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<article class="md-content__inner md-typeset">
<h1 id="_1">图神经网络<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>图神经网络通过在连接节点之间传递消息来学习图结构数据。本章涵盖消息传递框架、GCN、GraphSAGE、GIN、过平滑、图池化以及节点/边/图级别的任务；支撑分子性质预测、社交网络分析和推荐系统的核心架构。</em></p>
<ul>
<li>
<p>在前面的文件中，我们建立了数学基础：几何深度学习（文件1）告诉我们利用对称性，图论（文件2）提供了节点、边和邻接的语言。现在我们构建直接在图（graph）上操作的神经网络。</p>
</li>
<li>
<p>核心挑战：图数据是<strong>不规则</strong>的。与图像（固定网格）或序列（固定顺序）不同，图具有可变数量的节点、可变的连通性，并且没有规范的节点顺序。用于图的神经网络必须处理所有这些情况，同时保持置换等变性（重新标记节点不应改变输出）。</p>
</li>
</ul>
<h2 id="_2">消息传递框架<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>几乎所有的GNN都遵循同样的模式，称为<strong>消息传递</strong>（也称为邻域聚合）。这个想法简单而优雅：每个节点通过从邻居收集信息来更新其表示。</p>
</li>
<li>
<p>在每个层 <span class="arithmatex">\(l\)</span>，每个节点 <span class="arithmatex">\(i\)</span> 做三件事：</p>
<ol>
<li><strong>消息</strong>：节点 <span class="arithmatex">\(i\)</span> 的每个邻居 <span class="arithmatex">\(j\)</span> 基于其当前特征计算一条消息 <span class="arithmatex">\(\mathbf{m}_{j \to i}\)</span>。</li>
<li><strong>聚合</strong>：节点 <span class="arithmatex">\(i\)</span> 收集所有传入消息，并使用置换不变函数（求和、均值或取最大值）将它们组合。</li>
<li><strong>更新</strong>：节点 <span class="arithmatex">\(i\)</span> 将聚合的消息与其自身特征结合，产生一个新的表示。</li>
</ol>
</li>
<li>
<p>形式上：</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{m}_i^{(l)} = \bigoplus_{j \in \mathcal{N}(i)} \phi^{(l)}\left(\mathbf{h}_i^{(l)}, \mathbf{h}_j^{(l)}, \mathbf{e}_{ij}\right)\]</div>
<div class="arithmatex">\[\mathbf{h}_i^{(l+1)} = \psi^{(l)}\left(\mathbf{h}_i^{(l)}, \mathbf{m}_i^{(l)}\right)\]</div>
<ul>
<li>其中 <span class="arithmatex">\(\mathcal{N}(i)\)</span> 是节点 <span class="arithmatex">\(i\)</span> 的邻居集合，<span class="arithmatex">\(\bigoplus\)</span> 是一个置换不变的聚合操作（求和、均值、取最大值），<span class="arithmatex">\(\phi\)</span> 是消息函数，<span class="arithmatex">\(\psi\)</span> 是更新函数，<span class="arithmatex">\(\mathbf{e}_{ij}\)</span> 是可选的边特征。</li>
</ul>
<p><img alt="消息传递：邻居发送消息，置换不变函数聚合它们，然后节点更新其特征" src="../../images/message_passing_gnn.svg" /></p>
<ul>
<li>
<p>聚合操作 <span class="arithmatex">\(\bigoplus\)</span> 必须是置换不变的（邻居处理的顺序无关紧要），以确保整个函数是置换等变的。这直接实现了文件1中的对称性原理。</p>
</li>
<li>
<p>经过 <span class="arithmatex">\(k\)</span> 层消息传递后，每个节点的表示编码了其 <strong><span class="arithmatex">\(k\)</span> 跳邻域</strong>的信息：所有在 <span class="arithmatex">\(k\)</span> 条边内可达的节点。第1层看到直接邻居，第2层看到邻居的邻居，依此类推。这就是局部信息传播以建立全局理解的方式。</p>
</li>
<li>
<p>GNN的感受野随深度增长，就像CNN的感受野随层数增长一样（第8章）。但与规则网格上的CNN不同，感受野的形状根据图拓扑结构在每个节点上有所不同。</p>
</li>
</ul>
<h2 id="gcn">图卷积网络（GCN）<a class="headerlink" href="#gcn" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p><strong>GCN</strong>（Kipf &amp; Welling，2017）是基础性的GNN架构。它将谱域图卷积（来自文件2）简化为一个优雅、高效的公式。</p>
</li>
<li>
<p>从谱域卷积 <span class="arithmatex">\(g_\theta \star \mathbf{x} = U \, \text{diag}(\hat{g}_\theta) \, U^T \mathbf{x}\)</span> 出发，Kipf和Welling用一阶切比雪夫多项式近似谱域滤波器，这完全避免了计算特征分解。简化后，逐层更新变为：</p>
</li>
</ul>
<div class="arithmatex">\[H^{(l+1)} = \sigma\left(\hat{A} H^{(l)} W^{(l)}\right)\]</div>
<ul>
<li>
<p>其中：</p>
<ul>
<li><span class="arithmatex">\(H^{(l)} \in \mathbb{R}^{n \times d}\)</span> 是第 <span class="arithmatex">\(l\)</span> 层的节点特征矩阵</li>
<li><span class="arithmatex">\(W^{(l)} \in \mathbb{R}^{d \times d'}\)</span> 是可学习的权重矩阵</li>
<li><span class="arithmatex">\(\hat{A} = \tilde{D}^{-1/2} \tilde{A} \tilde{D}^{-1/2}\)</span> 是带自环的对称归一化邻接矩阵</li>
