Matrix decomposition is well known method for finding communities in the graph. In the iPython notebook, I experimented on using Singular Value Decomposition on different types graph matrices: adjacency matrix, normalized graph Laplacian matrix and normalized adjacency matrix.
The iPython notebook contains study of how the choice of graph matrix affects on learning good node representations which are helpful in finding communities. I used the complete bipartite graph with 3 nodes in one group and 4 nodes in other group of partition for the analysis.
The results show that normalized graph Laplacian is better at capturing the node features from the graph.