图论矩阵详解-优快云博客

文章目录

邻接矩阵 adjacency matrix

无权无平行边网络 $A$ , $N\times N$ , 若边 $(i, j)$ 存在, 则 $a_{ij}=1$ . 无自环网络 $a_{ii}=0$ , 无向网络 $a_{ij}=a_{ji}$

有向	无向
$A=\begin{pmatrix} 0 & 1 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 1\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}$	$A=\begin{pmatrix} 0 & 1 & 1 & 0 & 0 & 1\\ 1 & 0 & 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 & 0 & 1\\ 1 & 1 & 0 & 0 & 1 & 0 \end{pmatrix}$

关联矩阵 incidence matrix

$B$ , $N\times M$ , $M$ 为网络中当前所有存在的边. 边按字典升序 lexicographically ordered 排列.
有向 $N\times M$
$N = 6$ , $M = 9$ ,
$e_1=1\rightarrow 2$ , $e_2=1\rightarrow 3$ , $e_3=1\leftarrow 6$ ,
$e_4=2\rightarrow 3$ , $e_5=2\leftarrow 5$ , $e_6=2\rightarrow 6$ ,
$e_7=3\rightarrow 4$ ,
$e_8=4\leftarrow 5$ ,
$e_9=5\leftarrow 6$

$B_{oriented}=\begin{pmatrix} 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 1 & -1 & 1 &0 & 0 & 0\\ 0 & -1 & 0 & -1 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1\\ 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 1 \end{pmatrix}$
无向 (oriented) $N\times 2M$
$N = 6$ , $2 M = 18$ ,
$e_1=1\rightarrow 2$ , $e_2=1\leftarrow 2$ , $e_3=1\rightarrow 3$ , $e_4=1\leftarrow 3$ , $e_5=1\rightarrow 6$ , $e_6=1\leftarrow 6$ ,
$e_7=2\rightarrow 3$ , $e_8=2\leftarrow 3$ , $e_9=2\rightarrow 5$ , $e_{10}=2\leftarrow 5$ , $e_{11}=2\rightarrow 6$ , $e_{12}=2\leftarrow 6$ ,
$e_{13}=3\rightarrow 4$ , $e_{14}=3\leftarrow 4$ ,
$e_{15}=4\rightarrow 5$ , $e_{16}=4\leftarrow 5$ ,
$e_{17}=5\rightarrow 6$ , $e_{18}=5\leftarrow 6$
$B_{oriented}=\begin{pmatrix} 1 & -1 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & -1 \\ 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & -1 & 1\\ \end{pmatrix}$
$(oriented)~b_{im}= \begin{cases} 1, & \text{if link $e_m=i\rightarrow j$} \\ -1, & \text{if link $e_m=i\leftarrow j$} \\ 0, & \text{otherwise} \end{cases}$

无向 (unoriented) $N\times M$
$e_1=1 - 2$ , $e_2=1 - 3$ , $e_3=1 - 6$ ,
$e_4=2 - 3$ , $e_5=2 - 5$ , $e_6=2 - 6$ ,
$e_7=3 - 4$ ,
$e_8=4 - 5$ ,
$e_9=5 - 6$
$B_{unoriented}=\begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 1 & 1 & 1 &0 & 0 & 0\\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1\\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \end{pmatrix}$
$(unoriented)~b_{im}= \begin{cases} 1, & \text{if link $e_m=i - j$ incident} \\ 0, & \text{otherwise} \end{cases}$

度矩阵 degree matrix

$\Delta_{ii} =deg(i) = \sum_j A_{ij}$ .
$\Delta_{ij}=0$ , $i\neq j$

有向(出度＋入度)	无向
$\begin{pmatrix} 2＋1 & 0 & 0 & 0 & 0 & 0\\ 0 & 2＋2 & 0 & 0 & 0 & 0\\ 0 & 0 & 1＋2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0＋2 & 0 & 0\\ 0 & 0 & 0 & 0 & 2＋1 & 0\\ 0 & 0 & 0 & 0 & 0 & 2＋1 \end{pmatrix}$	$\begin{pmatrix} 3 & 0 & 0 & 0 & 0 & 0\\ 0 & 4 & 0 & 0 & 0 & 0\\ 0 & 0 & 3 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 3 & 0\\ 0 & 0 & 0 & 0 & 0 & 3 \end{pmatrix}$

