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<article class="md-content__inner md-typeset">
<h1 id="_1">图注意力网络<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>图注意力网络将均匀的邻居聚合替换为学习到的、依赖数据的加权。本章涵盖GAT、多头图注意力、GATv2、图Transformer、位置和结构编码以及可扩展性</em></p>
<ul>
<li>
<p>在GCN（文件3）中，每个节点使用由图结构确定的固定权重（归一化邻接矩阵）聚合其邻居特征。一个有三个邻居的节点会给每个邻居大致相等的权重（<span class="arithmatex">\(\approx 1/3\)</span>）。但并非所有邻居都同等重要：来自密切合作者的消息应比来自远方熟人的消息更重要。</p>
</li>
<li>
<p><strong>图注意力网络</strong>通过使用与Transformer（第7章）相同的注意力机制来学习<strong>关注哪些邻居</strong>，从而解决了这一问题。与固定的、基于结构的权重不同，每个节点在其邻居上计算动态的、基于内容的注意力分数。</p>
</li>
</ul>
<h2 id="gat">GAT：图注意力网络<a class="headerlink" href="#gat" title="Permanent link">&para;</a></h2>
<ul>
<li><strong>GAT</strong>（Veličković等，2018）计算每个节点与其邻居之间的注意力系数。对于节点 <span class="arithmatex">\(i\)</span> 和邻居 <span class="arithmatex">\(j\)</span>：</li>
</ul>
<div class="arithmatex">\[e_{ij} = \text{LeakyReLU}\left(\mathbf{a}^T \left[W\mathbf{h}_i \| W\mathbf{h}_j\right]\right)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(W \in \mathbb{R}^{d' \times d}\)</span> 是共享的线性变换，<span class="arithmatex">\(\|\)</span> 表示拼接，<span class="arithmatex">\(\mathbf{a} \in \mathbb{R}^{2d'}\)</span> 是可学习的注意力向量。分数 <span class="arithmatex">\(e_{ij}\)</span> 衡量节点 <span class="arithmatex">\(j\)</span> 的特征对节点 <span class="arithmatex">\(i\)</span> 的重要程度。</p>
</li>
<li>
<p>原始分数使用softmax在所有邻居之间进行归一化：</p>
</li>
</ul>
<div class="arithmatex">\[\alpha_{ij} = \text{softmax}_j(e_{ij}) = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal{N}(i)} \exp(e_{ik})}\]</div>
<ul>
<li>这确保了每个节点邻域上的注意力权重之和为1，就像Transformer注意力一样（第7章）。节点更新后的特征为：</li>
</ul>
<div class="arithmatex">\[\mathbf{h}_i' = \sigma\left(\sum_{j \in \mathcal{N}(i)} \alpha_{ij} W\mathbf{h}_j\right)\]</div>
<p><img alt="GCN为所有邻居分配固定的等权重；GAT学习依赖数据的注意力权重" src="../../images/gat_attention_weights.svg" /></p>
<ul>
<li>
<p>与GCN的关键区别：权重 <span class="arithmatex">\(\alpha_{ij}\)</span> 是<strong>从数据中学习</strong>的，而非由图结构固定。节点可以学会关注信息量最大的邻居，同时忽略噪声或无关的邻居。</p>
</li>
<li>
<p>注意，注意力仅在边上计算（节点 <span class="arithmatex">\(i\)</span> 只关注其邻居 <span class="arithmatex">\(\mathcal{N}(i)\)</span>），而不是在所有节点对之间。这使得计算量与边的数量成正比，而不是节点数的平方。</p>
</li>
</ul>
<h2 id="_2">多头图注意力<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h2>
<ul>
<li>正如在Transformer中（第7章），<strong>多头注意力</strong>并行运行 <span class="arithmatex">\(K\)</span> 个独立的注意力机制，每个都有自己的参数 <span class="arithmatex">\(W^k\)</span> 和 <span class="arithmatex">\(\mathbf{a}^k\)</span>。结果在中间层进行拼接，在最终层取平均：</li>
</ul>
<div class="arithmatex">\[\mathbf{h}_i' = \Big\|_{k=1}^{K} \sigma\left(\sum_{j \in \mathcal{N}(i)} \alpha_{ij}^k W^k \mathbf{h}_j\right)\]</div>
<ul>
<li>
<p>每个头可以关注邻域的不同方面：一个头可能关注结构特征，另一个关注语义相似性。这与Transformer中多头注意力的动机相同：不同的头捕获不同类型的关系。</p>
</li>
<li>
<p>使用 <span class="arithmatex">\(K\)</span> 个头和每个头输出维度 <span class="arithmatex">\(d'\)</span>，拼接后的输出维度为 <span class="arithmatex">\(K \times d'\)</span>。最后一层通常使用平均而不是拼接来产生固定大小的输出。</p>
</li>
</ul>
<h2 id="gatv2">GATv2：修复静态注意力<a class="headerlink" href="#gatv2" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>原始GAT有一个微妙的限制：其注意力函数是<strong>静态的</strong>（也称为基于排序的）。注意力分数取决于拼接 <span class="arithmatex">\([W\mathbf{h}_i \| W\mathbf{h}_j]\)</span>，但由于注意力向量 <span class="arithmatex">\(\mathbf{a}\)</span> 在拼接之后应用，它可以分解为两个独立的分量：<span class="arithmatex">\(\mathbf{a}^T [W\mathbf{h}_i \| W\mathbf{h}_j] = \mathbf{a}_1^T W\mathbf{h}_i + \mathbf{a}_2^T W\mathbf{h}_j\)</span>。</p>
</li>
<li>
<p>这意味着对于给定节点 <span class="arithmatex">\(i\)</span>，邻居的排序完全由邻居的特征 <span class="arithmatex">\(\mathbf{h}_j\)</span> 决定（项 <span class="arithmatex">\(\mathbf{a}_1^T W\mathbf{h}_i\)</span> 在 <span class="arithmatex">\(i\)</span> 的所有邻居中是常数）。注意力排名并不真正依赖于查询节点的特征。节点 <span class="arithmatex">\(i\)</span> 和节点 <span class="arithmatex">\(k\)</span> 将以完全相同的方式对同一组邻居进行排序，这限制了表达能力。</p>
</li>
<li>
<p><strong>GATv2</strong>（Brody等，2022）通过在注意力向量之前应用非线性函数来修复这个问题：</p>
</li>
</ul>
<div class="arithmatex">\[e_{ij} = \mathbf{a}^T \text{LeakyReLU}\left(W \left[\mathbf{h}_i \| \mathbf{h}_j\right]\right)\]</div>
<ul>
<li>将LeakyReLU移到计算内部意味着注意力分数是联合特征的非线性函数，不能分解为独立项。这使得注意力变为<strong>动态</strong>：邻居的排序现在依赖于特定的查询节点。GATv2严格比GAT更具表达能力，且没有额外的计算成本。</li>
