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<div class="md-content" data-md-component="content">
<article class="md-content__inner md-typeset">
<h1 id="_1">图论<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>图论为描述实体间关系提供了数学语言。本章涵盖节点、边、邻接矩阵、图类型、度和连通性、图拉普拉斯算子、谱图理论以及现实世界的图应用。我们将在纯计算机科学章节中更深入地讨论图</em></p>
<ul>
<li>
<p>到目前为止，本书中的数据都存在于规则结构上：<span class="arithmatex">\(\mathbb{R}^n\)</span> 中的向量（第1章）、数字网格形式的矩阵（第2章）、像素网格形式的图像（第8章）、有序列表形式的序列（第7章）。但许多现实世界的系统是<strong>不规则</strong>的：社交网络没有网格结构，分子没有从左到右的顺序，道路网络也不能整齐地平铺成行和列。</p>
</li>
<li>
<p><strong>图（Graph）</strong> 是表示这些不规则关系结构的数学工具。图捕获了<strong>实体</strong>（节点）及它们之间的<strong>关系</strong>（边）。一旦数据被表示为图，我们就可以应用文件1中的几何深度学习原理来从中学习。</p>
</li>
</ul>
<h2 id="_2">节点、边和邻接<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>一个<strong>图</strong> <span class="arithmatex">\(G = (V, E)\)</span> 由一组<strong>节点</strong>（或顶点）<span class="arithmatex">\(V = \{v_1, v_2, \ldots, v_n\}\)</span> 和一组连接节点对的<strong>边</strong> <span class="arithmatex">\(E \subseteq V \times V\)</span> 组成。</p>
</li>
<li>
<p>节点代表实体：人、原子、城市、网页、神经元。边代表关系：友谊、化学键、道路、超链接、突触。</p>
</li>
<li>
<p><strong>邻接矩阵</strong> <span class="arithmatex">\(A\)</span> 是图的矩阵表示。对于一个有 <span class="arithmatex">\(n\)</span> 个节点的图，<span class="arithmatex">\(A\)</span> 是一个 <span class="arithmatex">\(n \times n\)</span> 矩阵，其中如果存在从节点 <span class="arithmatex">\(i\)</span> 到节点 <span class="arithmatex">\(j\)</span> 的边，则 <span class="arithmatex">\(A_{ij} = 1\)</span>，否则 <span class="arithmatex">\(A_{ij} = 0\)</span>。</p>
</li>
<li>
<p>例如，一个三角形图（3个节点，全部相连）：</p>
</li>
</ul>
<div class="arithmatex">\[
A = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}
\]</div>
<p><img alt="一个三角形图及其邻接矩阵：边存在处为1，否则为0" src="../../images/graph_adjacency_matrix.svg" /></p>
<ul>
<li>
<p>对角线为零，因为节点默认不与自身相连（无自环）。邻接矩阵是我们在第2章中研究的布尔矩阵的直接应用：每个条目都是一个二元关系。</p>
</li>
<li>
<p>邻接矩阵完整地编码了图的结构。对 <span class="arithmatex">\(A\)</span> 的矩阵运算揭示了图的性质：<span class="arithmatex">\(A^2_{ij}\)</span> 计算节点 <span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 之间长度为2的路径数量（回顾第2章中的矩阵乘法：每个条目是经过中间节点的乘积之和）。更一般地，<span class="arithmatex">\(A^k_{ij}\)</span> 计算长度为 <span class="arithmatex">\(k\)</span> 的路径数量。</p>
</li>
<li>
<p>每个节点可以携带一个<strong>特征向量</strong> <span class="arithmatex">\(\mathbf{x}_i \in \mathbb{R}^d\)</span>。对于社交网络，这可能是用户的个人信息。对于分子，它编码原子类型、电荷和其他属性。全部节点特征的集合是一个矩阵 <span class="arithmatex">\(X \in \mathbb{R}^{n \times d}\)</span>，其中每一行是一个节点的特征。</p>
</li>
<li>
<p>边也可以携带特征：分子中的键类型、空间图中的距离、知识图谱中的关系类型。边 <span class="arithmatex">\((i, j)\)</span> 的<strong>边特征</strong>是一个向量 <span class="arithmatex">\(\mathbf{e}_{ij} \in \mathbb{R}^{d_e}\)</span>。</p>
</li>
</ul>
<h2 id="_3">图类型<a class="headerlink" href="#_3" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p><strong>无向图</strong>具有对称的边：如果 <span class="arithmatex">\(i\)</span> 连接到 <span class="arithmatex">\(j\)</span>，则 <span class="arithmatex">\(j\)</span> 也连接到 <span class="arithmatex">\(i\)</span>。邻接矩阵是对称的：<span class="arithmatex">\(A = A^T\)</span>（一个对称矩阵，见第2章）。友谊和化学键是无向的。</p>
</li>
<li>
<p><strong>有向图</strong>（digraph）具有带方向的边：从 <span class="arithmatex">\(i\)</span> 到 <span class="arithmatex">\(j\)</span> 的边不意味着从 <span class="arithmatex">\(j\)</span> 到 <span class="arithmatex">\(i\)</span> 的边。邻接矩阵是非对称的。Twitter关注、网页超链接和引文网络是有向的。</p>
</li>
<li>
<p><strong>加权图</strong>为每条边分配一个数值权重。邻接矩阵具有实数值而非二进制值：<span class="arithmatex">\(A_{ij} = w_{ij}\)</span>。道路网络中的距离、大脑连通性中的相关强度以及社交网络中的交互频率是加权的。</p>