拉普拉斯矩阵 Laplacian matrix

有向	无向
$B_{oriented}B_{oriented}^T$	$B_{oriented}B_{oriented}^T=2\Delta -2A$
$\begin{pmatrix} 3 & -1 & -1 & 0 & 0 & -1\\ -1 & 4 & -1 & 0 & -1 & -1\\ -1 & -1 & 3 & -1 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0\\ 0 & -1 & 0 & -1 & 3 & -1\\ -1 & -1 & 0 & 0 & -1 & 3 \end{pmatrix}$	$\begin{pmatrix} 6 & -2 & -2 & 0 & 0 & -2\\ -2 & 8 & -2 & 0 & -2 & -2 \\ -2 & -2 & 6 & -2 & 0 & 0 \\ 0 & 0 & -2 & 4 & -2 & 0 \\ 0 & -2 & 0 & -2 & 6 & -2 \\ -2 & -2 & 0 & 0 & -2 & 6 \end{pmatrix}$
…	$\Delta -A$
…	$\begin{pmatrix} 3 & -1 & -1 & 0 & 0 & -1\\ -1 & 4 & -1 & 0 & -1 & -1\\ -1 & -1 & 3 & -1 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0\\ 0 & -1 & 0 & -1 & 3 & -1\\ -1 & -1 & 0 & 0 & -1 & 3 \end{pmatrix}$
—	$B_{unoriented}B_{unoriented}^T=\Delta + A$
—	$\begin{pmatrix} 3 & 1 & 1 & 0 & 0 & 1\\ 1 & 4 & 1 & 0 & 1 & 1\\ 1 & 1 & 3 & 1 & 0 & 0\\ 0 & 0 & 1 & 2 & 1 & 0\\ 0 & 1 & 0 & 1 & 3 & 1\\ 1 & 1 & 0 & 0 & 1 & 3 \end{pmatrix}$

拉普拉斯矩阵定义在有向网络上, 并且邻接矩阵定义成对称形式, 即若边 $(i, j)$ 或者 $(j, i)$ 存在, 则 $a_{ij}=1$ , 则有 $L=\Delta-A=BB^T$
$L_{ij}=\Delta_{ij}-A_{ij}=(BB^T)_{ij}=\sum\limits_{m=1}^M b_{im}b_{jm}＝ \begin{cases} -1, & \text{if $(i, j)$ are linked $i\rightarrow j$, $i\neq j$} \\ 0, & \text{if $(i, j)$ are not linked $i\nrightarrow j$, $i\neq j$}\\ \sum_{m=1}^M b_{im}^2 = deg(i), & i=j \end{cases}$

随机游走归一化矩阵 Random walk normalized

邻接矩阵 $A$

归一化的邻接矩阵 normalized
$P=\Delta^{-1}A$ stochastic matrix 行随机矩阵. $\Delta$ 为对角线为度的对角矩阵即度矩阵, $A$ 为邻接矩阵. $P$ 的行元素之和为 1, $\sum_j P_{ij}=1$ . $P$ 是随机游走 walker 的跳转矩阵, pagerank 的转移矩阵, Markov 的转移矩阵, Markov 理论与代数结合能够有很多好的数学性质. $P_{ij}=\frac{A_{ij}}{k_i}=\frac{A_{ij}}{\sum_j A_{ij}}$ walker 从 $i$ 跳转到 $j$ 概率.

对称化 $P$ symmetric
$H=\Delta^{1/2}$ , $R=HPH^{-1}=\Delta^{1/2}P\Delta^{-1/2}=\Delta^{1/2}\Delta^{-1}A\Delta^{-1/2}=\Delta^{-1/2}A\Delta^{-1/2}$
$R=\Delta^{-1/2}A\Delta^{-1/2}$ 与 $P$ 有相同的特征值.