</ul>
<h2 id="transformer">图Transformer<a class="headerlink" href="#transformer" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>标准消息传递GNN受到图拓扑的限制：一个节点只能关注其直接邻居。经过 <span class="arithmatex">\(k\)</span> 层后，来自 <span class="arithmatex">\(k\)</span> 跳邻居的信息已通过多个聚合步骤混合，失去了保真度。这种局部瓶颈（再加上文件3中的过平滑）限制了捕获长距离依赖关系的能力。</p>
</li>
<li>
<p><strong>图Transformer</strong>通过将<strong>全局自注意力</strong>应用于所有节点对（无论它们之间是否有边）来突破这个瓶颈。每个节点可以在单层中关注每个其他节点，就像标准Transformer一样（第7章）。</p>
</li>
<li>
<p>基本思想：将所有节点视为标记（token），应用Transformer自注意力：</p>
</li>
</ul>
<div class="arithmatex">\[\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(Q = XW_Q\)</span>，<span class="arithmatex">\(K = XW_K\)</span>，<span class="arithmatex">\(V = XW_V\)</span> 是节点特征 <span class="arithmatex">\(X\)</span> 的查询、键和值投影（与第7章完全相同）。这是完全连接图（完全图 <span class="arithmatex">\(K_n\)</span>，文件2）上的GNN。</p>
</li>
<li>
<p>问题：完全连接图忽略了实际的图结构。边信息（谁实际连接到谁）丢失了。两种方法恢复了这一点：</p>
</li>
<li>
<p><strong>Graphormer</strong>（Ying等，2021）通过注意力分数中的<strong>偏置项</strong>将图结构注入Transformer：</p>
</li>
</ul>
<div class="arithmatex">\[A_{ij} = \frac{(\mathbf{h}_i W_Q)(W_K^T \mathbf{h}_j^T)}{\sqrt{d_k}} + b_{\text{spatial}}(i, j) + b_{\text{edge}}(i, j)\]</div>
<ul>
<li>
<p>空间偏置 <span class="arithmatex">\(b_{\text{spatial}}\)</span> 编码节点 <span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 之间的最短路径距离。边偏置 <span class="arithmatex">\(b_{\text{edge}}\)</span> 编码沿最短路径的边特征。此外，Graphormer使用<strong>中心性编码</strong>，将节点的度数添加到其输入嵌入中，为模型提供关于每个节点结构角色的信息。</p>
</li>
<li>
<p><strong>GPS</strong>（通用、强大、可扩展的图Transformer，Rampášek等，2022）在每一层中结合了局部消息传递和全局注意力：</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{h}_i' = \text{MLP}\left(\mathbf{h}_i^{\text{MPNN}} + \mathbf{h}_i^{\text{Attention}}\right)\]</div>
<ul>
<li>每一层同时应用标准GNN（用于局部结构）和Transformer（用于全局上下文），然后组合结果。这获得了两个世界的优点：来自消息传递的局部结构和来自注意力的长距离依赖关系。</li>
</ul>
<h2 id="_3">位置编码与结构编码<a class="headerlink" href="#_3" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>序列上的Transformer使用位置编码（第7章）来注入顺序信息。图没有规范的顺序，因此需要特定于图的编码。</p>
</li>
<li>
<p><strong>拉普拉斯特征向量编码</strong>使用图拉普拉斯算子（文件2）的特征向量作为位置特征。<span class="arithmatex">\(k\)</span> 个最小的非平凡特征向量提供了图的谱嵌入：在图中"附近"的节点具有相似的特征向量值。这些被拼接到节点特征中。</p>
</li>
<li>
<p>一个微妙之处：拉普拉斯特征向量有符号模糊性（如果 <span class="arithmatex">\(\mathbf{u}\)</span> 是特征向量，<span class="arithmatex">\(-\mathbf{u}\)</span> 也是）。模型必须对这些符号翻转保持不变。解决方案包括在训练期间使用随机符号翻转作为数据增强，或学习符号不变的变换。</p>
</li>
<li>
<p><strong>随机游走编码</strong>计算从节点 <span class="arithmatex">\(i\)</span> 开始的随机游走经过 <span class="arithmatex">\(k\)</span> 步后返回节点 <span class="arithmatex">\(i\)</span> 的概率，对于 <span class="arithmatex">\(k = 1, 2, \ldots, K\)</span>。这些概率编码了局部结构信息：密集簇中的节点具有高的返回概率，而稀疏区域中的节点返回概率低。着陆概率 <span class="arithmatex">\(p_{ii}^{(k)} = (A_{\text{rw}}^k)_{ii}\)</span>，其中 <span class="arithmatex">\(A_{\text{rw}} = D^{-1}A\)</span> 是随机游走转移矩阵。</p>
</li>
<li>
<p><strong>度数编码</strong>简单地将节点度数作为一个特征添加。这出奇地有效，因为度数是一个强大的结构信号：叶节点（度数为1）、桥接节点和枢纽节点的行为不同。</p>
</li>
<li>
<p>这些编码提供了普通Transformer所缺乏的结构信息，使图Transformer在需要长距离推理的任务上能够超越标准消息传递GNN。</p>
</li>
</ul>
<h2 id="_4">可扩展性<a class="headerlink" href="#_4" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>GNN的基本可扩展性挑战在于图可能拥有数百万个节点和数十亿条边。在完整图上训练GNN需要将所有节点特征和整个邻接矩阵存储在内存中，这通常是不可行的。</p>
</li>
<li>
<p>GNN的<strong>小批量训练</strong>比图像或序列更复杂，因为节点之间是相互连接的。朴素地采样一批节点需要它们的邻居（第1层）、邻居的邻居（第2层），依此类推。这种<strong>邻域爆炸</strong>意味着一个包含1000个目标节点的小批量可能需要计算图中数百万个节点。</p>
</li>
<li>
<p><strong>邻域采样</strong>（GraphSAGE风格，文件3）通过每层每个节点采样固定数量的邻居来限制爆炸。使用2层和每层15个样本，每个目标节点的子图最多有 <span class="arithmatex">\(15^2 = 225\)</span> 个节点，与完整图的大小无关。</p>
</li>
<li>
<p><strong>Cluster-GCN</strong>（Chiang等，2019）使用图聚类算法（例如METIS）将图划分为簇，然后一次在一个簇上训练。簇内边是密集的（大多数邻居在同一个簇内），因此子图捕获了相关结构。跨簇边通过偶尔包含簇之间的边来处理。</p>
</li>
<li>
<p><strong>图Transformer的可扩展性</strong>更困难，因为全局注意力是 <span class="arithmatex">\(O(n^2)\)</span> 的。对于具有数百万个节点的图，完整的注意力是不可行的。解决方案包括：</p>
<ul>
<li>稀疏注意力模式（只关注图中距离最近的 <span class="arithmatex">\(k\)</span> 个节点）</li>
<li>线性注意力近似</li>
<li>将局部消息传递（廉价，<span class="arithmatex">\(O(|E|)\)</span>）与粗化图上的全局注意力（更少的节点）相结合</li>
</ul>
</li>
</ul>
