This study utilized resting-state functional magnetic resonance imaging (rs-fMRI) data from the Autism Brain Imaging Data Exchange (ABIDE) database, which includes 539 individuals with ASD and 573 normal control (NC) subjects from 17 international imaging centers (Di Martino et al., 2014). The detailed scan procedures and protocols are described on the ABIDE website¹. Considering that several sites contain only a limited number of participants, we use data from 5 different sites, including NYU, Leuven, UCLA, UM and USM. Specifically, rs-fMRI scan data from 45 individuals with ASD and 47 typically developing controls were selected from the NUY site. Detailed demographic information is summarized in Table 1.

The data acquisition and preprocessing in this study follow a standard pipeline that include head movement correction, normalization, denoising, and other related processes and parameters, similar to previous literature (Satterthwaite et al., 2013; Yan et al., 2013; Washington et al., 2014; Lin et al., 2015; Ray et al., 2015; Urbain et al., 2016; Reinhart and Nguyen, 2019). Subsequently, the brain was segmented into 116 ROIs using the automatic anatomical labeling (AAL) map, and the mean value of the rs-fMRI time series for each ROI was calculated, generating the data matrix X ∈ R 170 × 116 for further analysis. It is important to note that the data matrix consists of 170 time points and 116 brain ROIs.

For the l -th subject, we use its corresponding low-order FCN matrices C _LON and H-C _HON as raw features. Since the C _LON and H-C _HON matrices are symmetric, including duplicate features would result in redundancy. Therefore, we only vectorize their lower off-diagonal triangular part to define the feature vectors, i.e., y 0 ( l ) , y 1 ( l ) , for representing the l -th subject’s C _LON and H-C _HON , respectively. The dimensionality of y i ( l ) is M ( M − 1 ) 2 , where M denotes the number of ROIs for y 0 ( l ) , and the number of hyperedges for y 1 ( l ) .

The feature vectors y 0 ( l ) , y 1 ( l ) extracted from C _LON and H-C _HON might include redundant or irrelevant features for ASD diagnosis. Thus, feature selection is necessary. For selecting a small subset of features most relevant to the pathology of ASD, we use the two-stage feature selection strategy. Step 1: Performing a two-sample t-test between normal controls (NCs) and ASD subjects for each feature in y 0 ( l ) , y 1 ( l ) . Those features whose p-values are smaller than a certain threshold are retained. At this point, we label the newly obtained feature set y ′ 0 ( l ) , y ′ 1 ( l ) . Step 2: Adopting L 1 -norm regularized least squares regression, known as LASSO (Tibshirani, 1996), due to its efficiency and simplicity (Jin et al., 2015; Jie et al., 2016; Wee et al., 2016). Specifically, let θ j = ( w i 1 , w i z , ⋯ , w i d ) T represent the weight vector for the feature selection task and K = ( k 1 , k 2 , … , k N ) T are the class labels of N training data (from N training subjects). Here, d is the number of features. Mathematically, the LASSO model can be described as follows:(8) 1 2 ∑ i = 1 N   k l − y i l T ω i 2 2 + λ ω i 1 where λ is a parameter for controlling the strength of L 1 norm regularization. The first term in Equation (8) is the empirical loss on the training data, and the second term is the L 1 norm regularization term that is used to enforce some elements of ω 1 to be zero (i.e., corresponding to non-discriminative features in our classification task). In this way, we can jointly achieve classification error minimization and sparse feature selection. Let y ˜ 0 ( i ) , y ˜ 1 ( i ) denote selected features from the original feature vectors in the first stage y ′ 0 ( l ) , y ′ 1 ( l ) .

After selecting the most important features by LASSO, we use SVM with a linear kernel for ASD classification (Cortes and Vapnik, 1995). SVM aims to find a hyperplane with the maximum margin to effectively separate the samples of one class from another.

