Weighted ordinal connection based functional network classification for schizophrenia disease detection using EEG signal

A brain connectivity network (BCN) is an advanced approach to examining brain functionality in various conditions. However, the predictability of the BCN is affected by the connectivity measure used for the network construction. Various connectivity measures available in the literature differ according to the domain of their working data. The application of random connectivity measures might result in an inefficient BCN that ultimately hampers its predictability. Therefore, selecting an appropriate functional connectivity metric is crucial in clinical as well as cognitive neuroscience. In parallel to this, an effective network identifier plays a vital role in distinguishing different brain states. Hence, the objective of this paper is two-fold, which includes identifying suitable connectivity measures and proposing an efficient network identifier. For this, the weighted BCN (WBCN) is constructed using multiple connectivity measures like correlation coefficient (r), coherence (COH), phase-locking value (PLV), and mutual information (MI) from electroencephalogram (EEG) signals. The most recent technique for feature extraction, i.e., weighted ordinal connections, has been applied to EEG-based BCN. EEG signals data has been taken from the schizophrenia disease database. Further, several classification algorithms such as k-nearest neighbours (KNN), support vector machine (SVM) with linear, radial basis function and polynomial kernels, random forest (RF), and 1D convolutional neural network (CNN1D) are used to classify the brain states based on extracted features. In classification, 90% accuracy is achieved by the CNN1D classifier with WBCN based on the coherence connectivity measure. The study also provides a structural analysis of the BCN.


Introduction
Schizophrenia (Sz) is one of the types of mental disorder affecting around 1% of the worldwide population [1,2]. In its active state, schizophrenia is characterized by symptoms like delusions, hallucinations, catatonia, disorganized speech, etc. [3]. Despite the fact that there is no permanent treatment for Sz, numerous patients do well with minor symptoms with it. The manual approach to diagnosing schizophrenia is lengthy and requires subjective feedback from the patient, leading to an incorrect diagnosis [4]. In order to improve the diagnosis accuracy, researchers have developed a computer-aided approach for schizophrenia detection using various neuroimaging techniques like magnetic resonance imaging (MRI), electroencephalogram (EEG) etc.
Even though various neuroimaging techniques are available in the literature, we considered EEG for this work. The EEG is an easily available and low-cost neuroimaging technique with high temporal resolution. The EEG data represents the electrical activity of the brain, recorded by placing electrodes over the scalp. Various studies have employed EEG data for schizophrenia detection. Das et al. [1] proposed an approach for multichannel EEG-based Schizophrenia detection. Here, the authors proposed a multivariate iterative filter (MIF) based approach, where EEG is divided 1 3 into multiple intrinsic mode functions (IMFs). The features are extracted from the different frequency components and are classified using different classifiers.
Similarly, Sun et al. [5] proposed an approach for EEGbased schizophrenia detection using a hybrid deep neural network. The approach starts with time and frequency domain feature extraction and converting them into the image form, which carries spatial data. The hybrid deep neural network model is constructed from the combination of convolutional neural network (CNN) and long short-term memory (LSTM) models. Baygin et al. [4] have considered EEG data for schizophrenia disease prediction. The study presents a collaz conjecture based automated approach for schizophrenia detection. The proposed technique starts with the feature generation using conjecture. The medically discriminant features are extracted using iterative neighbourhood component analysis, and finally, the classification is performed using the K-nearest neighbour classifier. Kutepov et al. [6] presented an approach for EEG-based schizophrenia detection. The approach starts with using EEG data. The technique considers Kantz, Rosenstein, and Wolf algorithms to calculate the Lyapunov exponent, which is the indicator of suffered brain regions due to mental disorders. The obtained results show that the Rosenstein based Lyapunov exponent characterizes schizophrenia more effectively.