</li>
<li>
<p><strong>二分图</strong>具有两个不相交的节点集合，边只存在于集合之间（集合内部没有边）。用户和产品构成一个二分图：用户评价产品，但用户之间不相互评价。二分图的邻接矩阵具有块结构：</p>
</li>
</ul>
<div class="arithmatex">\[
A = \begin{bmatrix} 0 & B \\ B^T & 0 \end{bmatrix}
\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(B\)</span> 是两个节点集之间的二分邻接矩阵。</p>
</li>
<li>
<p><strong>多重图</strong>允许同一对节点之间存在多条边和/或自环。知识图谱通常是多重图：两个实体之间可以有多种关系（例如"出生于"、"居住于"、"工作于"）。</p>
</li>
<li>
<p><strong>超图</strong>将边推广为一次连接两个以上节点。一条<strong>超边</strong>连接一组节点，表示高阶关系。一篇由五人合著的研究论文是一条连接五个作者节点的超边。</p>
</li>
<li>
<p><strong>完全图</strong> <span class="arithmatex">\(K_n\)</span> 在每一对节点之间都有边。这是全连接层的图类比，也是Transformer操作的结构（每个标记关注每个其他标记）。</p>
</li>
</ul>
<h2 id="_4">度、路径和连通性<a class="headerlink" href="#_4" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>一个<strong>节点</strong>的<strong>度</strong>是与它相连的边的数量。在无向图中，节点 <span class="arithmatex">\(i\)</span> 的度为 <span class="arithmatex">\(d_i = \sum_j A_{ij}\)</span>。高度节点是拥有大量连接的"枢纽"。</p>
</li>
<li>
<p><strong>度矩阵</strong> <span class="arithmatex">\(D\)</span> 是一个对角线元素为度的对角矩阵：<span class="arithmatex">\(D_{ii} = d_i\)</span>。这个矩阵出现在整个图论和GNN公式中。</p>
</li>
<li>
<p>两个节点之间的<strong>路径</strong>是连接它们的边序列。<span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 之间的<strong>最短路径</strong>（或测地线）是边数最少（或在加权图中总权重最小）的路径。<strong>迪杰斯特拉算法</strong>（Dijkstra's algorithm）在 <span class="arithmatex">\(O((|V| + |E|) \log |V|)\)</span> 时间内找到最短路径。</p>
</li>
<li>
<p>如果每对节点之间都存在路径，则图是<strong>连通的</strong>。否则，图有多个<strong>连通分量</strong>：相互之间没有边的孤立子图。</p>
</li>
<li>
<p>图的<strong>直径</strong>是任意一对节点之间最长最短路径的长度。它衡量图"分散"的程度。社交网络以直径小而闻名（"六度分隔"）。</p>
</li>
<li>
<p><strong>环</strong>是起点和终点在同一节点的路径。没有环的图是<strong>树</strong>。树是最简单的连通图：<span class="arithmatex">\(n\)</span> 个节点和恰好 <span class="arithmatex">\(n-1\)</span> 条边。</p>
</li>
<li>
<p><strong>中心性</strong>衡量节点的重要性。<strong>度中心性</strong>就是度数。<strong>介数中心性</strong>计算通过一个节点的最短路径数量。<strong>特征向量中心性</strong>根据节点邻居的重要性分配重要性，得到特征向量方程 <span class="arithmatex">\(A\mathbf{x} = \lambda \mathbf{x}\)</span>（第2章）。谷歌的PageRank是特征向量中心性在有向图上的变体。</p>
</li>
</ul>
<h2 id="_5">图拉普拉斯算子<a class="headerlink" href="#_5" title="Permanent link">&para;</a></h2>
<ul>
<li><strong>图拉普拉斯算子</strong>也许是图论中最重要的矩阵。定义如下：</li>
</ul>
<div class="arithmatex">\[L = D - A\]</div>
<ul>
<li>其中 <span class="arithmatex">\(D\)</span> 是度矩阵，<span class="arithmatex">\(A\)</span> 是邻接矩阵。对于我们的三角形示例：</li>
</ul>
<div class="arithmatex">\[
L = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} - \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix}
\]</div>
<ul>
<li>
<p>拉普拉斯算子具有显著的性质：</p>
<ul>
<li>它始终是<strong>对称的</strong>且<strong>半正定的</strong>（回顾第2章：所有特征值 <span class="arithmatex">\(\geq 0\)</span>）。对于任意向量 <span class="arithmatex">\(\mathbf{x}\)</span>：</li>
</ul>
</li>
</ul>
<div class="arithmatex">\[\mathbf{x}^T L \mathbf{x} = \sum_{(i,j) \in E} (x_i - x_j)^2\]</div>
<p><img alt="图拉普拉斯算子度量信号平滑度：平滑信号在连接节点上具有相似值，非平滑信号变化剧烈" src="../../images/graph_laplacian_smoothness.svg" /></p>
<div class="highlight"><pre><span></span><code>- 这个二次形式度量图上的信号 $\mathbf{x}$ 在边上的变化程度。如果相邻节点值相近，则 $\mathbf{x}^T L \mathbf{x}$ 较小。如果它们差异很大，则较大。拉普拉斯算子度量图上信号的**平滑度**。
- 最小特征值始终为0，特征向量为 $\mathbf{1} = [1, 1, \ldots, 1]^T$（常数信号没有变化）。零特征值的数量等于连通分量的数量。
- 第二小特征值 $\lambda_2$ 是**代数连通度**（Fiedler值）。它衡量图的连通程度：$\lambda_2 = 0$ 表示图不连通，大的 $\lambda_2$ 表示图紧密连通。
</code></pre></div>
<ul>
<li><strong>归一化拉普拉斯算子</strong>通过度进行缩放：</li>