拉普拉斯矩阵 $L$

归一化的拉普拉斯矩阵 $L^{rw}$ normalized
$L^{rw}=\Delta^{-1}L$
$L^{rw}=\Delta^{-1}L=\Delta^{-1}(\Delta - A)=\Delta^{-1}\Delta -\Delta^{-1}A=I-\Delta^{-1}A=I-P$ . 因 $P$ 是随机游走的转移矩阵, 所以 $L^{rw}=\Delta^{-1}L=I-\Delta^{-1}A$ 称为随机游走 (random walk) 归一化拉普拉斯矩阵, 也因此标记为 $L^{rw}$ .
$L^{rw}_{ij}= \begin{cases} -\frac{1}{deg(i)}, & \text{if $(i, j)$ are linked $i\rightarrow j$, $i\neq j$} \\ 0, & \text{if $(i, j)$ are not linked $i\nrightarrow j$, $i\neq j$}\\ 1, & i=j, deg(i)\neq 0 \end{cases}$
$L^{rw}_{ij}=\begin{cases} \delta_{ij}-\frac{A_{ij}}{deg(i)}, & \text{$deg(i)\neq 0$, (Kronecker delta: $\delta_{ij}=1$, if $i=j$; $\delta_{ij}=0$, if $i\neq j$) } \\ 0, & \text{otherwise} \end{cases}$

$L$ 与 $P$ 的数学关系
$P=I-\Delta^{-1}L$

归一化的拉普拉斯矩阵 $L^{sym}$
$L^{sym}=\Delta^{-1/2}L\Delta^{-1/2}=\Delta^{-1/2}(\Delta - A)\Delta^{-1/2}=(\Delta^{-1/2}\Delta - \Delta^{-1/2}A)\Delta^{-1/2}=(\Delta^{1/2} - \Delta^{-1/2}A)\Delta^{-1/2}=\Delta^{1/2}\Delta^{-1/2} - \Delta^{-1/2}A\Delta^{-1/2}=I - \Delta^{-1/2}A\Delta^{-1/2}$
$L^{sym}_{ij}= \begin{cases} -\frac{1}{\sqrt{deg(i)deg(j)}}, & \text{if $(i, j)$ are linked $i\rightarrow j$, $i\neq j$} \\ 0, & \text{if $(i, j)$ are not linked $i\nrightarrow j$, $i\neq j$}\\ 1, & i=j, deg(i)\neq 0 \end{cases}$
$L^{sym}_{ij}=\begin{cases} -\frac{A_{ij}}{\sqrt{deg(i)deg(j)}}, & \text{$deg(i)\neq 0$ and $deg(j)\neq 0$} \\ 1, & deg(i)\neq 0, i=j \\ 0, & \text{otherwise} \end{cases}$

$P$ , $L^{rw}$ , $L^{sym}$ 的数学关系
$L^{sym}=\Delta^{-1/2}L\Delta^{-1/2}=\Delta^{1/2}\Delta^{-1}L\Delta^{-1/2}=\Delta^{1/2}L^{rw}\Delta^{-1/2}$

$L^{rw}=\Delta^{-1/2}L^{sym}\Delta^{1/2}=\Delta^{-1/2}(I-I+L^{sym})\Delta^{1/2}=\Delta^{-1/2}(I-(I-L^{sym}))\Delta^{1/2}=I-\Delta^{-1/2}(I-L^{sym})\Delta^{1/2}$

The graph Laplacians, $L$ , $L^{sym}$ , and $L^{rw}$ are symmetric, positive, semide nite.
The graph Laplacians, $L$ and $L^{rw}$ have the same nullspace.
The vector $\mathbf{1}$ is in the nullspace of $L^{rw}$ , and $D_{1/2} \mathbf{1}$ is in the nullspace of $L^{sym}$ .

Ref:
[1]. 2013, Jean Gallier, Notes on Elementary Spectral Graph Theory Applications to Graph Clustering Using Normalized Cuts
[2]. 2007, L. W. Beineke, Topics in Algebraic Graph Theory
[3]. https://en.wikipedia.org/wiki/Laplacian_matrix
[4]. https://en.wikipedia.org/wiki/Line_graph
[5]. https://en.wikipedia.org/wiki/Incidence_matrix
[6]. http://mathworld.wolfram.com/LaplacianMatrix.html
[7]. 2011, P.V. Mieghem, Graph Spectra for Complex Networks