Subnetwork Mining on Functional Connectivity Network for Classification of Minimal Hepatic

Subnetwork Mining on Functional Connectivity Network for Classification of Minimal Hepatic Encephalopathy

Supplementary Resource

1. Algorithm

1.1 Frequent subgraph mining

1.1.1 Preliminary definition

This subsection will give some preliminaries which are used to derive the gSpan algorithm for frequent subgraph mining. Definition 1 (Labeled Undirected Graph) A labeled graph can be represented as , where is a set of nodes and is a set of edges where represents the edge between node . The is a label function to map and . Definition 2 (Subgraph) Given two graphs and , is the subgraph of if . is also called supergraph of . Definition 3 (Subgraph Frequency) Given a set of graphs , the frequency of a subgraph is defined as (1) Definition 4 (Frequent Subgraph) Given a set of graphs and a support parameter , where , a subgraph is a frequent subgraph if the frequency is larger than . Definition 5 (Frequent Subgraph Mining) Given a set of graphs and a support parameter , frequent subgraph mining is to find all undirected graphs that are subgraphs of at least of the input graphs. Definition 6 (Intersect-graph) Given two graphs and , the intersect-graph (denoted as ) is defined as , all the nodes in edges set form the nodes set . 1.1.2 DFS lexicographic order

The process of frequent subgraph mining is divided into growing of subgraphs and checking of frequent subgraphs. The gSpan algorithm proposed DFS lexicographic order and minimum DFS code and the problem of mining frequent subgraphs is converted to mine their corresponding minimum DFS code. Then the gSpan discovers all the frequent subgraphs without candidate generation and false positive pruning which combines the two procedures into one procedure. In this subsection, we introduce the manner by which gSpan maps each graph to a minimum DFS lexicographic order(Yan and Han 2002). gSpan first performs depth- first search strategy on graph and constructs several DFS trees which are isomorphic to each other. For each DFS tree, each node is labeled by subscripts. The node is discovered before if . Based on labeled node, a linear order is built among all the edges of graph by the following rules (assume , ): if (i) and , ; (ii) and , ; and (iii) if and . The edge sequence () is called a DFS code, denoted as For several DFS trees of graph , a DFS code set is obtained. The DFS lexicographic order is defined on the DFS code set as follows. If and , , then if either of the following two rule is satisfied: (i) , (ii) 0 Given a graph , based on DFS lexicographic order, the minimum DFS code is called minimum DFS code which is also a canonical label of Thus mining frequent subgraphs is equivalent to mining corresponding minimum DFS code. Given a DFS code , the DFS code is called child and is parent. To construct a valid DFS code, must be an edge which only grows from the vertices on the rightmost path. Based on the DFS lexicographic order, gSpan constructs the hierarchical search space of frequent subgraph which is called DFS code tree. In the DFS code tree, each node represents a DFS code. The relation between parent and child node compiles with the parent-child relation described above. The relation among siblings is consistent with the DFS lexicographic order. Through depth-first search method of the code tree, all the minimum DFS codes of frequent subgraph can be discovered. Figure S1 shows a DFS code tree where the level nodes contain DFS codes of graphs. It is worth noting that the blue nodes denotes the same subgraph with different DFS code. From Figure S1, we can see that the is on the left side which means its DFS code is larger than that of based on the DFS lexicographic order. Therefore, the branch of will be pruned since it doesn’t contain any frequent subgraph.

Figure S1. The DFS code tree In a word, gSpan defines the DFS lexicographic order on the graphs and maps each graph into a unique minimum DFS code as its canonical label which produces the hierarchical search space called a DFS code tree. The level of the tree has nodes which contain DFS codes for subgraphs which are generated by one edge expansion from the level of the tree. All subgraphs with non-minimal DFS codes are pruned so that redundant candidate generations are avoided.

1.2 Discriminative subgraphs minging

Graph kernel

Given two graphs, the basic process of the Weisfeiler-Lehman test is stated as following: If both graphs are unlabeled graphs (i.e., all vertexes have not been assigned labels), we first label each vertex with the number of edges connected to that vertex. Then, in each iteration process, we update the label of each vertex based on the original label of that node and the labels of its neighbors. Specifically, we parallel augment the label of each vertex in graph with labels of nodes connected to that vertex, and sort and compress those augmented labels into a new shorter label. The process is repeated until label sets of two graphs are different or the number of iteration reaches predefined maximum value. If the sets of new created labels are different, we can determine that those two graphs are non-isomorphism, otherwise, we cannot determine whether they are isomorphic or not. Given a pair of graphs and , let be the original set of node labels of and , and be the node label set of and at the iteration of the Weifeiler-Lehaman test of isomorphism. Let h be the number of iteration in Weisfeiler-Lehman test. Assume every is ordered to keep generality, then the Weisfeiler-Lehman subtree kernel of two graphs is defined as (Shervashidze et al. 2011) (2) where (3) and (4) Here, and are the number of occurrences of the node label in and at the -th iteration, respectively. The Figure S2 is a toy example of the process of computing graph kernel with one iteration (i.e., h=1). Here, is considered as the set of letters. The label of each vetex is augmented according to the neighboring nodes, and those augmented labels are compressed into a new shorter label (shown in Figure S2b-c). After that, we re-label all vertexes with the corresponding new label in Figure S2c. Finally, the kernel on graph and is computed according to Eq. (2). . Figure S2. A toy example of the process of computing graph kernel with for graph and . a): the original labeled graph and ; b): augmented label on graph G and H ; c): label compression; d) re-labeled graphs; e): computing the graph kernel on graph and .