</ul>
<div class="arithmatex">\[\hat{L} = D^{-1/2} L D^{-1/2} = I - D^{-1/2} A D^{-1/2}\]</div>
<ul>
<li>这种归一化确保拉普拉斯算子的性质不依赖于节点度的绝对尺度。项 <span class="arithmatex">\(D^{-1/2} A D^{-1/2}\)</span> 是<strong>对称归一化邻接矩阵</strong>，它直接出现在GCN公式中（文件3）。</li>
</ul>
<h2 id="_6">谱图理论<a class="headerlink" href="#_6" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>图拉普拉斯算子的特征值和特征向量定义了图的<strong>谱</strong>，它们充当图上的傅里叶变换的类似物。</p>
</li>
<li>
<p>在经典信号处理中，傅里叶变换将信号分解为频率分量（正弦和余弦）。在图上，拉普拉斯算子的特征向量扮演这些频率基的角色。小特征值的特征向量在图上变化缓慢（低频、平滑），而大特征值的特征向量变化迅速（高频、振荡）。</p>
</li>
<li>
<p>信号 <span class="arithmatex">\(\mathbf{x}\)</span> 在图上的<strong>图傅里叶变换（GFT）</strong> 为：</p>
</li>
</ul>
<div class="arithmatex">\[\hat{\mathbf{x}} = U^T \mathbf{x}\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(U\)</span> 是拉普拉斯算子特征向量的矩阵（回顾第2章中的特征分解：<span class="arithmatex">\(L = U \Lambda U^T\)</span>）。逆变换为 <span class="arithmatex">\(\mathbf{x} = U \hat{\mathbf{x}}\)</span>。</p>
</li>
<li>
<p>谱域中的<strong>图卷积</strong>是频域中的逐点乘法，正如空间域中的卷积对应于傅里叶域中的乘法（卷积定理，见第8章）：</p>
</li>
</ul>
<div class="arithmatex">\[g_\theta \star \mathbf{x} = U \left( (U^T g_\theta) \odot (U^T \mathbf{x}) \right) = U \, \text{diag}(\hat{g}_\theta) \, U^T \mathbf{x}\]</div>
<ul>
<li>
<p>滤波器 <span class="arithmatex">\(\hat{g}_\theta\)</span> 是特征值的可学习函数。这是谱域GNN的基础，我们将在文件3中将其简化为实用的GCN。</p>
</li>
<li>
<p>计算瓶颈是对 <span class="arithmatex">\(L\)</span> 进行特征分解，对于有 <span class="arithmatex">\(n\)</span> 个节点的图需要 <span class="arithmatex">\(O(n^3)\)</span> 时间。这对于大型图（数百万节点）是不可行的。多项式近似（切比雪夫多项式）完全避免了特征分解，而这种近似直接导致了GCN。</p>
</li>
</ul>
<h2 id="_7">社区检测<a class="headerlink" href="#_7" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>许多现实世界的图具有<strong>社区结构</strong>：紧密连接的节点簇，簇之间连接稀疏。社交网络有好友群组，生物网络有功能模块，引文网络有研究领域。</p>
</li>
<li>
<p><strong>谱聚类</strong>使用拉普拉斯算子特征向量来寻找社区。思路：使用 <span class="arithmatex">\(L\)</span> 的 <span class="arithmatex">\(k\)</span> 个最小的非平凡特征向量对每个节点进行嵌入，然后在这个嵌入空间中应用k-means（第6章）。同一社区中的节点在谱嵌入中最终彼此靠近。</p>
</li>
<li>
<p>这是可行的，因为Fiedler向量（<span class="arithmatex">\(\lambda_2\)</span> 的特征向量）自然地将图分成两组：正值的节点和负值的节点，沿着最稀疏的连接切开。更高的特征向量进一步细分为更多组。</p>
</li>
<li>
<p><strong>模块度</strong> <span class="arithmatex">\(Q\)</span> 衡量社区划分的质量。它将社区内边的数量与随机图中的期望数量进行比较：</p>
</li>
</ul>
<div class="arithmatex">\[Q = \frac{1}{2|E|} \sum_{ij} \left( A_{ij} - \frac{d_i d_j}{2|E|} \right) \delta(c_i, c_j)\]</div>
<ul>
<li>其中 <span class="arithmatex">\(c_i\)</span> 是节点 <span class="arithmatex">\(i\)</span> 的社区分配，如果节点在同一个社区则 <span class="arithmatex">\(\delta\)</span> 为1。<span class="arithmatex">\(Q\)</span> 的范围从 <span class="arithmatex">\(-0.5\)</span> 到 <span class="arithmatex">\(1\)</span>，值越高表示社区结构越强。</li>
</ul>
<h2 id="_8">现实世界中的图<a class="headerlink" href="#_8" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p><strong>社交网络</strong>：节点是人，边是友谊或互动。Facebook有数十亿节点和数千亿条边。这些图通常是稀疏的（每个人有几百个朋友，而不是几十亿），具有小世界性质（短的平均路径长度），以及重尾度分布（少数拥有数百万连接的枢纽节点）。</p>
</li>
<li>
<p><strong>分子图</strong>：节点是原子，边是化学键。每个原子有特征（元素类型、电荷、杂化方式），每条键有特征（单键、双键、三键、芳香键）。分子图很小（数十到数百个节点）但高度结构化。从图结构预测分子性质是GNN的一个重要应用。</p>
</li>
<li>
<p><strong>知识图谱</strong>：节点是实体（人、地点、概念），边是类型化的关系（"出生于"、"首都是"、"是……的实例"）。知识图谱为搜索引擎、推荐系统和问答系统提供支持。它们通常是具有数百万实体和数十亿关系的有多重图。</p>
</li>
<li>
<p><strong>引文网络</strong>：节点是论文，边是引用（有向的）。聚类揭示研究社区。节点特征包括标题、摘要和出版年份。</p>
</li>
<li>
<p><strong>蛋白质相互作用网络</strong>：节点是蛋白质，边表示物理相互作用或功能关联。理解这些图有助于识别药物靶点和疾病机制。</p>
</li>
<li>
