduced as above, and Equations 5 and 6 can be rewritten in the following matrix forms:

where W is the signed adjacency matrix constructed by SR, |W is the absolute adjacency matrix of W. Here, =D-1 W where D is the degree diagonal matrix of |W| + contains only positive edge weights in W̃, W̃- only contains the absolute values of negative edge weights in .

As mentioned earlier, due to the complexity of the interactions among different brain regions, the structural balance theory does not always hold. Therefore, two parameters, γ and β, are introduced in Equations 7 and 8 to improve the flexibility of SRW, which results in a parametric variant of SRW. Its mathematical formula is expressed as follows:

where γ(or &#xx03B2;) is the probability that the sign of the surfer is unchanged (change) according to the balance theory. The iterative algorithm computes the r+ and r, as shown in [Table 1]. Specifically, we set q to the sth unit vector and initialized r+to qand r to 0. Thereafter, we iterate Equations 9 and 10 by updating r into r’until δ < Є.
Table 1: Algorithm

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  Results Top

Based on the estimated BFNs or HOBFNs, in this section, we design experiments for identifying participants with MCI from normal controls, and then show the experimental results under several different performance measures.

Feature selection and classification

In this study, we selected SR and PC2 as baseline methods and used the BFN edge weights as features for MCI identification. As a result, a high feature dimension (116×115/2 = 6670) was formed from the BFN matrix, and it usually causes the curse of dimensionality. To address this, we used the t-test to select features before MCI identification.

We applied the linear support vector machine (Chang and Lin, 2011), with default parameter C = 1 (it is relatively immune to high dimensions), to perform the classification task. A 10-fold cross-validation was adopted in our experiment to estimate the classification performance of different methods. Specifically, all participants were grouped into 10 subsets (with each subset having a roughly equal size). One subset was adopted for testing, whereas the remaining samples were used for training a classifier. The final performance was obtained by averaging results of all the runs. Three parameters were involved in the HOBFN-based classification method. Specifically, for regularization parameter λ, its candidate range was, [26, 25, 24,..., 24, 25, 26], and the candidate range of parameters γ and β that could measure the imbalance of sign edges were [0.90, 0.91,..., 0.99].

To obtain the optimal parameters for each method, an inner 10-fold cross-validation was conducted based on training data, as shown in [Figure 2]. That is, we randomly selected 10% of the training data as the validation set to determine the optimal parameters for each method. To prevent bias caused by random selection, this process was repeated 10 times independently, and the test data were not involved in such process. Additionally, to evaluate our proposed methods, we adopted accuracy, sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV) as performance metrics. These metrics are defined as follows: PPV = TP/(TP + FP) and NPV = TN/(TN + FN), where TP, TN, FP, and FN indicate true positive, true negative, false positive, and false negative, respectively. Notably, in this study, we treated patients with MCI as the positive class and the normal controls as the negative class.
Figure 2: The pipeline of mild cognitive impairment identifications used in this study
Note: BFN:brain functional network; BOLD: blood oxygen level-dependent; CV: cross validation;HOBFN:high-order brain functional network; SR:sparse representation; SRW: signed random walk; SRW-P: parametric variant of signed random walk; SVM:support vector machine.

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Classification results

As described earlier, we first constructed initial BFNs with different sparsity using SR. Thereafter, SRW and SRW-P were performed to refine the novel HOBFNs based on the initial BFNs. In [Table 2], we report the classification results of four different methods (i.e., SR, PC2, SRW, and SRW-P) for MCI classification under different P-values. For the SRW and SRW-P, we empirically set the tolerance ϵ to 10−15.
Table 2: The MCI classification results of four methods (i.e., SR, PC2,SRW, SRW-P) based on five performance metrics

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As shown in [Table 2], the following observations were made for the MCI classification task. First, the experimental results were affected significantly by the number of selected features (or the P-values in the t-test). For example, with P-value of 0.005, all the estimated BFNs cannot work well, even though SR obtains the best classification accuracy. In contrast, when more features were selected with an increased P-value of 0.05, the performance was improved for all the methods. Second, compared with the traditional methods (i.e., SR and PC2), the proposed SRW-P achieved the best classification performance when P-value is 0.05. It was about 9% and 2% higher than SR and SRW, respectively.

  Discussion Top

Significance of the results

Compared with traditional SR and PC2, the proposed method, by considering the global network structure, produced better classification performance, especially at P = 0.01 and 0.05 [Table 2]. This further shows that the structure of the network is indeed a kind of valuable information in the classification of MCI. In other words, the structure of BFN might influence the cognitive activity of the brain. Additionally, SRW-P generally performs better than SRW, meaning that the assumption that uncertainty existing in the relationship between ROIs is plausible. However, as discussed earlier, SRW and SRW-P do not perform well with P-values of 0.005 and 0.001, which illustrates that more features are needed in (especially SRW- and SRW-P-based) HOBFNs.

Influence of network modelling parameters

In fact, besides the P-values in feature selection, the regularization parameters in SR (responsible for initializing BFN) may affect the classification results. To further investigate the influence of regularization parameters on the classification results, we conducted an experiment for identifying MCI under different regularization parameters. The results are reported in [Figure 3].
Figure 3: The mild cognitive impairment classification accuracy (ACC) based on three brain functional network estimation methods (i.e., SR, SRW, SRW-P) with different regularization parameters from 2-6 to 26.
Note: ACC:Accuracy; SR:sparse representation; SRW:signed random walk; SRW-P:parameterized variant of signed random walk.

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Based on the experimental results, we can observe that as the regularization parameter increases, the classification performance of three methods (i.e., SR, SRW, SRW-P) tends to increase in most cases. Additionally, under different regularization parameters, the classification performance of SRW and SRW-P are generally higher than that of SR, indicating that the global structure of BFN may be linked with cognitive functions of the human brain, or at least can provide more discriminative information for identifying brain disorders.


Although our proposed methods performed better than SR and PC2 in MCI discrimination in most cases, this study still had several limitations. First, our proposed methods have only been used in MCI discrimination and have not been tested in other brain diseases, such as autism spectrum disorder (Wee et al., 2014, 2016) and major depressive disorder. Second, brain parcellation (i.e., the definition of brain regions) is a basic step for constructing BFN. In this study, we applied the AAL template to brain parcellation, while different brain parcellations will produce different FC (Zalesky et al., 2010). Therefore, we will evaluate our proposed methods using different brain parcellations to identify brain disorders and further improve our model to make it more flexible for simulating the dynamics of complex brain activity.


In this study, we proposed a novel method that can capture the global structure of BFN and improve the SRW model to better simulate the complex relationships between different regions of the human brain. More specifically, two main steps could successfully estimate the HOBFNs. First, we estimated initial BFNs based on SR with different regularization parameters. Thereafter, we applied the SRW and SRW-P to estimate HOBFNs. The experiments have proven that compared with traditional methods, the proposed SRW-P achieved the best classification accuracy under certain P-values.

Author contributions

LMZ proposed the idea for estimating HOBFN by SRW. TTG and HS realized the codes and conducted experiments. XW wrote the manuscript. ML revised the manuscript and performed the result analysis with constructive discussions. All authors approved the final version of the manuscript.

Conflicts of interest

No potential conflict of interest was reported by the authors.

Open access statement

This is an open access journal, and articles are distributed under the terms of the Creative Commons AttributionNonCommercial-ShareAlike 4.0 License, which allows others to remix, tweak, and build upon the work non-commercially, as long as appropriate credit is given and the new creations are licensed under the identical terms.

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  [Figure 1], [Figure 2], [Figure 3]

  [Table 1], [Table 2]


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