graph shift operator

Data

subject:: Data Science Methods for Large Scale Graphs parent:: Graph Signals and Graph Signal Processing theme:: math notes

Graph Shift Opterator

We can define more general diffusion processes by defining the matrix $S\in {\mathbb{R}}^{n\times n}$ such that $S_{ij}\ne 0⟺(i,j)\in E$ (can only equal 0 along the diagonal?)

This matrix is called the graph shift operator (GSO). We way $z=Sx$ is a (graph) “shift” or diffusion of $x$ by $S$.

The most common operators are the adjacency matrix and the graph laplacian

Note that we can always recover the local implementations: $z_i={\sum }_j{S}_{ij}{x}_j={\sum }_{j\in N_i}{S}_{ij}{x}_j$

Examples of Graph Shift Operators

• adjacency matrix

• degree matrix

• graph laplacian

• random walk matrix

• random walk laplacian

• normalized adjacency matrix

• normalized graph laplacian

directed graphs.

We usually don’t use the last 3 with

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