<h2 id="_5">时序图与动态图<a class="headerlink" href="#_5" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>我们迄今为止研究的图是<strong>静态</strong>的：节点、边和特征都是固定的。但许多现实世界的图会<strong>随时间演化</strong>：新用户加入社交网络、金融交易创建边、交通模式全天变化、分子相互作用发生波动。</p>
</li>
<li>
<p><strong>时序图</strong>为每条边增加一个时间戳：<span class="arithmatex">\((i, j, t)\)</span> 表示节点 <span class="arithmatex">\(i\)</span> 在时间 <span class="arithmatex">\(t\)</span> 与节点 <span class="arithmatex">\(j\)</span> 发生了交互。挑战在于学习同时捕获图结构和时序动态的表示。</p>
</li>
<li>
<p>存在两种范式：</p>
</li>
<li>
<p><strong>离散时间动态图（DTDG）</strong>：图被表示为一系列快照 <span class="arithmatex">\(G_1, G_2, \ldots, G_T\)</span>，每个时间步一个。GNN处理每个快照，RNN或时序注意力机制捕获快照间的演化。这很简单，但丢失了精细的时间信息（快照之间的事件丢失了），并且需要选择快照频率。</p>
</li>
<li>
<p><strong>连续时间动态图（CTDG）</strong>：事件被建模为带时间戳的交互流。每个事件 <span class="arithmatex">\((i, j, t)\)</span> 在其发生的准确时间更新节点 <span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 的表示。这保留了所有时序信息。</p>
</li>
<li>
<p><strong>时序图网络（TGN）</strong>（Rossi等，2020）是领先的CTDG架构。每个节点维护一个<strong>记忆状态</strong> <span class="arithmatex">\(\mathbf{s}_i(t)\)</span>，每当节点参与交互时更新：</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{s}_i(t^+) = \text{GRU}\left(\mathbf{s}_i(t^-), \; \mathbf{m}_i(t)\right)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(\mathbf{m}_i(t)\)</span> 是从交互中计算出的消息（结合了两个节点的特征、边特征和时间编码）。GRU（第6章）选择性地保留和遗忘过去的信息，使记忆能够捕获长期模式，同时适应近期事件。</p>
</li>
<li>
<p><strong>时间编码</strong>表示自上次交互以来经过的时间，类似于Transformer中的位置编码（第7章）。常用方法使用可学习的傅里叶特征：</p>
</li>
</ul>
<div class="arithmatex">\[\Phi(t) = \left[\cos(\omega_1 t), \sin(\omega_1 t), \ldots, \cos(\omega_d t), \sin(\omega_d t)\right]\]</div>
<ul>
<li>
<p>这为模型提供了时间间隔的丰富表示："该用户上次活跃是5分钟前"与"3个月前"以不同的方式嵌入。</p>
</li>
<li>
<p><strong>时序图注意力（TGAT）</strong>在节点的时间邻域上应用自注意力：一组最近的交互，每个交互同时按特征相关性（如GAT）和时间近度加权。来自遥远过去的交互自然地被降低权重。</p>
</li>
<li>
<p>应用包括欺诈检测（金融图中的异常交易模式）、交通预测（从历史流量模式预测拥堵）、社交网络动态（预测病毒内容传播）以及随时间推移的药物相互作用预测。</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>从头实现一个单头GAT注意力。计算节点与其邻居之间的注意力权重，并验证权重之和为1。
<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="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-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">k1</span><span class="p">,</span> <span class="n">k2</span><span class="p">,</span> <span class="n">k3</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">split</span><span class="p">(</span><span class="n">rng</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">n_nodes</span><span class="p">,</span> <span class="n">d_in</span><span class="p">,</span> <span class="n">d_out</span> <span class="o">=</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="c1"># 随机节点特征</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">H</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">k1</span><span class="p">,</span> <span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="n">d_in</span><span class="p">))</span>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="c1"># 可学习参数</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></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">k2</span><span class="p">,</span> <span class="p">(</span><span class="n">d_in</span><span class="p">,</span> <span class="n">d_out</span><span class="p">))</span> <span class="o">*</span> <span class="mf">0.5</span>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="n">a</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">k3</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">d_out</span><span class="p">,))</span> <span class="o">*</span> <span class="mf">0.5</span>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="c1"># 邻接（节点0连接到1, 2, 3）</span>
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="n">neighbours_of_0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</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"># 变换特征</span>
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="n">Wh</span> <span class="o">=</span> <span class="n">H</span> <span class="o">@</span> <span class="n">W</span> <span class="c1"># (n_nodes, d_out)</span>
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="c1"># 计算节点0的注意力分数</span>