To assess the performance of the H-C_HON, we train classifiers for ASD diagnosis using the conventional C_LON, H-C_HON and their fusion (FUSION), respectively. It is worth noting that FUSION is fused by a linear combination of C_LON and H-C_HON. Specifically, the two SVMs are trained using C_LON and H-C_HON, and the output of each SVM is used as the classification result. And then the final classification results are obtained by fusing the decision scores of all SVMs. Since the combination coefficients are difficult to determine in practical applications, they are simply fused by 0.5 × (C_LON + H-C_HON) in this paper.

In this experiment, we adopt a nested tenfold cross-validation strategy consisting of two nested loops to evaluate the classification performance. The outer loop involves dividing the 92 subjects into 10 subsets of comparable size, where one subset is designated as the test set, and the other nine subsets are used as the training set. In the inner loop, the train set is combined and redistributed into 10 subsets of similar size, with nine subsets used for tuning the hyperparameters and one for model evaluation. Our method’s performance is primarily affected by three hyperparameters: p and λ in feature selection and γ in the SVM model. The optimal hyperparameters are determined when the average classification accuracy reaches its maximum. We determine the optimal values for the parameters in the following ranges: p ∈ [ 0.01 : 0.01 : 0.1 ] , λ ∈ [ 0.1 : 0.1 : 0.9 ] and γ ∈ [ 2 − 4 , ⋯ , 2 4 ] . Once the optimal hyperparameters are selected in the inner loop, they are returned to the outer loop where the model is trained on the training dataset and evaluated on the test set.

In our experiments, we adopt six metrics to evaluate different FC construction methods: classification accuracy (ACC), sensitivity or true positive rate (TPR), specificity or true negative rate (TNR), positive predictive value (PPV), negative predictive value (NPV), and F1 score. Denote TP, TN, FP and FN as True Positive, True Negative, False Positive, and False Negative, respectively. Those evaluation metrics can be defined as follows: ACC = (TP + TN)/(TP + TN + FP + FN), SEN = TP/(TP + FN), SPE = TN/(TN + FP), BAC = (SEN + SPE)/2, PPV = TP/(TP + FP), and NPV = TN/(TN + FN). For these metrics, higher values indicate better classification performance. In addition, we performed the statistical significance test (t-test) on the accuracy obtained by three comparison methods and FUSION, and the value of ps of the test are also listed in Table 2. When the value of p is less than 0.05, it indicates that there is a significant difference between the two methods.

To demonstrate the robustness of the test results, we conducted experiments on the real multi-site ASD dataset with five imaging sites (NYU, Leuven, UCLA, UM, and USM). The experimental results are shown in Table 2. The results from each site consistently indicate that the proposed high-order FCN (H-C_HON) outperforms compared to the two baseline methods, C_LON and C_HON. The best performance is highlighted in bold. It is worth noting that for the experiments conducted on C_LON and C_HON, no free parameters were involved. For the proposed method, we set the threshold to 0.7, which yielded the best performance in ASD identification.

Based on the experimental results shown in Table 2, we can draw the following conclusions: (1) The proposed high-order FCN has better performance than the conventional high-order FCN, indicating that constructing a high-order FCN using the idea of hypergraphs likely reduces redundant information and related noise, thereby improving the accuracy of brain analysis. Additionally, setting a threshold allows for the elimination of weak connections between brain regions, which helps improve computational efficiency and reduce complexity. (2) The fusion of H-C_HON and C_LON is better than any single FCN, suggesting that different levels of brain networks contain distinct features. Feature fusion potentially enables the integration of complementary information, enhancing the comprehensiveness of discriminative features and facilitating the identification of brain disorders such as ASD.

In general, the selection of free parameters in FCN construction methods plays a crucial role in determining the final classification performance. In the proposed method, we investigate the influence of the threshold t that constitutes the set of vertices of the hyperedges on the performance of C_HON using data from the NUY site.

To evaluate the sensitivity of our method to t , we repeat the identification experiments based on threshold steps [0.1:0.05:0.95], and discuss the effect of different thresholds on the final classification performance. Table 3 reports the individual evaluation metrics for different thresholds, and the best results are shown in bold. From Table 3, we find that the choice of threshold is crucial to the classification performance, and different thresholds determine different network topologies, which can provide different useful information for ASD identification and obtain different classification performance. We observe that the proposed method exhibits the highest performance when the threshold is set to 0.7 across all evaluation metrics.

