在图论中，**调和矩阵**（**harmonic matrix**），也称**拉普拉斯矩阵**或**拉氏矩阵**（**Laplacian matrix**）、**离散拉普拉斯**（**discrete Laplacian**），是图的矩阵表示。

调和矩阵也是拉普拉斯算子的离散化。换句话说，调和矩阵的缩放极限是拉普拉斯算子。它在机器学习和物理学中有很多应用。

定义
--

若G是简单图，G有n个顶点，A是邻接矩阵，D是度数矩阵，则**调和矩阵**是

{\displaystyle L=D-A}

{\displaystyle L\_{i,j}:={\begin{cases}\deg(v\_{i})&{\mbox{if}}\ i=j\\-1&{\mbox{if}}\ i\neq j\ {\mbox{and}}\ v\_{i}{\mbox{ is adjacent to }}v\_{j}\\0&{\mbox{otherwise}}\end{cases}}}

### 动机

这跟拉普拉斯算子有什么关系？若f 是加权图G的顶点函数，则

{\displaystyle E(f)=\sum w(uv)(f(u)-f(v))^{2}/2=f^{t}Lf}

w是边的权重函数。u、v是顶点。f = (f(1), ... f(n)) 是n维的矢量。上面泛函也称为Dirichlet泛函。

### 接续矩阵

而且若K是接续矩阵（incidence matrix），则

{\displaystyle K\_{ev}=\left\{{\begin{array}{rl}1,&{\text{if }}v=i\\-1,&{\text{if }}v=j\\0,&{\text{otherwise}}.\end{array}}\right.}

{\displaystyle L=K^{t}K}

Kf 是f 的图梯度。另外，特征值满足

{\displaystyle {\begin{aligned}\lambda \_{i}&=\mathbf {v} \_{i}^{\textsf {T}}L\mathbf {v} \_{i}\\&=\mathbf {v} \_{i}^{\textsf {T}}M^{\textsf {T}}M\mathbf {v} \_{i}\\&=\left(M\mathbf {v} \_{i}\right)^{\textsf {T}}\left(M\mathbf {v} \_{i}\right)\\\end{aligned}}}

举例
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| 图 | 度矩阵 | 邻接矩阵 | 调和矩阵 |
| --- | --- | --- | --- |
|  | {\textstyle \left({\begin{array}{rrrrrr}2&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&0\\0&0&0&0&0&1\\\end{array}}\right)} | {\textstyle \left({\begin{array}{rrrrrr}0&1&0&0&1&0\\1&0&1&0&1&0\\0&1&0&1&0&0\\0&0&1&0&1&1\\1&1&0&1&0&0\\0&0&0&1&0&0\\\end{array}}\right)} | {\textstyle \left({\begin{array}{rrrrrr}2&-1&0&0&-1&0\\-1&3&-1&0&-1&0\\0&-1&2&-1&0&0\\0&0&-1&3&-1&-1\\-1&-1&0&-1&3&0\\0&0&0&-1&0&1\\\end{array}}\right)} |

其他形式
----

### 对称正規化调和矩阵

{\displaystyle L^{\text{sym}}:=D^{-{\frac {1}{2}}}LD^{-{\frac {1}{2}}}=I-D^{-{\frac {1}{2}}}AD^{-{\frac {1}{2}}}}

{\displaystyle L\_{i,j}^{\text{sym}}:={\begin{cases}1&{\mbox{if }}i=j{\mbox{ and }}\deg(v\_{i})\neq 0\\-{\frac {1}{\sqrt {\deg(v\_{i})\deg(v\_{j})}}}&{\mbox{if }}i\neq j{\mbox{ and }}v\_{i}{\mbox{ is adjacent to }}v\_{j}\\0&{\mbox{otherwise}}.\end{cases}}}

