matrix-based methods for graph analysis

 Here are common matrix-based methods for graph analysis.

Matrix / Method	Main idea	Typical use
Adjacency matrix (A)	(A_{ij}=1) or weight if node (i) connects to node (j)	Basic graph representation, path counting, connectivity
Degree matrix (D)	Diagonal matrix where (D_{ii}) is node degree	Used with Laplacian methods
Graph Laplacian (L=D-A)	Captures graph structure through degree minus adjacency	Clustering, graph cuts, diffusion, smoothness
Normalized Laplacian	(L_{\text{sym}}=I-D^{-1/2}AD^{-1/2})	Spectral clustering, comparing graphs with unequal degrees
Incidence matrix (B)	Node-edge matrix showing which nodes belong to which edges	Flow analysis, electrical networks, graph Laplacian construction
Transition matrix (P=D^{-1}A)	Row-normalized adjacency matrix	Random walks, Markov chains, PageRank
Powers of adjacency matrix (A^k)	Counts walks of length (k) between nodes	Path analysis, reachability, motif detection
Eigenvalue / spectral decomposition	Analyze eigenvalues and eigenvectors of (A) or (L)	Community detection, graph partitioning, centrality
Singular value decomposition	Factorize adjacency or feature matrices	Low-rank approximation, graph embeddings
Matrix factorization	Approximate graph as (A \approx UV^\top)	Link prediction, recommender systems, node embeddings
PageRank matrix method	Uses a modified random-walk transition matrix	Ranking important nodes
Katz centrality	Uses ((I-\alpha A)^{-1})	Measures influence through all walks, with decay
Eigenvector centrality	Important nodes connect to important nodes	Social-network influence analysis
Laplacian eigenmaps	Use small eigenvectors of Laplacian	Dimensionality reduction, node embedding
Spectral clustering	Cluster nodes using eigenvectors of Laplacian	Community detection
Modularity matrix	(B=A-\frac{kk^\top}{2m})	Community detection
Graph diffusion kernels	Uses functions such as (e^{-tL})	Similarity, diffusion, semi-supervised learning
Resistance distance matrix	Based on pseudoinverse of Laplacian (L^+)	Network robustness, electrical-network analogy
Graph convolution matrices	Use normalized adjacency/Laplacian in neural networks	Graph neural networks, node classification

A simple way to organize them:

Representation matrices: adjacency, degree, incidence, Laplacian.
Spectral methods: eigenvectors, graph Fourier transform, spectral clustering.
Random-walk methods: transition matrix, PageRank, diffusion.
Embedding methods: SVD, matrix factorization, Laplacian eigenmaps.
Centrality methods: eigenvector centrality, Katz centrality, PageRank.
Community methods: modularity matrix, spectral partitioning.