<li><span class="arithmatex">\(\tilde{A} = A + I\)</span> 添加了自环（因此每个节点也接收自己的消息）</li>
<li><span class="arithmatex">\(\tilde{D}\)</span> 是 <span class="arithmatex">\(\tilde{A}\)</span> 的度矩阵</li>
<li><span class="arithmatex">\(\sigma\)</span> 是一个非线性激活函数（ReLU，如第6章所述）</li>
</ul>
</li>
<li>
<p>矩阵乘法 <span class="arithmatex">\(\hat{A} H^{(l)}\)</span> 是聚合步骤：对于每个节点，它计算其邻居特征（加上自身特征，通过自环）的加权平均。权重矩阵 <span class="arithmatex">\(W^{(l)}\)</span> 是可学习的变换，在所有节点间共享。激活函数增加了非线性。</p>
</li>
<li>
<p>这非常简单：它只是矩阵乘法后接一个学习到的线性映射和激活函数。整个GCN层可以用一行代码实现。通过 <span class="arithmatex">\(\tilde{D}^{-1/2}\)</span> 的归一化防止具有许多邻居的节点占主导地位：高度节点的消息被按比例缩小。</p>
</li>
<li>
<p>在消息传递框架中，GCN使用：</p>
<ul>
<li>消息：<span class="arithmatex">\(\phi(\mathbf{h}_j) = \mathbf{h}_j\)</span>（只发送你的特征）</li>
<li>聚合：归一化和（按度加权）</li>
<li>更新：线性变换 + 激活函数</li>
</ul>
</li>
</ul>
<h2 id="graphsage">GraphSAGE<a class="headerlink" href="#graphsage" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>GCN是<strong>直推式</strong>的：它在训练时需要完整的图，无法处理新出现的未知节点。如果新用户加入社交网络，GCN必须对整个图重新训练。<strong>GraphSAGE</strong>（Hamilton等，2017）通过<strong>归纳式</strong>方法解决了这个问题。</p>
</li>
<li>
<p>关键思想是<strong>邻域采样</strong>：不是使用所有邻居，而是采样一个固定大小的子集。这使得计算独立于完整的图结构，并允许推广到未见过的节点和图。</p>
</li>
<li>
<p>节点 <span class="arithmatex">\(i\)</span> 的GraphSAGE更新：</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{h}_i^{(l+1)} = \sigma\left(W^{(l)} \cdot \text{CONCAT}\left(\mathbf{h}_i^{(l)}, \text{AGG}\left(\{\mathbf{h}_j^{(l)} : j \in \mathcal{S}(i)\}\right)\right)\right)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(\mathcal{S}(i)\)</span> 是一个<strong>采样</strong>的邻居子集（例如，从500个邻居中随机采样10个）。CONCAT操作显式地将节点自身的特征与聚合后的邻居特征分开，让网络学习"自身"和"邻域"的不同变换。</p>
</li>
<li>
<p>GraphSAGE支持多种聚合函数：</p>
<ul>
<li><strong>均值（Mean）</strong>：<span class="arithmatex">\(\text{AGG} = \frac{1}{|\mathcal{S}|} \sum_{j \in \mathcal{S}} \mathbf{h}_j\)</span>（简单，有效）</li>
<li><strong>LSTM</strong>：将采样的邻居通过LSTM（但这引入了顺序依赖，一定程度上违反了置换不变性）</li>
<li><strong>池化（Pool）</strong>：<span class="arithmatex">\(\text{AGG} = \max(\{\sigma(W_{\text{pool}} \mathbf{h}_j + \mathbf{b})\})\)</span>（非线性变换后取最大值）</li>
</ul>
</li>
<li>
<p>采样策略使GraphSAGE可扩展到非常大的图。训练使用节点的小批量：对于每个目标节点，在第1层采样 <span class="arithmatex">\(k_1\)</span> 个邻居，然后对于其中每个邻居在第2层采样 <span class="arithmatex">\(k_2\)</span> 个邻居。使用 <span class="arithmatex">\(k_1 = k_2 = 10\)</span> 和2层，每个节点的计算树最多有 <span class="arithmatex">\(10 \times 10 = 100\)</span> 个节点，与图的大小无关。</p>
</li>
</ul>
<h2 id="gin">图同构网络（GIN）<a class="headerlink" href="#gin" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>不同的GNN架构具有不同的<strong>表达能力</strong>：它们区分结构不同之图的能力。GCN和GraphSAGE虽然在实践中有效，但理论上在能区分哪些图结构方面是受限的。</p>
</li>
<li>
<p>衡量GNN表达能力的理论工具是<strong>Weisfeiler-Lehman（WL）测试</strong>，这是一个用于测试图同构（两个图是否结构相同）的经典算法。WL测试通过将每个节点的标签与其邻居标签的多重集一起哈希，迭代地精炼节点标签。</p>
</li>
<li>
<p><strong>GIN</strong>（Xu等，2019）被设计为具有与WL测试同等的表达能力，使其成为最强大的消息传递GNN（在消息传递的理论限制内）。关键洞察：聚合函数必须在多重集上是<strong>单射</strong>的（不同的邻居特征多重集必须产生不同的聚合值）。</p>
</li>
<li>
<p>求和聚合在多重集上是单射的（求和 <span class="arithmatex">\(\{1, 1, 2\}\)</span> 得到4，而 <span class="arithmatex">\(\{1, 3\}\)</span> 也得到4，但在具有足够维度的特征向量上，不同多重集的和一般而言是不同的）。均值和取最大值不是单射的：均值无法区分 <span class="arithmatex">\(\{1, 1\}\)</span> 和 <span class="arithmatex">\(\{2, 2\}\)</span>，取最大值无法区分 <span class="arithmatex">\(\{1, 2, 3\}\)</span> 和 <span class="arithmatex">\(\{1, 1, 3\}\)</span>。</p>