The recent trends for EEG-based automated detection of mental disorders consider whole-brain connectivity rather than individual brain regions [7]. The motivation for the development of brain connectivity comes from the fact that the human brain can be visualized as a complex network with multiple interconnected functional areas. The interaction between different brain regions can be modelled using the connectivity graph/network called brain connectivity network (BCN) [8]. Construction of any network requires two entities, i.e., node and edge. In the context of functional BCN, the individual discrete functional region is considered a node, and the functional connectivity between different brain regions is regarded as an edge [9]. The BCN provides more insights into the organization of the brain. Brain functionality analysis using the BCN approach requires the construction of BCN for each subject and extracting features from corresponding BCN for automated network identification [10].
The application of the BCN approach to the clinical field has been supported by various studies stating that pathology in the brain is analyzed by its network architecture [11]. For example, neurodegeneration arises in statically and anatomically connected networks [12,13], and pathology accumulates in densely connected brain regions [14,15]. The cognitive sequelae of brain disorders are closely related to the connectivity network of the affected brain area [16].
Many researchers have applied BCN to investigate brain conditions for neuro-generative diseases. The brain-computer interface has employed different connectivity measures to construct the BCN. For example, Hassan et al. [17] have constructed EEG-based BCN using phaselocking value (PLV) as a connectivity measure to identify the brain network changes related to the cognitive phenotypes in Parkinson's disease. Shim et al. [18] utilized EEGbased functional BCN using PLV as a connectivity measure to identify the alterations in the brain connectivity network corresponding to schizophrenia patients. Yin et al. [19] employed EEG-based BCN constructed using mutual information (MI) as a connectivity measure. The study aims to differentiate schizophrenia patients with positive and negative syndrome.
The literature considers a particular similarity/connectivity measure to find the relationship/ connection among functionally discrete brain regions [20][21][22]. The selection of suitable connectivity is essential to precisely model brain connectivity. Hence, a proper analysis must be performed with several connectivity measures to identify the best suitable similarity measure for BCN. Along with the connectivity measure, an efficient feature extraction over constructed BCN is also essential. The study [23] suggested that ordinal connections are most suitable for features that revile structural patterns inside BCNs. Ordinal-connection-based features have not been integrated with EEG signal-based Weighted BCN (WBCN). Therefore, this study proposed an approach for selecting appropriate connectivity measures to construct the EEGbased WBCN. The proper selection of connectivity measures helps construct a more efficient connectivity network that accurately represents connectivity among different brain regions. In addition, the study developed a network characterizer for the BCN analysis called the weighted ordinal connection (WOC). The WOC helps to characterize the BCN corresponding to schizophrenia diseased and healthy subjects. Its ability to incorporate the edge weight information and the ordered relation among different weighted edges in the network makes it more effective for network characterization.
The contribution of this study is two-fold; the first aims to identify the most suitable connectivity measure, while the second is to utilize and extend the WOC-based feature extraction using multiple subgraphs with EEG signals for schizophrenia disease detection.

