Different types of graphs and their corresponding adjacency matrix representations. The first row from (A) to (D) are, respectively, directed, undirected, knowledge and weighted graph examples. The main difference between (A) and (B) is that edges are directed in (A) but undirected in (B). (C) is a knowledge graph consisting of two different types of nodes (in "brown" and "blue" colors) and two different types of edges ("teach" and "is team leader"). Graph (C) is an instance of directed and heterogeneous graph. (D) shows a weighted graph where every edge is weighted with a specific value. The second row from (E) to (H) shows the corresponding 4 × 4 adjacency matrices for graphs (A)-(D).

Different types of graphs and their corresponding adjacency matrix representations. The first row from (A) to (D) are, respectively, directed, undirected, knowledge and weighted graph examples. The main difference between (A) and (B) is that edges are directed in (A) but undirected in (B). (C) is a knowledge graph consisting of two different types of nodes (in "brown" and "blue" colors) and two different types of edges ("teach" and "is team leader"). Graph (C) is an instance of directed and heterogeneous graph. (D) shows a weighted graph where every edge is weighted with a specific value. The second row from (E) to (H) shows the corresponding 4 × 4 adjacency matrices for graphs (A)-(D).

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Graph analytics can lead to better quantitative understanding and control of complex networks, but traditional methods suffer from high computational cost and excessive memory requirements associated with the high-dimensionality and heterogeneous characteristics of industrial size networks. Graph embedding techniques can be effective in converting...

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... element in the matrix can be filled with ("1"-the column edge is one outgoing edge from the row node; "0"-the column edge is not connected with the row node; "-1"-the column edge is one incoming edge to the row node (for undirected graphs, elements with "-1" are all filled with "'1'). Figure 2 (E-H), a static graph can be also represented as an "adjacency list" or "incidence matrix". ...
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... advantages are apparently superior to the deterministic embeddings applied in the previous study [42]. Figure 12 presents quantitative results for evaluating the MDCT effects for aMCI patients. In Figure 12 (A), different colors of the "dots" (i.e., 264 brain regions in the fMRI brain network) correspond to the computed Wasserstein-2 (W2) distance values, which measures the learned low-dimensional stochastic resting state fMRI brain network embedding distance before and after cognitive intervention in the latent probabilistic space. ...
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... 12 presents quantitative results for evaluating the MDCT effects for aMCI patients. In Figure 12 (A), different colors of the "dots" (i.e., 264 brain regions in the fMRI brain network) correspond to the computed Wasserstein-2 (W2) distance values, which measures the learned low-dimensional stochastic resting state fMRI brain network embedding distance before and after cognitive intervention in the latent probabilistic space. "Links" represent the sampled brain connectivity among the detected the top 50 affected brain regions (mostly concentrated in the default mode, visual, and memory retrieval functional brain systems). ...
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... represent the sampled brain connectivity among the detected the top 50 affected brain regions (mostly concentrated in the default mode, visual, and memory retrieval functional brain systems). In Figure 12 (B) we compare a stochastic (MG2G) and a deterministic (node2vec) graph embedding method; the plot shows the top 15 most affected brain regions for all patients measured by the W2-distance, and identified with the brain systems they belong to. Figure 12 (C) shows a violin plot of the W2-distance distributions and probability densities of all 264 regions for all 12 different patients included in this aMCI study [62]. ...
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... Figure 12 (B) we compare a stochastic (MG2G) and a deterministic (node2vec) graph embedding method; the plot shows the top 15 most affected brain regions for all patients measured by the W2-distance, and identified with the brain systems they belong to. Figure 12 (C) shows a violin plot of the W2-distance distributions and probability densities of all 264 regions for all 12 different patients included in this aMCI study [62]. ...
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... intercromosome matrix based on which the interaction graph is constructed. A schematic of this workflow is shown in Figure 13 (A). Areas (genomic bins) with the same chromosome are represented by the same color nodes. Based on the constructed interaction graph, the graph embedding algorithm projects the graph into a lower-dimensional space for k- Fig. 12. Example of applying stochastic (MG2G) and deterministic (node2vec) graph embedding methods for cognitive training effects using fMRI brain networks. (A) Visualization of the top 50 affected brain regions with the corresponding brain connectivity due to multi-domain cognitive training (MDCT) detected by fMRI brain network embedding for ...

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