注意

{\displaystyle \lambda \ =\ {\frac {\langle g,L^{\text{sym}}g\rangle }{\langle g,g\rangle }}\ =\ {\frac {\left\langle g,D^{-{\frac {1}{2}}}LD^{-{\frac {1}{2}}}g\right\rangle }{\langle g,g\rangle }}\ =\ {\frac {\langle f,Lf\rangle }{\left\langle D^{\frac {1}{2}}f,D^{\frac {1}{2}}f\right\rangle }}\ =\ {\frac {\sum \_{u\sim v}(f(u)-f(v))^{2}}{\sum \_{v}f(v)^{2}d\_{v}}}\ \geq \ 0,}

### 随机漫步

{\displaystyle L^{\text{rw}}:=D^{-1}L=I-D^{-1}A}

{\displaystyle L\_{i,j}^{\text{rw}}:={\begin{cases}1&{\mbox{if }}i=j{\mbox{ and }}\deg(v\_{i})\neq 0\\-{\frac {1}{\deg(v\_{i})}}&{\mbox{if }}i\neq j{\mbox{ and }}v\_{i}{\mbox{ is adjacent to }}v\_{j}\\0&{\mbox{otherwise}}.\end{cases}}}

动力学和微分方程
--------

例如，离散的冷却定律使用调和矩阵

{\displaystyle {\begin{aligned}{\frac {d\phi \_{i}}{dt}}&=-k\sum \_{j}A\_{ij}\left(\phi \_{i}-\phi \_{j}\right)\\&=-k\left(\phi \_{i}\sum \_{j}A\_{ij}-\sum \_{j}A\_{ij}\phi \_{j}\right)\\&=-k\left(\phi \_{i}\ \deg(v\_{i})-\sum \_{j}A\_{ij}\phi \_{j}\right)\\&=-k\sum \_{j}\left(\delta \_{ij}\ \deg(v\_{i})-A\_{ij}\right)\phi \_{j}\\&=-k\sum \_{j}\left(\ell \_{ij}\right)\phi \_{j}.\end{aligned}}}

使用矩阵矢量

{\displaystyle {\begin{aligned}{\frac {d\phi }{dt}}&=-k(D-A)\phi \\&=-kL\phi \end{aligned}}}

{\displaystyle {\frac {d\phi }{dt}}+kL\phi =0}

解是

{\displaystyle {\begin{aligned}{\frac {d\left(\sum \_{i}c\_{i}\mathbf {v} \_{i}\right)}{dt}}+kL\left(\sum \_{i}c\_{i}\mathbf {v} \_{i}\right)&=0\\\sum \_{i}\left[{\frac {dc\_{i}}{dt}}\mathbf {v} \_{i}+kc\_{i}L\mathbf {v} \_{i}\right]&=\\\sum \_{i}\left[{\frac {dc\_{i}}{dt}}\mathbf {v} \_{i}+kc\_{i}\lambda \_{i}\mathbf {v} \_{i}\right]&=\\{\frac {dc\_{i}}{dt}}+k\lambda \_{i}c\_{i}&=0\\\end{aligned}}}

{\displaystyle c\_{i}(t)=c\_{i}(0)e^{-k\lambda \_{i}t}}

{\displaystyle c\_{i}(0)=\left\langle \phi (0),\mathbf {v} \_{i}\right\rangle }

### 平衡举动

当{\textstyle \lim \_{t\to \infty }\phi (t)}的时候，

{\displaystyle \lim \_{t\to \infty }e^{-k\lambda \_{i}t}=\left\{{\begin{array}{rlr}0&{\text{if}}&\lambda \_{i}>0\\1&{\text{if}}&\lambda \_{i}=0\end{array}}\right\}}

{\displaystyle \lim \_{t\to \infty }\phi (t)=\left\langle c(0),\mathbf {v^{1}} \right\rangle \mathbf {v^{1}} }

{\displaystyle \mathbf {v^{1}} ={\frac {1}{\sqrt {N}}}[1,1,...1]}

{\displaystyle \lim \_{t\to \infty }\phi \_{j}(t)={\frac {1}{N}}\sum \_{i=1}^{N}c\_{i}(0)}