Materials and methodology
This section explains the process flow of the proposed methodology in Fig. 1, along with the cross-validation method. The methodology starts with data distribution into k-folds, which means the whole data is divided into k equal parts. 1 to k−1 parts of data are used for training, and kth part is used for testing purposes. This process is repeated k time, each time a different part is used as testing data, called as k-fold cross-validation method. First of all, training and testing data are used to create WBCN using different connectivity measures. After that, subgraphs are constructed from WBCN, and weighted ordinal connections are extracted. Now, from the training data, (1) discriminant weighted ordinal connections are identified and based on that, (2) a feature extraction framework is designed. After the completion of the framework design, (3) feature extraction is performed, followed by (4) classification. From the above four steps, only the third and fourth steps were executed over testing data because steps one and two are analysis phases in which identification of discriminant feature is made followed by feature extraction framework design. All four steps are explained in detail in later sections.

Dataset
The proposed approach has been validated on an EEG signal dataset containing EEG records of schizophrenia diseased and healthy subjects. The dataset is obtained from Laboratory for Neurophysiology and Neuro-Computer Interface, Moscow, Russia [24]. The dataset includes 1 min long 16-channel EEG with a 128 Hz sampling frequency. The study considered a total of 78 EEG records to validate the proposed approach, where 39 EEG records correspond to schizophrenia patients, and 39 EEG records correspond to healthy subjects. Total 16 channels used for recording EEG signals are F7 (

Weighted brain connectivity networks (WBCN)
WBCN is modelled as graph G = {V, E, W}, where V, E, and W represent a set of vertices, edges, and a weight matrix, respectively. Each value in the weight matrix W corresponds to a single edge of G. In EEG-based WBCN, a vertex represents functional brain location over the scalp from where the EEG signals are recorded. The edge between two vertices is the functional relationship between those brain regions, calculated using connectivity measures [26]. Connectivity measure is the quantitative means to find the strength of the connection/relationship between two different brain regions. Multiple measures are available in the literature, like correlation coefficient, coherence, phase-locking value (PLV), mutual information (MI), etc., to calculate the functional relationship between two brain regions [27]. The W ∈ R n×n is obtained by calculating the connectivity measure between EEG signals corresponding to all the pairs of electrodes [9]. The value at i th row and j th column of W represent the value of connectivity measure between EEG signals corresponding to i th and j th electrode, where i, j ⊆ n.
This study considered four different connectivity measures, namely, correlation coefficient [28], coherence [29], PLV [27], and MI [27], to compare and select the most suitable connectivity measure for schizophrenia. A brief description of all the connectivity measures used in this study is as follows:

Correlation coefficient
It is a time-domain connectivity measure responsible for calculating the relationship strength between two EEG signals corresponding to discrete functional brain regions. The mathematical formulation for calculating r is given in Eq. (1),  where X and Y represent the EEG signals. N represents the length of the EEG signal. X(i) and Y(i) represents the i th sample of the corresponding signals. X and Y represents the mean of signal X and Y , respectively.

Coherence
It is the most used frequency-domain measure to calculate the similarity between two signals. The value of coherence between two signals ranges from 0 to 1. The value 0 represents two signals that are not similar, whereas value 1 represents the complete similarity between two signals [30]. The mathematical formula for calculating coherence is given in where, S XY (f ) represents the cross-spectral density of signal X and Y . S XX (f ) and S YY (f ) depicts the auto spectral density of signal X and Y, respectively. The spectral density of the signals is estimated using the fast Fourier transform (FFT) inbuilt function in Matlab. The parameters for the calculation of FFT include the input signal, the transform length value ' n ', and the dimension value ' dim ', which represents the value (1 or 2) to represent the dimension to operate along. For this study, the value of ' n ' is considered as a default value which is equal to the length of input signal and the value for 'dim' is considered as 2 to operate along row, which calculates the FFT for each signal.

Phase locking value
It is a connectivity measure that calculates the phase synchronization between two brain regions. The value of PLV ranges between 0 and 1, where a value near 1 represents the exact phase-locking, and a value near 0 represents random phase distribution [27,31]. The mathematical formula for calculating PLV is given in Eq.
where the Δ�(t) represents the difference in phase between signal X and Y at time instant t, which is calculated using Eq. (4). (1) Here, X an , Y an represents the analytical signals obtained by applying Hilbert transform on signals X and Y, respectively.

