## 2 Materials and Methods
### Computational Methods
#### 2.3.2 Net-Net AutoML Models  
Once the values of the Markov-Shannon entropies were obtained for the BCN nodes and ANNs, twelve Machine Learning classifiers from scikit-learn (python) were tested to find the best Net-NetAutoML classifier able to determine the optimal ANN topology to predict BNCs (co-activations between brain regions):  
* KNeighborsClassifier = KNN--k-nearest neighbors: A nonparametric classifier that assigns an unclassified sample to the same class as the nearest of k samples in the training set [50].
* LinearDiscriminantAnalysis = LDA--linear discriminant analysis [51]: A statistical supervised method that projects the input data to a lower dimension in order to maximize the scatter between classes versus the scatter within each class.
* GaussianNB = GBN--Gaussian Naive Bayes, a simple \"probabilistic classifier\" [52].
* SVC(kernel = 'rbf') = SVM_RBF--support-vector machines with nonlinear radial basis functions [53].
* LogisticRegression = LogR--Logistic regression [54] is a linear model that estimates the probability of a binary response using different factors.
* MLPClassifier = MLP--multilayer perceptron (artificial neural network) using 20 neurons in a hidden layer [55].
* DecisionTreeClassifier = DT--Decision Tree (DT) represents a set of decision rules inferred from the features as a tree of rules (the paths from root to leaf represent classification rules) [56].
* RandomForestClassifier = RF--Random Forest [57] aggregates several decision trees (parallel trees). Each tree is generated using a bootstrap sample randomly drawn from the original dataset.
* XGBClassifier = XGB--an optimized distributed gradient boosting library based on serial trees [58].
* GradientBoostingClassifier = GB--gradient boosting library [59].
* AdaBoostClassifier = Ada--is a meta-estimator that starts the fitting with a classifier based on the original dataset and then adds additional copies of the original classifier to the adjusted weights for the incorrectly classified instances [60].
* BaggingClassifier = Bagging--similar with Ada but the additional classifiers are based on subsets of the original dataset [61].  
LDA could represent the simplest model equation. Let \({}^{q}\)S(L\({}_{ij}\)) be the output variable of a Net-Net AutoML classifier used to score the ability of a given q ANN to correctly predict the link or brain region co-activation L\({}_{ij}\) between two nodes/brain regions, A\({}_{i}\)-B\({}_{j}\) (L\({}_{ij}=1\)). Equation (3) describes the general formula for the LDA model. \(k\) means Sh\({}_{k}\) codifies information for nodes placed at least at a topological distance \(k\) (from the reference node).  
\[{}^{q}S\big{(}L_{ij}\big{)}=\sum_{k=1}^{5}a_{ki}\cdot Sh_{k}(A_{i})+\sum_{k=1}^ {5}b_{ki}\cdot Sh_{k}\big{(}B_{j}\big{)}+\sum_{k=1}^{5}c_{ki}\cdot Sh_{k}\big{(} A_{i}-B_{j}\big{)}+\sum_{k=1}^{5}d_{ki}\cdot Sh_{k}(\text{ANN}) \tag{3}\]  
Therefore, 20 features were obtained for the final dataset: Sh\({}_{1}\)(A), Sh\({}_{1}\)(B), Sh\({}_{1}\)(A-B), Sh\({}_{1}\)(ANN), Sh\({}_{2}\)(A), Sh\({}_{2}\)(B), Sh\({}_{2}\)(A-B), Sh\({}_{2}\)(ANN), Sh\({}_{3}\)(A), Sh\({}_{3}\)(B), Sh\({}_{3}\)(A-B), Sh\({}_{3}\)(ANN), Sh\({}_{4}\)(A), Sh\({}_{4}\)(B), Sh\({}_{4}\)(A-B), Sh\({}_{4}\)(ANN), Sh\({}_{5}\)(A), Sh\({}_{5}\)(B), Sh\({}_{5}\)(A-B), and Sh\({}_{5}\)(ANN). In order to avoid overfitting, a 10-fold cross-validation was performed. The performance of the models was measured using the Area Under the Receiver Operating Characteristics (AUROC) [62]. The scripts and dataset are available at an open repository in GitHub from one of the authors, [PERSON] [63].