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Graph Theory 图论

Laplacian matrix

Categories of graphs:

• directed/undirected.
• homogeneous/heterogeneous.
• static/dynamic. A dynamic graph is a graph whose topology varies with time.

It is a matrix representation of a graph.

It can be used:
(1) to construct low dimentional graph node embeddings.
(2) to find sparsest \(K\) subgraphs of a graph through the \(K\) smallest eigenvalue of its laplacian matrix.
(3) to calculate the number of spanning trees.
(4) ...

Given a simple graph \(G\) with vertices \(V\) , the Laplacian matrix \(L\in\R^{|V|\times |V|}\) of \(G\) is given by

\[L := D-A \]

, where \(D\) is the degree matrix, which is diagonal with entries \(D_{ii}\) the degree of node \(i\) , and \(A\) is the adjacency matrix. Since \(G\) is a simple graph, \(A\) only contains 1 or 0 and its diagonal elements are all 0s.

(Symmetric) normalized Laplacian matrix

\[L_{\rm sym} := D^{-\frac12}LD^{-\frac12} = I-D^{-\frac12}AD^{-\frac12} \]

The elements of \(L_{\rm sym}\) are given by

\[L_{\rm sym}[i,j] := \begin{dcases} 1 & \text{ if } i=j \wedge \deg(i)\ne 0 \\ -\frac{1}{\sqrt{\deg(i)\deg(j)}}& \text{ if } i\ne j \text{ and i is adjacent to j} \\ 0 & \text{ otherwise} \end{dcases} \]

Random Walk normalized Laplacian matrix

\[L_{\rm rw}[i,j] := D^{-1}L = I- D^{-1}A \]

The elements of \(L_{\rm rw}\) are given by

\[L_{\rm rw}[i,j] := \begin{dcases} 1 & \text{ if } i=j \wedge \deg(i)\ne 0 \\ -\frac{1}{\deg(i)}& \text{ if } i\ne j \text{ and i is adjacent to j} \\ 0 & \text{ otherwise} \end{dcases} \]

Properties of Laplacian matrix

• \(\forall \bm x\in \R^{|V|}: \bm x^T L \bm x=\sum_{i,j}^{|V|} A_{ij}\|x_i-x_j\|^2\)
• \(L\) is symmetric, positive semi-definite, diagonally dominant.
• \(L\) is a M-matrix (its off-diagonal entries are non-positive, and the eigenvalues are non-negative ( on real parts for complex numbers).
• The smallest eigenvalue is \(0\) , and the corresponding eigenvector is \(\bm 1\) (all elements are 1s).
• \(L\) has non-negative eigenvalues, \(0\le \lambda_1 \le \lambda_2 \le ... \le \lambda_n\) .

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Considerations of Graph Representation Learning

1. Permutation Invariance. Permutation invariance means that the function does not depend on the arbitary ordering of the row/columns vectors of the matrix.

\[f(PAP^T)=f(A) \implies \text{ Permutation Invariance} f(PAP^T)=Pf(A) \implies \text {Permutation Equivariance} \]

where \(P\) is a permutation matrix.
2.

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