Mutual information
It is a nonlinear, information-based measure used to evaluate the information between two signals [27]. The MI identifies the mutual dependence of two signals [32]. The mutual information can be calculated between signals X and Y using mathematical formula given in Eq. (5), where N represents the length of the signal X and Y ; P XY is the joint probability density function between signal X and Y ; P X and P Y represents the marginal probability density function of X and Y, respectively [33].
The MI is estimated using the Kullback-Leibler divergence D KL with the input parameter P X , P Y , and P XY [33]. Figure 2 represents the plot of channels 1 and 2 corresponding to healthy subjects 1 and diseased subjects 1. Table 1 lists the values of different connectivity measures  between EEG signals from channel 1 and channel 2 of the healthy subject 1 and diseased subject 1. From Fig. 2 and Table 1, we can observe that even though channel 1 and channel 2 represent overlapping structures for diseased and healthy subjects, the level of similarity between those channels can be quantified using similarity measures. Figure 3 presents an example of the WBCN connectivity constructed using the correlation coefficient as a similarity measure, corresponding to healthy subject 1. The edges in Fig. 3 are coloured blue, red, and green to depict different weight value ranges associated with edges. The blue-coloured edges have correlation values of less than 60%; the green-coloured edges have a correlation value between 61 to 80%, and the red-coloured edges have a value of correlation greater than 80%. Figure 3 comprises 16 nodes corresponding to 16 channels of EEG data corresponding to the healthy subject 1, out of which 13 are visible, and three are hidden in the background due to the 3D view of WBCN.
The concept of WOC-based feature extraction is motivated by the subgraph mining technique [23]. However, one demerit of considering a subgraph for WOC extraction is the neglect of a few connections that might have provided helpful information. This study reduced this problem by obtaining four different subgraphs from each WBCN using four subgraph extraction algorithms including depth-first search (DFS), breadth first search (BFS), Dijkstra's shortest path (DSP) and minimum spanning tree (MST) algorithm, covering almost all links for WOC-based feature extraction. Finally, the WOC-based features obtained corresponding to subgraph DFS, BFS, DSP and MST are merged in all the possible combinations to collaborate the possible knowledge for disease diagnosis. Total 15 combinations of features (F1 to F15) are generated and used for classification, as explained below in "Feature extraction" section.
WOC extraction is an essential step of the proposed methodology, which considers both the weight information and ordinal connection information for feature extraction. The discriminate WOC represents the connectivity patterns that differentiate the WBCN for diseased and healthy subjects. With the help of discriminate WOC, the framework for feature extraction is constructed. Finally, Feature extraction and classification is performed for disease diagnosis and the selection of appropriate connectivity measure.
Four subgraphs were obtained from BCN for ordinal patterns extraction using well-known algorithms as described below: -

Depth-first search (DFS)
It is an algorithm for exploring or traversing trees. The traversal starts at the root or any other arbitrary node, marks the node as traversed, go on to the next unmarked node as far as possible until there are no more unmarked nearby nodes. Then go back and look for more unmarked nodes and visit them [34].

Breadth-first search (BFS)
It is another algorithm for exploring or traversing trees. Here, all the nodes at the same level are traversed before moving on to the next level. This cycle is repeated until no unvisited node is left [35].

Dijkstra's shortest path (DSP)
The algorithm is used to find the shortest path/minimum cost tree. The shortest path tree is the tree having the minimum distance from the source node to all the other nodes [36].

Minimum spanning tree (MST)
It is an algorithm for constructing a tree from the graph that connects all the nodes of the graph using a minimum number of edges with the smallest weight value [37].
The source node must be selected first for the construction of DFS, BFS, and DSP. Here, a weighted centralitybased measure is used to determine the root node; instead of randomly selecting a root node [38]. The weighted centrality for node i of the network is calculated using Eq. (6).
Here, S i is the sum of the weights of the edges connected to the i th node; K i is the number of nodes connected to the i th node. The tuning parameter has a value between 0 and 1. The large value of gives maximum importance to the weight of the connected nodes, and the small value of gives more importance to the connectedness of the node. As the study considers weighted BCN, all the nodes will be connected to each other with different weight values. Hence, this study considers the large value of to rely more on the weight value than the number of connected nodes for the Fig. 3 Weighted BCN constructed using a correlation coefficient calculation of the weighted centrality of a node. The value of is 0.8 considered in this study.
In general, the construction of DSP and MST requires the selection of edges with minimum cost value so that the cost of traversing all the nodes remains minimum. In the case of WBCN, the edges with maximum weight need to be retained while ignoring the edges with minimum weight. The edges with maximum connectivity strength represent a strong relationship between two different brain regions. Hence, to ensure the selection of the maximum weighted edge, the cost matrix is constructed using the connectivity strength of the edges using Eq. (7), The costmat(i, j) represents the cost of travelling between node i and j . W i,j is the connectivity strength between node i and j.
All the respective subgraphs corresponding to diseased and healthy subjects might represent different topological structures and different connectivity strengths for some brain regions.