</li>
<li>
<p>GIN更新：</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{h}_i^{(l+1)} = \text{MLP}^{(l)}\left((1 + \epsilon^{(l)}) \cdot \mathbf{h}_i^{(l)} + \sum_{j \in \mathcal{N}(i)} \mathbf{h}_j^{(l)}\right)\]</div>
<ul>
<li>其中 <span class="arithmatex">\(\epsilon\)</span> 是一个可学习的标量（或固定为0），MLP提供非线性、单射的映射。求和聚合保留了多重集结构，MLP可以学会区分任意两个不同的聚合值。</li>
</ul>
<h2 id="_3">过平滑<a class="headerlink" href="#_3" title="Permanent link">&para;</a></h2>
<ul>
<li>GNN的一个主要挑战是<strong>过平滑</strong>：随着层数增加，所有节点表示收敛到相同的值，失去区分不同节点的能力。</li>
</ul>
<p><img alt="过平滑：在第1层各不相同的节点特征在更深层逐渐融合为统一特征" src="../../images/over_smoothing_gnn.svg" /></p>
<ul>
<li>
<p>其机制是直观的。每个消息传递层将节点的特征与其邻居的特征进行平均。经过多轮平均后，每个节点已经"看到"（并混合了）其连通分量中的每个其他节点。这些特征变成了统一的平均值，相当于将图像模糊太多次直到变成纯色的图类比。</p>
</li>
<li>
<p>形式上，重复应用归一化邻接矩阵 <span class="arithmatex">\(\hat{A}\)</span> 收敛到一个秩为1的矩阵（每一行都变得与图上随机游走的平稳分布成正比）。这与幂迭代收敛到主特征向量的过程相同（第2章）。</p>
</li>
<li>
<p>过平滑将GNN限制在很浅的深度（通常2-4层），而CNN和Transformer可以从几十或数百层中受益。这意味着每个节点只能看到有限的邻域，这对于需要长距离信息的任务来说是有问题的。</p>
</li>
<li>
<p>缓解方法包括：</p>
<ul>
<li><strong>残差连接</strong>（来自ResNet，第8章）：<span class="arithmatex">\(\mathbf{h}_i^{(l+1)} = \mathbf{h}_i^{(l+1)} + \mathbf{h}_i^{(l)}\)</span>，保留来自较早层的信息。</li>
<li><strong>跳跃知识（Jumping Knowledge）</strong>：拼接或注意力池化来自所有层的表示，而不仅仅是最后一层。</li>
<li><strong>DropEdge</strong>：训练期间随机移除边，减缓信息传播。</li>
<li><strong>图Transformer（Graph Transformer）</strong>（文件4）：用全局注意力绕过局部消息传递的瓶颈。</li>
</ul>
</li>
</ul>
<h2 id="_4">图池化<a class="headerlink" href="#_4" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>对于<strong>图级别任务</strong>（预测整个图的属性，如分子的毒性），我们需要将所有节点表示折叠成一个单一的图级别向量。这就是<strong>图池化</strong>，是CNN中全局平均池化的图类比（第8章）。</p>
</li>
<li>
<p>最简单的方法是<strong>读出（readout）</strong>：对所有节点特征应用一个置换不变函数：</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{h}_G = \text{READOUT}(\{\mathbf{h}_i^{(L)} : i \in V\}) = \sum_i \mathbf{h}_i^{(L)} \quad \text{或} \quad \frac{1}{|V|} \sum_i \mathbf{h}_i^{(L)} \quad \text{或} \quad \max_i \mathbf{h}_i^{(L)}\]</div>
<ul>
<li>
<p>这就是文件1中的DeepSets聚合，应用于最终的GNN层之后。求和保留了大小信息（一个有100个节点的图会比只有10个节点的图具有更大的和），而均值对大小进行了归一化。</p>
</li>
<li>
<p><strong>分层池化</strong>逐步粗化图，模仿CNN逐步下采样图像的方式。在每个层级，节点组被合并为"超节点"：</p>
</li>
<li>
<p><strong>DiffPool</strong>（可微分池化）学习一个软分配矩阵 <span class="arithmatex">\(S^{(l)} \in \mathbb{R}^{n_l \times n_{l+1}}\)</span>，将每个节点分配到一个簇：</p>
</li>
</ul>
<div class="arithmatex">\[X^{(l+1)} = S^{(l)T} H^{(l)}, \quad A^{(l+1)} = S^{(l)T} A^{(l)} S^{(l)}\]</div>
<ul>
<li>
<p>分配矩阵由一个单独的GNN预测，使聚类变得端到端可微分。这创建了一个层次结构：原始图 → 具有较少节点的粗化图 → 更粗的图 → 单个节点（图表示）。</p>
</li>
<li>
<p><strong>TopKPool</strong>采用更简单的方法：为每个节点学习一个标量分数，保留得分最高的 top-<span class="arithmatex">\(k\)</span> 个节点，丢弃其余节点。这是一种硬选择（而非软分配），计算上比DiffPool更廉价。</p>
</li>
</ul>
<h2 id="_5">异构图<a class="headerlink" href="#_5" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>截至目前的所有GNN都假设一个<strong>同构图</strong>：一种节点类型，一种边类型。但大多数现实世界的图是<strong>异构</strong>的：多种节点类型和多种边类型。知识图谱有人物节点、组织节点和位置节点，由"工作于"、"出生于"和"位于"边连接。推荐系统有用户节点和物品节点，由"已购买"、"已浏览"和"已评价"边连接。</p>
</li>
<li>
<p>异构图有一个<strong>模式</strong>（也称为元图），定义了允许的节点类型和边类型。每个边类型连接特定的源类型到特定的目标类型。例如，"工作于"连接 Person → Organisation。</p>
</li>
<li>