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="n">h_i</span> <span class="o">=</span> <span class="n">Wh</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="n">scores</span> <span class="o">=</span> <span class="p">[]</span>
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">neighbours_of_0</span><span class="p">:</span>
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a> <span class="n">h_j</span> <span class="o">=</span> <span class="n">Wh</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a> <span class="n">e_ij</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">jnp</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">h_i</span><span class="p">,</span> <span class="n">h_j</span><span class="p">]))</span>
<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a> <span class="n">e_ij</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">leaky_relu</span><span class="p">(</span><span class="n">e_ij</span><span class="p">,</span> <span class="n">negative_slope</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a> <span class="n">scores</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">e_ij</span><span class="p">))</span>
<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a>
<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a><span class="n">scores</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="n">scores</span><span class="p">)</span>
<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="n">alpha</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">softmax</span><span class="p">(</span><span class="n">scores</span><span class="p">)</span>
<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a>
<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;原始分数: </span><span class="si">{</span><span class="n">scores</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;注意力权重: </span><span class="si">{</span><span class="n">alpha</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;权重之和: </span><span class="si">{</span><span class="n">alpha</span><span class="o">.</span><span class="n">sum</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-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a>
<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a><span class="c1"># 加权聚合</span>
<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a><span class="n">h_new</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">alpha</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">*</span> <span class="n">Wh</span><span class="p">[</span><span class="n">neighbours_of_0</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span> <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="nb">len</span><span class="p">(</span><span class="n">neighbours_of_0</span><span class="p">)))</span>
<a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;更新后的节点0特征: </span><span class="si">{</span><span class="n">h_new</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>比较GCN（固定权重）和GAT（学习权重）的聚合。展示GAT可以为邻居分配不同的权重，而GCN统一对待它们。
<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</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></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-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="c1"># 4个节点：节点0连接到1, 2, 3</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></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">1</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">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-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></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">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></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">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-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"># 特征：节点1非常相关，节点2是噪声，节点3中等</span>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></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="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">],</span> <span class="c1"># 节点0</span>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">],</span> <span class="c1"># 节点1（信号）</span>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a> <span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">],</span> <span class="c1"># 节点2（噪声）</span>