<p><strong>道路网络与交通</strong>：节点是交叉路口，边是具有距离/时间权重的道路段。这些图上的最短路径算法为导航系统提供动力。自动驾驶运动预测（第11章）将智能体交互表示为图。</p>
</li>
</ul>
<h2 id="colabnotebook">编程任务（使用CoLab或notebook）<a class="headerlink" href="#colabnotebook" title="Permanent link">&para;</a></h2>
<ol>
<li>
<p>构建一个小型图的邻接矩阵，计算基本性质：每个节点的度、长度为2的路径数量以及图是否连通。
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="c1"># 一个简单图：5个节点</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># 0-1, 0-2, 1-2, 2-3, 3-4</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">A</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">float</span><span class="p">)</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="c1"># 度</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="n">degrees</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;度数: </span><span class="si">{</span><span class="n">degrees</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="c1"># 长度为2的路径</span>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="n">A2</span> <span class="o">=</span> <span class="n">A</span> <span class="o">@</span> <span class="n">A</span>
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;长度为2的路径（节点0到3）: </span><span class="si">{</span><span class="nb">int</span><span class="p">(</span><span class="n">A2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">])</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="c1"># 是否连通？检查 A^(n-1) 是否所有条目非零</span>
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="n">An</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_power</span><span class="p">(</span><span class="n">A</span> <span class="o">+</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="mi">4</span><span class="p">)</span> <span class="c1"># (A+I)^4 用于可达性</span>
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="n">connected</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">An</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">)</span>
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;连通: </span><span class="si">{</span><span class="n">connected</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>计算图拉普拉斯算子及其特征值。验证最小特征值为0且对应的特征向量为常数。
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="n">A</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">float</span><span class="p">)</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">D</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">L</span> <span class="o">=</span> <span class="n">D</span> <span class="o">-</span> <span class="n">A</span>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="n">eigenvalues</span><span class="p">,</span> <span class="n">eigenvectors</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">L</span><span class="p">)</span>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;特征值: </span><span class="si">{</span><span class="n">eigenvalues</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;最小特征向量: </span><span class="si">{</span><span class="n">eigenvectors</span><span class="p">[:,</span><span class="w"> </span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Fiedler值（代数连通度）: </span><span class="si">{</span><span class="n">eigenvalues</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a>
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="c1"># 验证: x^T L x 度量平滑度</span>
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="n">x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">])</span> <span class="c1"># 两个组</span>