Concept of WOC generation
Here, the WOC represents the sequence of two interconnected edges having an ordinal relationship. For example, suppose a weighted graph G , containing interconnected edges e a−b and e b−c such that weight e a−b > weight e b−c then the sequence WC 1 = {e a−b , e b−c } is considered a WOC for G . The same procedure is followed for all the interconnected edges starting from the root node edge to the leaf node edge.
The WOC extraction process is illustrated in Fig. 4. Where the weighted graph G composed of seven nodes and six edges. The WOC extraction starts from the root node and progresses towards the leaf node. If the sequence of connected edges follows the relationship mentioned in the above paragraph, then those sequences of edges are considered as WOC. In Fig. 4, the edges e 1−2 and e 2−4 are related in the ordinal manner i.e. weight e 1−2 > weight e 2−4 , therefore the WOC WC 1 = {e 1,2 , e 2,4 } is constructed. Similarly, edges e 1−2 , e 2−5 and edges e 2−5 and e 5−7 follows the same relationship. Hence, from graph G in Fig. 4, three WOCs are constructed. The pseudocode for the WOC extraction is presented in Algorithm 1.
Multiple WOCs can be extracted from each of the subgraphs (DFS, BFS, DSP, MST) constructed from the WBCN. The multiple WOCs extracted from a subgraph corresponding to train data to characterize the network by preserving its local topological property. The WOCs obtained from the training data are further processed to identify the most discriminative WOCs (explained below) which are used to construct a framework for feature extraction. The process from WOC extraction to the construction of framework for feature extraction analyze the topological characteristics of the networks from different category and based on this information the feature vector is constructed.

Discriminative WOC identification
The WBCN is considered to be accurately modelled using a diffusion model of communication [8,39]. According to the diffusion process, the flow of information spreads along multiple paths, being biased by edge weights. There may be an ordinal relation in the path followed by information [40]. Therefore the WOC is used to model these possible paths based on the ordinal relation between them. Which ultimately creates the connection topology with the collection of all the extracted WOCs.
The discriminative WOC represents the topology of WBCN activation for one class which is different from that of the other class. For example, in the presented study, the discriminative WOCs for the diseased class means the set of links active in the WBCN corresponding to the diseased subject and inactive in the WBCN corresponding to the healthy subject. Similarly, the discriminative WOC for the healthy class depicts the set of active links in the WBCN corresponding to healthy subjects and inactive in the WBCN corresponding to diseased subjects.
The discriminative WOCs for each class are extracted from the most frequent WOCs of the corresponding class. The Frequent WOCs for a particular class represent the set of ordinal connections present in the maximum number of WBCNs corresponding to the subjects belonging to that class. Hence, it is necessary to identify frequent WOCs corresponding to each belonging to that class. The frequent WOCs for a particular class ∈ {Diseased, Healthy} is obtained by calculating the frequency ratio (FqRt) for each WOC obtained from WBCN corresponding to subjects belonging to that class. The mathematical expression for calculating the frequency ratio for a WOC belonging to class is given in Eq. (8).
Here, D is the set of WBCN corresponding to the subjects belonging to class . woc p is WOC belongs to the set of WOC extracted from all the WBCNs in D . H is the total number of subjects from group ; ∝ h is equal to 1 if woc p is present in the WBCN corresponding to h th subject from group D , otherwise 0. Arrange the WOCs belonging to the class in descending order of frequency ratio value and select the top T number of WOCs as frequent WOC. There is no standard approach available to identify the optimal value for T ; it is decided using a trial-and-error approach.
The discriminative WOCs for each class are extracted by calculating the ratio score ( R t S c ) for each frequent WOC belonging to the same class. The mathematical expression for calculating the ratio score is given in Eq. (9).
Here, in , ot ∈ {Diseased, Healthy} , represents the class of interest and other class, respectively; ('in' is an abbreviation for interest and 'ot' is an abbreviation for other).
In any instance, if the class of interest is Healthy, then the other class will be Diseased and vice versa. D in represents the collection of graphs belonging to the class in . H in is the count of graphs belonging to the in category and. H ot is the count of graphs belonging to the ot category. The value of Φ h1, int = 1 if wopc is present in the h1 th graph belonging to class int otherwise Φ h1, int = 0. Similarly, the value of β h2, ot = 1 if OP is present in the h2 th graph belonging to class ot otherwise β h2, ot = 0. represents a small value to avoid the denominator being zero. The ratio score value represents the discriminatory power of WOC. The top K WOCs with maximum ratio score values are considered the most discriminative WOC for each class. The value for K is decided based on a trial-and-error approach.