<p><strong>关系GCN（R-GCN）</strong>（Schlichtkrull等，2018）通过为每种边类型使用单独的权重矩阵来处理异构边：</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{h}_i^{(l+1)} = \sigma\left(\sum_{r \in \mathcal{R}} \sum_{j \in \mathcal{N}_r(i)} \frac{1}{|\mathcal{N}_r(i)|} W_r^{(l)} \mathbf{h}_j^{(l)} + W_0^{(l)} \mathbf{h}_i^{(l)}\right)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(\mathcal{R}\)</span> 是边类型的集合，<span class="arithmatex">\(\mathcal{N}_r(i)\)</span> 是通过关系 <span class="arithmatex">\(r\)</span> 连接到节点 <span class="arithmatex">\(i\)</span> 的邻居集合，<span class="arithmatex">\(W_r\)</span> 是关系 <span class="arithmatex">\(r\)</span> 特有的权重矩阵。自连接 <span class="arithmatex">\(W_0\)</span> 单独处理节点自身的特征。</p>
</li>
<li>
<p>问题：当关系类型很多时，参数数量爆炸（每种关系一个 <span class="arithmatex">\(d \times d\)</span> 矩阵）。R-GCN通过<strong>基分解</strong>缓解这一问题：<span class="arithmatex">\(W_r = \sum_{b=1}^{B} a_{rb} V_b\)</span>，其中 <span class="arithmatex">\(V_b\)</span> 是共享的基矩阵，<span class="arithmatex">\(a_{rb}\)</span> 是每个关系的标量系数。这类似于低秩分解（第2章）：关系特定的矩阵生活在一个低维子空间中。</p>
</li>
<li>
<p><strong>异构图表Transformer（HGT）</strong>（Hu等，2020）将注意力机制应用于异构图。关键洞察：注意力应同时依赖于节点类型和连接它们的边类型。HGT为查询、键和值使用类型特定的投影矩阵：</p>
</li>
</ul>
<div class="arithmatex">\[\text{Attention}(i, j) = \left(W_{\tau(i)}^Q \mathbf{h}_i\right)^T \cdot \frac{W_{\phi(i,j)}^{\text{ATT}}}{\sqrt{d}} \cdot \left(W_{\tau(j)}^K \mathbf{h}_j\right)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(\tau(i)\)</span> 是节点 <span class="arithmatex">\(i\)</span> 的类型，<span class="arithmatex">\(\phi(i,j)\)</span> 是它们之间的边类型。这确保了模型对不同的关系类型使用不同的注意力权重：一篇论文关注其作者时，应使用与关注其参考文献时不同的注意力权重。</p>
</li>
<li>
<p><strong>基于元路径的方法</strong>定义通过模式的含义路径（例如，作者 → 论文 → 作者表示合著关系），并沿着这些路径聚合信息。<strong>HAN</strong>（异构图注意力网络）在两个层次应用注意力：在每个元路径内（沿此路径哪些邻居重要？）和跨元路径（哪些关系模式重要？）。</p>
</li>
</ul>
<h2 id="_6">链接预测与知识图谱补全<a class="headerlink" href="#_6" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p><strong>链接预测</strong>提出的问题是：给定现有边，哪些缺失的边可能存在？这是知识图谱补全（预测缺失的事实）、推荐（预测用户会喜欢哪些物品）和社交网络分析（预测未来的友谊）的核心任务。</p>
</li>
<li>
<p><strong>基于嵌入的方法</strong>为每个实体学习一个向量，为每个关系学习一个变换，然后通过实体和关系的匹配程度对潜在边进行评分：</p>
</li>
<li>
<p><strong>TransE</strong>将关系建模为嵌入空间中的平移：如果 <span class="arithmatex">\((h, r, t)\)</span> 是一个有效的三元组（头实体，关系，尾实体），那么 <span class="arithmatex">\(\mathbf{h} + \mathbf{r} \approx \mathbf{t}\)</span>。评分函数为 <span class="arithmatex">\(f(h, r, t) = -\|\mathbf{h} + \mathbf{r} - \mathbf{t}\|\)</span>。直观地说，关系向量在嵌入空间中将头实体"移动"到尾实体。</p>
</li>
<li>
<p><strong>RotatE</strong>将关系建模为复空间中的旋转：<span class="arithmatex">\(\mathbf{t} = \mathbf{h} \circ \mathbf{r}\)</span>，其中 <span class="arithmatex">\(\circ\)</span> 是逐元素复数乘法，<span class="arithmatex">\(|\mathbf{r}_i| = 1\)</span>（单位复数就是旋转）。这可以建模TransE无法处理的对称性、反对称性、反转和复合模式。</p>
</li>
<li>
<p><strong>ComplEx</strong>使用复数值嵌入和埃尔米特点积，使其能够建模非对称关系（如果A是B的老板，B不是A的老板）。</p>
</li>
<li>
<p>基于GNN的链接预测通过消息传递计算节点嵌入，然后使用端点嵌入对边进行评分。这结合了GNN的结构推理能力和嵌入方法的关系建模能力。GNN编码器捕获了单嵌入方法所遗漏的多跳邻域结构。</p>
</li>
</ul>
<h2 id="_7">任务类型<a class="headerlink" href="#_7" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>GNN解决三类任务：</p>
</li>
<li>
<p><strong>节点级别任务</strong>：为每个节点预测一个属性。示例：对社交网络中的用户进行分类（机器人还是人类），预测相互作用网络中每个蛋白质的功能，半监督节点分类（标记少数节点，预测其余节点）。输出是节点嵌入 <span class="arithmatex">\(\mathbf{h}_i^{(L)}\)</span> 经过一个分类器。</p>
</li>
<li>
<p><strong>边级别任务</strong>：为每条边预测一个属性或预测边是否存在。示例：链接预测（这两个用户会成为朋友吗？），知识图谱补全（这个关系在这些实体间成立吗？），药物-药物相互作用预测。输出通常使用两个端点节点的嵌入：<span class="arithmatex">\(\hat{y}_{ij} = f(\mathbf{h}_i, \mathbf{h}_j)\)</span>，其中 <span class="arithmatex">\(f\)</span> 是点积、拼接+MLP或其他组合。</p>