From Table 3, we can conclude two conclusions: (1) When the threshold is less than the optimal threshold 0.7, the H-C_HON lacks specific expressiveness and loses much relevant information, which is not conducive to the auxiliary diagnosis of ASD and other brain disorders. (2) When the threshold is larger than the optimal threshold of 0.7, it represents that most of the connections among brain regions are preserved. Whereas, the lower diagnostic accuracy of brain diseases suggests that the information representation of the H-C_HON may be redundant and contain a large amount of non-essential noise-related information. Therefore, choosing an appropriate threshold setting is crucial to improve the performance of the proposed higher-order FCN.

To identify the most discriminative features in the H-C_HON, we utilized two-sample t-test and LASSO. In this study, we quantify the correlation between features and target classification by using the frequency of feature selection across all cross-validation cases.

The H-C_HON with the highest frequency in tenfold cross-validation was selected as the most discriminative connection. The reported results were based on the original Automated Anatomical Labeling (AAL) atlas, which comprises 116 brain regions (Tzourio-Mazoyer et al., 2002). In this section, we only analyze the H-C_HON with the best classification accuracy threshold at 0.7. The five most discriminative hyperedge connections identified were hyperedge e 1 and e 18 , hyperedge e 1 and e 48 , hyperedge e 1 and e 56 , hyperedge e 1 and e 94 , hyperedge e 2 and hyperedge e 67 . Since one hyperedge represents multiple connections for two groups of ROIs, we trace back to the connections between the two groups of ROIs based on the hyperedge connections.

Figure 4 shows the most discriminative hyperedge connections in five sets of hyperedge connections, which were the hyperedges connected by hyperedge e 1 and hyperedge e 94 . Figures 4A,B respectively represent the ROIs connected by hyperedge e 1 and hyperedge e 94 in ASD and NC, where the areas surrounded by blue dashed lines and purple dashed lines are the ROIs connected by hyperedge e 1 and hyperedge e 94 respectively. As shown in Figure 4, there are obvious differences in the ROIs connected by hyperedges in the most discriminative hyperedge connections of ASD and NC.

Hyperedge e 1 connects several brain regions centered around left Precentral gyrus (PreCG.L). In NC, e 1 connects to PreCG. L, right Precentral gyrus (PreCG.R), left Postcentral gyrus (PoCG.L), and left Inferior parietal (IPL.L). In ASD, e 1 connects to PreCG. L, PreCG.R, PoCG. L, IPL.L, left supplementary motor area (SMA.L), left Inferior frontal gyrus (opercular) (IFGoperc.L) and left Inferior frontal gyrus (triangular) (IFGtriang.L). By comparing e 1 in NC and ASD, it is suggested that there may be abnormal connections between PreCG. L and SMA. L, PreCG.L and IFGoperc. L, as well as PreCG. L and IFGtriang.L in individuals with ASD. Nebel et al. (2014) found that the strength of connectivity within and between distinct functional subregions of the PreCG was related to ASD diagnosis and to the severity of ASD traits. Zhang et al. (2020) found differences in convergence in the SMA.L when comparing the NC sample with the ASD sample. IFGoperc.L and IFGtriang.L belong to the Inferior frontal gyrus. Rudie et al. (2012) indicate that the inferior frontal gyrus, especially its posterior portion, has an important role in imitation and social reciprocity and in the pathophysiology of their disturbance in ASD. Hyperedge e 2 connects several brain regions centered around right Cerebellar Crus 2 (CRBL Crus 2.R). In NC, e 2 connects to right CRBL Crus 2, left Cerebellar3 and right Cerebellar3. In ASD, e 2 connects to CRBL Crus 2.R, right Cerebellar8 and right Cerebellar7b. It can be observed that there are abnormalities in the Cerebellar of between NC and ASD. Kelly et al. (2021) that cerebellar dysfunction is increasingly associated with ASD.