Framework for WOC-based feature extraction
The discriminative WOCs corresponding to each class is used to construct the framework for feature extraction. The construction of the framework for feature extraction is based on the assumption that the WBCN corresponding to schizophrenia disease shows a different connectivity pattern than the WBCN for healthy subjects. The most discriminative WOCs are extracted from diseased and healthy classes are arranged in one row to construct a framework fm ∈ R 1×2K for feature extraction. The WOC from 1 to K represents discriminative WOC corresponding to the disease class, and WOC from K + 1 to 2K represents discriminative WOCs from the healthy class.

Feature extraction
The feature matrix fmat ∈ R 78×2K is constructed for this study using the framework obtained in the previous step.
The value 78 represents this study's total number of subjects, i.e., 39 belong to the schizophrenia disease class, and 39 belong to healthy subjects. In fmat the first 39 rows represent features for the schizophrenia diseased subjects, and the next 39 rows represent features for healthy subjects. The WOC obtained from each WBCN is compared with the discriminative WOCs of the framework such that if the WOC from i th WBCN is matched with discriminative WOC at j th column then fmat(i, j) = 1 , else fmat(i, j) = 0 . In this way, all the WOCs from each WBCN are compared with the framework's discriminative WOCs to construct the final feature matrix. The structure of the final feature matrix is shown in Fig. 5.
If the feature matrix is structurally divided into four quadrants, it should contain the following properties for maximum performance.
(1) Quadrants II and IV must have a dense value of 1's.

Disease detection/classification
The feature matrices extracted correspond to DFS, BFS, DSP, and MST subgraphs, and all the possible combinations of these features are classified using various classification algorithms.

1-Dimensional convolutional neural network (1DCNN)
A convolutional neural network (CNN) represents a deep learning system primarily used for image or video datasets [41]. This study implements the 1D CNN for the classification of features extracted in the previous section. The working of 2-D CNN and 1-D CNN is similar; the only difference lies in the dimension of the applied input. The 1-DCNN works on the elements of the lower dimension [42]. The structure of the proposed 1D-CNN model for WOC-based feature classification is represented in Fig. 6.
The structure of the 1D CNN model includes two 1D convolution layers with rectified linear unit (ReLU) activation function followed by a MaxPooling layer. After passing through the pooling layer, features are flattened to a one-dimensional vector and passed through two dense layers with.the ReLU function. The dropout layer with a 0.5 value is applied before the second dense layer. Finally, the output layer with the SoftMax function provides the class label's probability for the input test sample [43]. The class label with maximum probability is considered a predicted class for the input sample. In addition to the 1DCNN, this study also applied traditional machine learning algorithms as described below.
In addition to the deep learning model, the study also employed traditional classification algorithms like K-nearest neighbours [44], support vector machine with linear, radial basis function and polynomial kernels [45,46] and the random forest classifier [47]. The classification performance of the deep learning model is compared with the performance of the traditional classifier.
The performance of each classification algorithm over all the combinations of features is evaluated using performance parameters like accuracy, precision, recall and Cohen's kappa [53]. The description and mathematical formulation of the performance parameters are given in Table 2.