2. Results

The important brain regions

This subsection will discuss the important brain regions based on the discriminative subnetworks in each fold of LOO cross-validation. Similar to the selection criteria of the most important subnetworks, the number of occurrences of each ROI is computed and the top 13 ROIs with the highest occurrences are selected. Table S1 lists these brain regions.

Table S1. Top 13 important brain regions Top 13 discriminative regions R supramarginal gyrus L inferior frontal gyrus, opercular part R inferior frontal gyrus, opercular part L inferior frontal gyrus, orbital part L lingual gyrus R rolandic operculum R superior temporal gyrus R insular R superior frontal gyrus, orbital part L gyrus rectus R olfactory cortex L and R denote Left and Right, respectively.

3. Discussion

3.1 Results with multiple thresholds combination

For each subject, we obtain multiple threshold connectivity networks corresponding to multiple thresholds which reflect the different level of topological properties of original network. According to previous researches based on brain networks (Jie et al. 2014; Zanin et al. 2012), the combination of multiple threshold networks may further improve the identification ability. Accordingly, we combine the discriminative subnetworks mined from different threshold networks, i.e., the set is denoted as where corresponds to the discriminative subnetworks mined from MHE groups which are calculated by . In the experiment, we combine the threshold networks corresponding to the thresholds in Table 3, i.e., 0.2, 0.29, 0.35, 0.41, 0.47. The experiment results are also evaluated by accuracy, sensitivity, specificity and AUC measurements. The multiple thresholds combination method achieves the accuracy of 84.42%, sensitivity of 89.47%, specificity of 79.49%, and AUC of 0.88. Figure S3 plots the results of the each thresholds and multiple thresholds combination. The better results indicate that combination strategy is beneficial to the classification task. Figure S3. The results of each thresholds and multiple thresholds combination.

3.2 Function connectivity analysis

As shown in the Figure 5, the frequent subnetworks in MHE group and NHE group are different. The frequent subnetworks in NHE group are mainly related to the function of language and communication ability. To further explore the topology difference between MHE and NHE, we compute the average weights of the edges which appear in the frequent subnetworks mined from NHE group over the MHE group and NHE group, respectively. Then we show the same subnetworks in the MHE and NHE group in Figure S4. The right column is the discriminative subnetworks mined from NHE group while the left column is subnetworks in the MHE group corresponding to right column. The thickness of edges is proportional to the average weights of edges between ROI pairs. According to Figure S4, we can observe significant decrease in connections in the MHE group compared to NHE group. These observations suggest the connectivity in these subnetworks may exists possible disruptions. In the threshold networks of MHE group, these network may disappear which explain the significant differences of the discriminative subnetworks mined from MHE and NHE groups, respectively. We also performed the two sample t-test (Longjiang Zhang et al. 2012) on the average edge weights between the subnetworks in Figure S6. The p-values between each subnetwork are 0.0120, 5.0697e-05, 9.8136e-06 and 0.0438, respectively. As shown in Figure S4, the second and third subnetworks show significant difference between MHE and NHE groups which is consistent with previous researches (Longjiang Zhang et al. 2012). Figure S4. The comparison of same subnetwork in MHE and NHE groups 3.3 The effect of thresholds

Threshold-based methods have been widely employed for classification and exploring the topological properties in functional network researches (Supekar et al. 2008; Sanz-Arigita et al. 2010). However, there is no golden standard to determine an appropriate threshold. In the experiment, we chose five thresholds (i.e., 0.2, 0.29, 0.35, 0.41, 0.47) from a range of thresholds according to their classification results and the best performance is achieved on the threshold of 0.35. In this subsection, we investigate how our proposed method changes with respect to thresholds. We use more delicate interval partition of threshold which ranges from 0.2 to 0.5 with an increment of 0.04. The classification results with different thresholds are plotted in the Figure S5. As seen from Figure S5, when the thresholds are less than 0.35, the classification accuracy increase when the thresholds increases. The reason is that small thresholds may leave more edges and generate some redundancy information for classification. While the thresholds are larger than 0.35, the classification accuracy change slightly as the redundancy information is eliminated. Besides, small thresholds may lead to long experiment time since many edges are preserved. For example, the experiment time of each fold for threshold of 0.35 is around 10m while 0.2 is 1h. Figure S5. The effect of thresholds.

3.4 The effect of the number of discriminative subnetworks

In our experiments, we fixed the number of selected discriminative subnetworks (i.e., , which means 50 discriminative subnetworks were selected from MEH group and NHE group, respectively). However, the number of discriminative subnetwork is also an important parameter which has significant impact on classification accuracy. To evaluate the effect of the number of the selected discriminative subnetworks for classification accuracy, we test the classification accuracy of our proposed method with different numbers of selected subnetworks, ranging from 40 to 200 with an increment of 20. Figure S6 presents the corresponding result of each number of discriminative subnetworks. As we can observe from Figure S8, the accuracy rate shows obvious change with the increase of number of discriminative subnetworks.

Figure S6. The effect of the number of discriminative subnetworks

Reference

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