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="n">smoothness</span> <span class="o">=</span> <span class="n">x</span> <span class="o">@</span> <span class="n">L</span> <span class="o">@</span> <span class="n">x</span>
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;两组信号的平滑度: </span><span class="si">{</span><span class="n">smoothness</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>对具有两个社区的图执行谱聚类。使用Fiedler向量嵌入节点，并按符号分离。
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a>
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="c1"># 两个社区，各5个节点，弱连接</span>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">A</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="c1"># 社区1：节点0-4（密集）</span>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">):</span>
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a> <span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="c1"># 社区2：节点5-9（密集）</span>
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">):</span>
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">):</span>
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a> <span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="c1"># 一条桥接边</span>
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a>
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="n">D</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="n">L</span> <span class="o">=</span> <span class="n">D</span> <span class="o">-</span> <span class="n">A</span>
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="n">eigenvalues</span><span class="p">,</span> <span class="n">eigenvectors</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">L</span><span class="p">)</span>
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a>
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="c1"># Fiedler向量（第二小特征值）</span>
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="n">fiedler</span> <span class="o">=</span> <span class="n">eigenvectors</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span>
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="n">communities</span> <span class="o">=</span> <span class="p">(</span><span class="n">fiedler</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a>
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Fiedler向量: </span><span class="si">{</span><span class="n">fiedler</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;聚类: </span><span class="si">{</span><span class="n">communities</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a>
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">),</span> <span class="n">fiedler</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;#3498db&quot;</span> <span class="k">if</span> <span class="n">c</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">else</span> <span class="s2">&quot;#e74c3c&quot;</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">communities</span><span class="p">])</span>
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;节点&quot;</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Fiedler向量值&quot;</span><span class="p">)</span>
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;通过Fiedler向量进行谱聚类&quot;</span><span class="p">)</span>
<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></p>
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