Results
The section provides the quantitative and statistical comparison of performance achieved corresponding to different connectivity measures. The classification performance of the ordinal pattern-based features from various similarity measures is compared using ROC curves for the quantitative analysis. The ROC curve is the plot between the truepositive rate (TPR) and the false-positive rate (FPR) at different threshold values. TPR is nothing but the Recall value presented in Table 2, and the FPR is calculated using Eq. (10). The curve line near the top-left corner represents more accurate classification performance, and the curve near the diagonal line represents the least accurate classification performance [54].   Table 2. AUC provides an effective way for the analysis of accurate predictions. The value of AUC ranges from 0 to 1. The value 0 represents inaccurate predictions, and value 1 is for accurate prediction. The value 0.5 of the AUC means that the ROC curve will fall on the diagonal. The ROC plot and AUC values have been used to show the performance of features [55].
After comparison, the performance from the correlation coefficient and coherence connectivity measure found to be better than that from the PLV and MI. Hence, the ROC plot corresponding to the correlation coefficient and coherence are shown and explained.
The X-axis of the ROC plots represent the FPR value, and the Y-axis depicts the TPR value in Figs. 5 and 6. In the end, Table A1 provided in the supplementary materials is listed with AUC values corresponding to each of the connectivity measures used.
From Fig. 7(a)-(f), it is observed that for WOC-based features obtained from WBCN constructed using correlation coefficient, the KNN, SVM(Linear), SVM(RBF), SVM(Polynomial), random forest, and CNN1D classifiers achieve maximum performance using the feature F4, F7, F9, F12, F7, and F12 respectively. Figure 8(a)-(f) show the ROC plot for the eightfold classification performance for all WOC features obtained from coherence-based WBCN. From these figures, it is observed that the classification algorithms, i.e., KNN, SVM (Linear), SVM (RBF), SVM (Polynomial), random forest, and CNN1D, achieve maximum performance using the features F9, F15, F12, F4, F14, and F4, respectively. Table A2 is listed with the average performance of each WOC-based feature extracted using different connectivity measures. Table A2 shows that for the features extracted from the WBCN constructed based on correlation (10) FPR = FP TN + FP coefficient, the F9 features to achieve maximum average performance with 83% accuracy. For the coherence-based WBCN, maximum average performance can be achieved using F9 features with 85% average accuracy. For PLVbased WBCN, the F7 feature achieves maximum average performance with 79% average accuracy. Finally, among the features extracted from MI-based WBCN, the maximum average performance is achieved using F12 and F15 features with 62% average accuracy.
Further, the statistical analysis using the Friedman test is performed with the help of KEEL (Knowledge Extraction based on Evolutionary Learning) software [56] to validate the selection of the appropriate connectivity measure. The Friedman test is performed on the AUC values obtained after plotting the ROC curve. The Friedman test provides the lowest rank to the best-performing algorithm [57]. Table 3 presents the average ranking obtained using the Friedman test for each connectivity measure. The bold value in table shows that the lowest rank is assigned to coherence; hence, coherence is considered the most suitable connectivity measure for schizophrenia disease detection using EEGbased WBCN [58].
The study further performed the structural analysis of the proposed approach. The structural analysis aims to differentiate the connectivity pattern for healthy and diseased subjects. The structural analysis is performed based on the most discriminative WOCs, representing the connectivity patterns that are not common in two different categories of data. Figure 9 represents the most discriminative WOCs corresponding to the different subgraphs (i.e., DFS, BFS, DSP and MST) of a WBCN constructed for diseased and healthy subjects using coherence connectivity measure (as it is the most suitable measure). Figure 9(a) and (b) represent the most discriminative patterns from DFS subgraph corresponding to diseased and healthy subjects, respectively. It shows that the diseased subject is missing connectivity at the frontal region of the brain. Figure 9(c) and (d) present It is the measure of the predictability of the classifier. It is used to identify out of total input samples how many samples are correctly classified [

TP TP+FP
It is also called a positive predicted value. It is used to identify that out of the total samples predicted as a positive class, how many belong to the positive class [

TP TP+FN
It is also called a true positive rate or sensitivity. It is used to identify the positive class predictability of the classifier, i.e., out of positive class samples, how many are correctly classified [50] Cohen's kappa OA−EA 1−EA It is one of the essential performance parameters used to identify how much your model is better than the random classifier that predicts based on the frequency of class [51,52]     the most discriminative patterns from BFS subgraph corresponding to the diseased and healthy subjects, respectively. From these figures, it can be observed that the patterns in the diseased subject show maximum connectivity at left brain region than that of the healthy subject and the midline parietal and left brain region of the healthy subject show maximum connectivity as compared to the diseased subject. The most discriminative patterns from the DSP subgraph corresponding to the diseased and healthy subjects are presented in Fig. 9(e) and (f), respectively. Figures show that the diseased subject shows maximum connectivity at the central left and parietal brain regions, whereas the healthy subject contains maximum connectivity in the midline central and parietal region than the diseased subject. Figure 9(g) and (h) represent the most discriminative patterns from MST subgraph corresponding to the diseased and healthy subjects, respectively. The patterns of the diseased subject shows maximum connectivity at the left temporal-parietal and occipital region of the brain, whereas the patterns corresponding to healthy subject shows maximum connectivity at the left brain region as compared to that of the diseased subject.

