调和矩のり阵

在ざい图论中なか，调和矩のり阵（harmonic matrix），也称拉ひしげ普ひろし拉ひしげ斯矩阵或ある拉ひしげ氏し矩のり阵（Laplacian matrix）、离散拉ひしげ普ひろし拉ひしげ斯（discrete Laplacian），是ぜ图的てき矩のり阵表示ひょうじ。

调和矩のり阵也是ぜ拉ひしげ普ひろし拉ひしげ斯算子こ的てき离散化か。换句话说，调和矩のり阵的缩放极限是これ拉ひしげ普ひろし拉ひしげ斯算子こ。它在机つくえ器き学がく习和わ物理ぶつり学がく中有ちゅうう很多应用。

定てい义

若わかG是ぜ简单图，G有ゆうn个顶点，A是ぜ邻接矩のり阵，D是ぜ度数どすう矩のり阵，则调和矩のり阵是これ

$L=D-A$

$L_{i,j}:={\begin{cases}\deg(v_{i})&{\mbox{if}}\ i=j\\-1&{\mbox{if}}\ i\neq j\ {\mbox{and}}\ v_{i}{\mbox{ is adjacent to }}v_{j}\\0&{\mbox{otherwise}}\end{cases}}$

动机

这跟拉ひしげ普ひろし拉ひしげ斯算子こ有ゆう什么关系？若わかf 是ぜ加か权图G的てき顶点函数かんすう，则

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

w是ぜ边的权重函数かんすう。u、v是ぜ顶点。f = (f(1), ..., f(n)) 是ぜn维的矢や量りょう。上面うわつら泛函也称为Dirichlet泛函。

接せっ续矩阵

而且若わかK是ぜ接せっ续矩阵（incidence matrix），则

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

$L=K^{t}K$

Kf 是ぜf 的てき图梯はしご度ど。另外，特とく征せい值满足あし

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

举例

图	度ど矩のり阵	邻接矩のり阵	调和矩のり阵
	${\textstyle \left({\begin{array}{rrrrrr}2&0&0&0&0&0\\0&3&0&0&0&0\\0&0&2&0&0&0\\0&0&0&3&0&0\\0&0&0&0&3&0\\0&0&0&0&0&1\\\end{array}}\right)}$	${\textstyle \left({\begin{array}{rrrrrr}0&1&0&0&1&0\\1&0&1&0&1&0\\0&1&0&1&0&0\\0&0&1&0&1&1\\1&1&0&1&0&0\\0&0&0&1&0&0\\\end{array}}\right)}$	${\textstyle \left({\begin{array}{rrrrrr}2&-1&0&0&-1&0\\-1&3&-1&0&-1&0\\0&-1&2&-1&0&0\\0&0&-1&3&-1&-1\\-1&-1&0&-1&3&0\\0&0&0&-1&0&1\\\end{array}}\right)}$

其他形式けいしき

对称正規せいき化か调和矩のり阵

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

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

注意ちゅうい

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

随ずい机つくえ漫步まんぽ

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

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

动力学がく和わ微分びぶん方かた程ほど

例れい如，离散的てき冷却れいきゃく定律ていりつ使用しよう调和矩のり阵

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

使用しよう矩のり阵矢量りょう

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

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

解かい是ぜ

${\begin{aligned}{\frac {d\left(\sum _{i}c_{i}\mathbf {v} _{i}\right)}{dt}}+kL\left(\sum _{i}c_{i}\mathbf {v} _{i}\right)&=0\\\sum _{i}\left[{\frac {dc_{i}}{dt}}\mathbf {v} _{i}+kc_{i}L\mathbf {v} _{i}\right]&=\\\sum _{i}\left[{\frac {dc_{i}}{dt}}\mathbf {v} _{i}+kc_{i}\lambda _{i}\mathbf {v} _{i}\right]&=\\{\frac {dc_{i}}{dt}}+k\lambda _{i}c_{i}&=0\\\end{aligned}}$

$c_{i}(t)=c_{i}(0)e^{-k\lambda _{i}t}$

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

平衡へいこう举动

当とう${\textstyle \lim _{t\to \infty }\phi (t)}$的てき时候，

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

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

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

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

MATLAB代だい码

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：离散拉ひしげ普ひろし拉ひしげ斯过程ほど，使用しよう拉ひしげ普ひろし拉ひしげ斯矩阵

应用

• Kirchoff定理ていり：调和矩のり阵的行列ぎょうれつ式しき
• 人工じんこう智能ちのう
• 非ひ线性降くだ维
• 谱聚类
• Cheeger不等式ふとうしき：调和矩のり阵的第だい二に大だい的てき特とく征せい值跟最大さいだい割わり問題もんだい有ゆう关
• 图Signal processing

阅读

• T. Sunada. Chapter 1. Analysis on combinatorial graphs. Discrete geometric analysis. P. Exner, J. P. Keating, P. Kuchment, T. Sunada, A. Teplyaev (编). 'Proceedings of Symposia in Pure Mathematics 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).