标签：图论 Theory Graph Laplacian graph text rm deg matrix
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, \(0\le \lambda_1 \le \lambda_2 \le ... \le \lambda_n\) .
Considerations of Graph Representation Learning
- Permutation Invariance. Permutation invariance means that the function does not depend on the arbitary ordering of the row/columns vectors of the matrix.
where \(P\) is a permutation matrix.
2.
标签：图论,Theory,Graph,Laplacian,graph,text,rm,deg,matrix 来源： https://www.cnblogs.com/xmaples/p/16446764.html
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