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">]])</span> <span class="c1"># 节点3（中等）</span>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a>
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="c1"># GCN：归一化邻接权重</span>
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></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">4</span><span class="p">)</span>
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="n">D_inv</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">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-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="n">gcn_weights</span> <span class="o">=</span> <span class="p">(</span><span class="n">D_inv</span> <span class="o">@</span> <span class="n">A_hat</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="c1"># 节点0的权重</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;GCN中节点0的权重: </span><span class="si">{</span><span class="n">gcn_weights</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="nb">print</span><span class="p">(</span><span class="s2">&quot; → 所有邻居获得大致相等的权重&quot;</span><span class="p">)</span>
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a>
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="c1"># GAT：学习到的注意力（模拟）</span>
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="c1"># 假设注意力机制学会关注节点1</span>
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="n">gat_weights</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">0.1</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">])</span> <span class="c1"># 学习到的</span>
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">GAT中节点0的权重: </span><span class="si">{</span><span class="n">gat_weights</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="nb">print</span><span class="p">(</span><span class="s2">&quot; → 最具信息量的节点1获得最多关注&quot;</span><span class="p">)</span>
<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a>
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="n">gcn_output</span> <span class="o">=</span> <span class="n">gcn_weights</span> <span class="o">@</span> <span class="n">H</span>
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="n">gat_output</span> <span class="o">=</span> <span class="n">gat_weights</span> <span class="o">@</span> <span class="n">H</span>
<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">GCN输出: </span><span class="si">{</span><span class="n">gcn_output</span><span class="si">}</span><span class="s2"> （被噪声稀释）&quot;</span><span class="p">)</span>
<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;GAT输出: </span><span class="si">{</span><span class="n">gat_output</span><span class="si">}</span><span class="s2"> （聚焦于信号）&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>演示位置编码的益处。计算图的拉普拉斯特征向量编码，展示结构相似的节点获得相似的编码。
<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"># 杠铃图：两个团由一条桥连接</span>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">n</span> <span class="o">=</span> <span class="mi">10</span>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></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="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="c1"># 团1：节点0-4</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></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-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></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-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></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-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="c1"># 团2：节点5-9</span>
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></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-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></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-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></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-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="c1"># 桥</span>