</li>
<li>
<p><strong>图级别任务</strong>：为整个图预测一个属性。示例：分子性质预测（这个分子有毒吗？），图分类（这个社交网络是机器人网络吗？），图生成（设计一个具有期望性质的分子）。输出使用图池化产生 <span class="arithmatex">\(\mathbf{h}_G\)</span>，然后进行分类或回归。</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>使用归一化邻接矩阵从头实现一个单层GCN。应用于一个小型图，观察节点特征如何被平滑。
<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</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></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-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># 图：5个节点，简单链带分支</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">0</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">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> <span class="mi">1</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">0</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">1</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="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"># 添加自环</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="n">A_hat</span> <span class="o">=</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>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="n">D_hat</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_hat</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-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="n">D_inv_sqrt</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="mf">1.0</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">A_hat</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-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="n">A_norm</span> <span class="o">=</span> <span class="n">D_inv_sqrt</span> <span class="o">@</span> <span class="n">A_hat</span> <span class="o">@</span> <span class="n">D_inv_sqrt</span>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="c1"># 节点特征：one-hot 单位阵</span>
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="n">H</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>
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="c1"># 权重矩阵（随机初始化）</span>
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="n">rng</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</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="n">W</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">rng</span><span class="p">,</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span> <span class="o">*</span> <span class="mf">0.5</span>
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="c1"># GCN层：H&#39; = ReLU(A_norm @ H @ W)</span>
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="n">H_new</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="n">A_norm</span> <span class="o">@</span> <span class="n">H</span> <span class="o">@</span> <span class="n">W</span><span class="p">)</span>
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>
<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;原始特征（one-hot）:&quot;</span><span class="p">)</span>
<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a><span class="nb">print</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">经过GCN层后:&quot;</span><span class="p">)</span>