Discussion
This study presents an automated technique for schizophrenia disease detection using EEG-based BCN. BCN is an advanced approach for modelling the brain as a graph to analyze its functional behaviour in different conditions. The modelling of the human brain as a graph helps to visually inspect the changes in connectivity between discrete brain regions while performing any physical or mental task. The key objectives of this study are: (i) Selection of suitable connectivity measures to construct a BCN for schizophrenia detection. (ii) Development of an advanced network identifier to characterize the BCN corresponding to the diseased and healthy subject.
This study explores four categories of connectivity measures and selects the best one to construct the most informative BCN for schizophrenia detection. Here, connectivity measures considered are correlation coefficient, coherence, PLV, and MI. The correlation coefficient is a The BCN constructed with different connectivity measures is analyzed using an advanced network identifier proposed as WOC. The WOC utilizes the strength of connections and the ordinal relation information to characterize the BCN. Each BCN is divided into four subgraphs, i.e., DFS, BFS, DSP, and MST, before WOC extraction to reduce the complexity of the WOC extraction process. The WOCs are extracted corresponding to each subgraph separately to obtain four sets of feature vectors. These feature vectors are then classified independently and in combinations to collaborate their abilities to enhance the classification performance. In this way, total 15 feature vectors are obtained from each BCN, where 4 feature vectors correspond to the DFS, BFS, DSP, and MST subgraphs, and the remaining 11 are the combinations of these four feature vectors as mentioned in the "Feature extraction" section.
The extracted features are classified using various classification algorithms like KNN, SVM, RF, and CNN. From the quantitative and statistical analysis of the classification performance presented in the "Results" section, coherence is inferred to be the most suitable connectivity measure for EEG-based schizophrenia detection. The study performed the structural analysis of the performed in the proposed approach.
The classification performance achieved using the proposed approach is compared with other state-of-the-art studies. Table 4 presents a comparative analysis of the proposed study with the existing studies on schizophrenia disease detection. From the table, it is observed that the proposed approach outperforms the existing studies. The important reason behind the enhanced performance of the proposed approach is the ability of the network characterizer to utilize the weight and ordinal relation information of the links to characterize the network.
This work is an advancement in the research toward identifying the most suitable connectivity measure to construct an efficient BCN for schizophrenia detection and developing an advanced network identifier for its characterization.

Conclusion
The proposed study shows the importance of connectivity measures while constructing WBCN. The results show that coherence is the best-suited connectivity measure for constructing weighted BCN to predict schizophrenia disease through EEG signals. The proposed approach utilizes both the weight and ordinal relationship information among the edges of the WBCN constructed using multiple similarity measures. Several classification algorithms classify the extracted features corresponding to diseased and healthy subjects. The average performance of each WOC-based feature obtained from BCN constructed using different similarity measures is also estimated and compared. Combining WOC-based features extracted from DFS and MST subgraphs with coherence as a similarity measure achieves maximum average performance (85% accuracy) among all the features and connectivity measures. It is also observed that WBCN constructed using coherence as a similarity measure is more informative than others.
In addition to the statistical analysis of classification results, the structural analysis of the weighted BCN is also performed. The most discriminative WOCs from MST subgraph of coherence connectivity-based weighted BCN are analyzed. The WOCs of the diseased subjects show maximum connectivity at the left temporal-parietal and occipital regions of the brain. In comparison, the WOCs of healthy subjects show maximum connectivity in the right hemisphere of the brain.
Acknowledgements The authors gratefully acknowledge researchers at Laboratory for Neurophysiology and Neuro-Computer Interfaces for providing public access to the EEG records database used in this study.
Author contributions All authors contributed to the study conception and design. Coding and analysis were performed by Mangesh Ramaji