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></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">4</span><span class="p">,</span><span class="mi">5</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">5</span><span class="p">,</span><span class="mi">4</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-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a>
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></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-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></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-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></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-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a>
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="c1"># 使用前3个非平凡特征向量作为位置编码</span>
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="n">pe</span> <span class="o">=</span> <span class="n">eigenvectors</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">:</span><span class="mi">4</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="s2">&quot;拉普拉斯位置编码:&quot;</span><span class="p">)</span>
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></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="n">n</span><span class="p">):</span>
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a> <span class="n">group</span> <span class="o">=</span> <span class="s2">&quot;团1&quot;</span> <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="mi">5</span> <span class="k">else</span> <span class="s2">&quot;团2&quot;</span>
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a> <span class="n">bridge</span> <span class="o">=</span> <span class="s2">&quot; (桥)&quot;</span> <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span> <span class="k">else</span> <span class="s2">&quot;&quot;</span>
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot; 节点 </span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2"> (</span><span class="si">{</span><span class="n">group</span><span class="si">}{</span><span class="n">bridge</span><span class="si">}</span><span class="s2">): </span><span class="si">{</span><span class="n">pe</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a>
<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">scatter</span><span class="p">(</span><span class="n">pe</span><span class="p">[:</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">pe</span><span class="p">[:</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;#3498db&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">80</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;团1&quot;</span><span class="p">)</span>
<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">pe</span><span class="p">[</span><span class="mi">5</span><span class="p">:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">pe</span><span class="p">[</span><span class="mi">5</span><span class="p">:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;#e74c3c&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">80</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;团2&quot;</span><span class="p">)</span>
<a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">pe</span><span class="p">[[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span> <span class="mi">0</span><span class="p">],</span> <span class="n">pe</span><span class="p">[[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">120</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;*&quot;</span><span class="p">,</span>
<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;桥节点&quot;</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span> <span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;拉普拉斯特征向量位置编码&quot;</span><span class="p">)</span>
<a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;特征向量 1&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;特征向量 2&quot;</span><span class="p">)</span>
<a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></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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