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Statistical and Machine Learning Approaches for Network Analysis by Subhash C. Basak, Matthias Dehmer

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6.3 A Simple Worked Example

After the fairly theoretical previous section, we aim at giving the reader a better intuition behind the WSD with a simple example. Figure 6.2 shows a small network, called G1, with seven nodes and eight links. As can be seen there are two cycles of length 3 in this network and one of length 4. We will take N = 3 in this example for convenience and without loss of generality. The random walk probabilities are labeled in Figure 6.2. For example, node 3 has a degree of 5 resulting in a probability of 1/5 th for each edge. The total probability of taking a random walk around each 3-cycle is 6 × 1/2 × 1/3 × 1/3 = 0.33, also shown in Figure.6

Figure 6.2 A simple example network G1.

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Figure 6.3 shows a 3D plot of the absolute value (for clarity) of the eigenvectors of the normalized Laplacian. The corresponding eigenvalues are shown in Table 6.1.

Figure 6.3 Eigenvectors of the simple example network.

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Table 6.1 Eigenvalues, WSD, and Dominant Nodes of Example Network.

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The eigenvectors of the normalized Laplacian can be used to form a partitioning of the nodes in a graph. In this example, nodes 4 and 5 are grouped into eigenvector 3, nodes 1, 2, and 6 into eigenvectors ...

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