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An example connected graph, with 6 vertices. Partitioning into two connected graphs. In multivariate statistics, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is provided as an input and ...
Spectral graph theory emerged in the 1950s and 1960s. Besides graph theoretic research on the relationship between structural and spectral properties of graphs, another major source was research in quantum chemistry, but the connections between these two lines of work were not discovered until much later. [15]
Global approaches rely on properties of the entire graph and do not rely on an arbitrary initial partition. The most common example is spectral partitioning, where a partition is derived from approximate eigenvectors of the adjacency matrix, or spectral clustering that groups graph vertices using the eigendecomposition of the graph Laplacian ...
The Laplacian matrix of a directed graph is by definition generally non-symmetric, while, e.g., traditional spectral clustering is primarily developed for undirected graphs with symmetric adjacency and Laplacian matrices. A trivial approach to applying techniques requiring the symmetry is to turn the original directed graph into an undirected ...
Representative hypergraph learning techniques include hypergraph spectral clustering that extends the spectral graph theory with hypergraph Laplacian, [14] and hypergraph semi-supervised learning that introduces extra hypergraph structural cost to restrict the learning results. [15]
Conductance also helps measure the quality of a Spectral clustering. The maximum among the conductance of clusters provides a bound which can be used, along with inter-cluster edge weight, to define a measure on the quality of clustering. Intuitively, the conductance of a cluster (which can be seen as a set of vertices in a graph) should be low.
There are a couple of software tools available that are able to compute clusterings in graphs with good modularity. Original implementation of the multi-level Louvain method. [14] The Leiden algorithm which additionally avoids unconnected communities. [15] The Vienna Graph Clustering (VieClus) algorithm, a parallel memetic algorithm. [16]
Being able to identify these sub-structures within a network can provide insight into how network function and topology affect each other. Such insight can be useful in improving some algorithms on graphs such as spectral clustering. [8] Importantly, communities often have very different properties than the average properties of the networks.