<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a><span class="nb">print</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">H_new</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">注意：连接的节点现在具有相似的表示&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>实现具有求和聚合（GIN风格）和均值聚合（GCN风格）的消息传递。展示求和能区分均值无法区分的多重集。
<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="c1"># 两个具有相同均值的不同邻居多重集</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="c1"># 节点A：邻居特征为 [1, 1, 1, 1] （四个邻居，都是1）</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="c1"># 节点B：邻居特征为 [2, 2] （两个邻居，都是2）</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">neighbours_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="mf">1.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">]])</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">neighbours_B</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">2.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.0</span><span class="p">]])</span>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="c1"># 均值聚合</span>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="n">mean_A</span> <span class="o">=</span> <span class="n">neighbours_A</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="n">mean_B</span> <span class="o">=</span> <span class="n">neighbours_B</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</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;均值 A: </span><span class="si">{</span><span class="n">mean_A</span><span class="si">}</span><span class="s2">, 均值 B: </span><span class="si">{</span><span class="n">mean_B</span><span class="si">}</span><span class="s2">, 相同: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">mean_A</span><span class="p">,</span><span class="w"> </span><span class="n">mean_B</span><span class="p">)</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>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="c1"># 求和聚合</span>
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="n">sum_A</span> <span class="o">=</span> <span class="n">neighbours_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">0</span><span class="p">)</span>
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="n">sum_B</span> <span class="o">=</span> <span class="n">neighbours_B</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">0</span><span class="p">)</span>
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;求和 A: </span><span class="si">{</span><span class="n">sum_A</span><span class="si">}</span><span class="s2">, 求和 B: </span><span class="si">{</span><span class="n">sum_B</span><span class="si">}</span><span class="s2">, 相同: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">sum_A</span><span class="p">,</span><span class="w"> </span><span class="n">sum_B</span><span class="p">)</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">求和能区分这些多重集；均值不能！&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>演示过平滑。重复应用归一化邻接矩阵，观察节点特征收敛。
<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"># 随机图</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">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><span class="mi">0</span><span class="p">],</span>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></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><span class="mi">0</span><span class="p">],</span>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></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><span class="mi">0</span><span class="p">],</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></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><span class="mi">1</span><span class="p">],</span>
