拉普拉斯、图与扩散Laplacians, Graphs & Diffusion
局部与邻居的差异怎样推动信号逐步扩散？How do local differences from neighbors drive diffusion?
前置知识：Prerequisites: G01：散度；B05：矩阵；O01：二阶导数。G01: divergence; B05: matrices; O01: second derivatives.
复习前置：Review prerequisites: G01 · 标量场、向量场与散度Scalar Fields, Vector Fields & Divergence · B05 · 矩阵与批量计算Matrices & Batch Computation · O01 · 二阶导数与泰勒近似Second Derivatives & Taylor Approximation
学完后你能：After this lesson you can: 按图拉普拉斯单步扩散，检查守恒、非负性与孤立点。Step through graph-Laplacian diffusion and check conservation, nonnegativity and isolated nodes.
动手观察Explore
在一维网格或小图上单步扩散，比较总量、非负性和孤立节点；切换常量信号。Step through diffusion on a one-dimensional grid or small graph, comparing total mass, nonnegativity and an isolated node; try a constant signal.
从观察到解释From observation to explanation
与邻居比较Compare with neighbors
图拉普拉斯在一个节点处累加“自己减邻居”的边权差。常量信号处处没有差，因而 Lx=0。连续拉普拉斯是各坐标二阶导数之和，差分近似需包含网格尺度。At a node, the graph Laplacian adds weighted differences “self minus neighbor.” A constant signal has no differences, so Lx=0. The continuous Laplacian sums coordinate second derivatives, and a finite-difference approximation includes grid scale.
负号推动平滑The minus sign smooths
采用 L=D−W 时，用 x′=x−τLx 更新，信号从高值向邻居传递。对于无向非负边权，若 τ 不超过最大加权度的倒数，非负初值保持非负且总量守恒。With L=D−W, x′=x−τLx transfers signal from higher values toward neighbors. For undirected nonnegative weights, if τ does not exceed the reciprocal of the maximum weighted degree, nonnegative initial values remain nonnegative and total mass is conserved.
孤立点An isolated node
孤立点没有边，度为 0，更新时保持原值。不同连通分量各自守恒；把所有节点都当作必然收敛到同一个平均值会忽略图的连通性。An isolated node has no edges and degree 0, so it keeps its value. Each connected component conserves its own total; assuming every node must converge to one common average ignores graph connectivity.
公式与符号Formula and symbols
W 是非负对称边权矩阵，D 的对角线是各节点的边权和，L 是组合图拉普拉斯；x 是节点信号，τ 是步长。W is a symmetric nonnegative edge-weight matrix, D has weighted degrees on its diagonal, L is the combinatorial graph Laplacian, x is the node signal, and τ is the step size.
留意这个区别Watch for this
组合拉普拉斯 L=D−W 的扩散有负号；步长过大时不能保证非负性或稳定。谱分析需额外学习 A03。Diffusion with combinatorial L=D−W uses a minus sign. Oversized steps do not guarantee nonnegativity or stability. Spectral analysis additionally requires A03.
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