### MATLAB代码

```
N = 20;%The number of pixels along a dimension of the image
A = zeros(N, N);%The image
Adj = zeros(N\*N, N\*N);%The adjacency matrix

%Use 8 neighbors, and fill in the adjacency matrix
dx = [-1, 0, 1, -1, 1, -1, 0, 1];
dy = [-1, -1, -1, 0, 0, 1, 1, 1];
for x = 1:N
 for y = 1:N
 index = (x-1)\*N + y;
 for ne = 1:length(dx)
 newx = x + dx(ne);
 newy = y + dy(ne);
 if newx > 0 && newx <= N && newy > 0 && newy <= N
 index2 = (newx-1)\*N + newy;
 Adj(index, index2) = 1;
 end
 end
 end
end

%%%BELOW IS THE KEY CODE THAT COMPUTES THE SOLUTION TO THE DIFFERENTIAL
%%%EQUATION
Deg = diag(sum(Adj, 2));%Compute the degree matrix
L = Deg - Adj;%Compute the laplacian matrix in terms of the degree and adjacency matrices
[V, D] = eig(L);%Compute the eigenvalues/vectors of the laplacian matrix
D = diag(D);

%Initial condition (place a few large positive values around and
%make everything else zero)
C0 = zeros(N, N);
C0(2:5, 2:5) = 5;
C0(10:15, 10:15) = 10;
C0(2:5, 8:13) = 7;
C0 = C0(:);

C0V = V'\*C0;%Transform the initial condition into the coordinate system 
%of the eigenvectors
for t = 0:0.05:5
 %Loop through times and decay each initial component
 Phi = C0V.\*exp(-D\*t);%Exponential decay for each component
 Phi = V\*Phi;%Transform from eigenvector coordinate system to original coordinate system
 Phi = reshape(Phi, N, N);
 %Display the results and write to GIF file
 imagesc(Phi);
 caxis([0, 10]);
 title(sprintf('Diffusion t = %3f', t));
 frame = getframe(1);
 im = frame2im(frame);
 [imind, cm] = rgb2ind(im, 256);
 if t == 0
 imwrite(imind, cm, 'out.gif', 'gif', 'Loopcount', inf, 'DelayTime', 0.1); 
 else
 imwrite(imind, cm, 'out.gif', 'gif', 'WriteMode', 'append', 'DelayTime', 0.1);
 end
end

```

GIF：离散拉普拉斯过程，使用拉普拉斯矩阵

应用
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* Kirchoff定理：调和矩阵的行列式
* 人工智能
* 非线性降维
* 谱聚类
* Cheeger不等式：调和矩阵的第二大的特征值跟最大割問題有关
* 图Signal processing

参考文献
----

1. 1 2  Weisstein, Eric W. (编).. at *MathWorld*--A Wolfram Web Resource. Wolfram Research, Inc.  [2020-02-14]. （原始内容存档于2019-12-23） （英语）.
2. 1 2 . www.quora.com.  [2020-02-14].
3. 1 2  Shuman, David I.; Narang, Sunil K.; Frossard, Pascal; Ortega, Antonio; Vandergheynst, Pierre.. IEEE Signal Processing Magazine. 2013-05, **30** (3): 83–98  [2020-02-14]. ISSN 1053-5888. doi:10.1109/MSP.2012.2235192. （原始内容存档于2020-01-11）.
4. ↑  Chung, Fan.. American Mathematical Society. 1997 [1992]  [2020-02-14]. ISBN 978-0821803158. （原始内容存档于2020-02-14）.
5. ↑  Newman, Mark.. Oxford University Press. 2010. ISBN 978-0199206650.

阅读
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* T. Sunada.. P. Exner, J. P. Keating, P. Kuchment, T. Sunada, A. Teplyaev (编).  **77**. 2008: 51–86. ISBN 978-0-8218-4471-7.
* B. Bollobás, *Modern Graph Theory*, Springer-Verlag (1998, corrected ed. 2013), ISBN 0-387-98488-7, Chapters II.3 (Vector Spaces and Matrices Associated with Graphs), VIII.2 (The Adjacency Matrix and the Laplacian), IX.2 (Electrical Networks and Random Walks).