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></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="mi">1</span><span class="p">],</span>
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></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">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-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a>
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="n">A_hat</span> <span class="o">=</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">6</span><span class="p">)</span>
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="n">D_inv_sqrt</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="mf">1.0</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">A_hat</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-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="n">A_norm</span> <span class="o">=</span> <span class="n">D_inv_sqrt</span> <span class="o">@</span> <span class="n">A_hat</span> <span class="o">@</span> <span class="n">D_inv_sqrt</span>
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a>
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="c1"># 初始特征：每个节点各不相同</span>
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="n">H</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">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</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="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</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-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a>
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="n">distances</span> <span class="o">=</span> <span class="p">[]</span>
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">):</span>
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a> <span class="n">H</span> <span class="o">=</span> <span class="n">A_norm</span> <span class="o">@</span> <span class="n">H</span>
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a> <span class="c1"># 衡量特征的区别程度（节点间的标准差）</span>
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a> <span class="n">spread</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a> <span class="n">distances</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">spread</span><span class="p">))</span>
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a>
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">distances</span><span class="p">,</span> <span class="s2">&quot;o-&quot;</span><span class="p">)</span>
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;消息传递轮数&quot;</span><span class="p">)</span>
<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">ylabel</span><span class="p">(</span><span class="s2">&quot;特征分散度（节点间标准差）&quot;</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">title</span><span class="p">(</span><span class="s2">&quot;过平滑：特征随深度增加而收敛&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">show</span><span class="p">()</span>
</code></pre></div></p>
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