超图上的表示学习
定义
超图 (也可以称为无向超图) 可以表示为 \(\mathcal{G} = \{\mathcal{V}, \mathcal{E}\}\)。
\(\mathcal{V}\) 是 顶点 集(也可以称为 节点 或者 点);
\(\mathcal{E} \subseteq \{ \mathcal{P}(\mathcal{V}) \}\) 是 超边 集(也可以称为 边), 其中 \(\mathcal{P}(\mathcal{V})\) 是 \(\mathcal{V}\) 的 幂集 。 每一条超边 \(e \in \mathcal{E}\) 可以包含两个或更多顶点。
图内的边仅可以连接2个顶点，而超边可以连接任意数量的顶点。 然而，通常需要研究每条超边均连接相同个数顶点的超图; k-uniform超图是所有超边连接的顶点数均为k的超图。 (换句话说，这样的超图是一组集合，每个集合包含k个顶点。) 所以2-uniform超图就是图， 3-uniform超图就是无序三元组的集合，依此类推。 无向超图也被称为集合系统或者从全集中提取的集合族。
结构构建
超图的关联结构可以通过以下方法构建。详细参考 这里。
超边列表 (默认)
dhg.Hypergraph使用特征的k近邻
dhg.Hypergraph.from_feature_kNN()从低阶关联结构得到
在如下的例子中，我们随机生成一个超图关联结构和一个特征矩阵，并对此结构进行一些基本的学习操作。
>>> import torch >>> import dhg >>> # Generate a random hypergraph with 5 vertices and 4 hyperedges >>> hg = dhg.random.hypergraph_Gnm(5, 4) >>> # Generate a vertex feature matrix with size 5x2 >>> X = torch.rand(5, 2) >>> # Print information about the hypergraph and feature >>> hg Hypergraph(num_v=5, num_e=4) >>> # Print edges in the hypergraph >>> hg.e[0] [(2, 3), (0, 2, 4), (2, 3, 4), (1, 2, 3, 4)] >>> # Print vertex features >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]])
基于谱域的学习
使用HGNN的拉普拉斯进行平滑
>>> # Print the vertex features befor feautre smoothing >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]]) >>> X_ = hg.smoothing_with_HGNN(X) >>> # Print the vertex features after HGNN-based smoothing >>> X_ tensor([[0.2257, 0.4890], [0.3745, 0.3443], [0.5411, 0.7403], [0.4945, 0.5725], [0.4888, 0.6728]])
基于空域的学习
从顶点到超边的消息传递
>>> # Print the vertex messages >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]]) >>> # Message propagation from vertex to hyperedge >>> Y_ = hg.v2e(X, aggr="mean") >>> # Print the new hyperedge messages >>> Y_ tensor([[0.4098, 0.5702], [0.2955, 0.6381], [0.4280, 0.5911], [0.5107, 0.5386]])
从顶点到超边依赖边权的消息传递
>>> # Print the vertex messages >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]]) >>> hg.v2e_weight tensor([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]) >>> # Generate random edge weights for the first stage >>> v2e_weight = torch.rand(len(hg.v2e_weight)) >>> v2e_weight tensor([0.6689, 0.2302, 0.8003, 0.7353, 0.7477, 0.5585, 0.6226, 0.8429, 0.6105, 0.1248, 0.8265, 0.2117]) >>> # Message propagation from vertex to hyperedge >>> Y_ = hg.v2e(X, v2e_weight=v2e_weight, aggr="mean") >>> # Print the new hyperedge messages >>> Y_ tensor([[0.7326, 1.1010], [0.5229, 1.4678], [2.5914, 3.5052], [1.2437, 1.4487]])
从超边到顶点的消息传递
>>> # Print current hyperedge messages >>> Y_ tensor([[0.4098, 0.5702], [0.2955, 0.6381], [0.4280, 0.5911], [0.5107, 0.5386]]) >>> # Message propagation from hyperedge to vertex >>> X_ = hg.e2v(Y_, aggr="mean") >>> # Print the new vertex messages >>> X_ tensor([[0.2955, 0.6381], [0.5107, 0.5386], [0.4110, 0.5845], [0.4495, 0.5667], [0.4114, 0.5893]])
从超边到顶点依赖边权的消息传递
>>> # Print current hyperedge messages >>> Y_ tensor([[0.4098, 0.5702], [0.2955, 0.6381], [0.4280, 0.5911], [0.5107, 0.5386]]) >>> hg.e2v_weight tensor([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]) >>> # Generate random edge weights for the second stage >>> e2v_weight = torch.rand(len(hg.e2v_weight)) >>> e2v_weight tensor([0.8574, 0.4282, 0.3964, 0.1440, 0.0034, 0.9504, 0.2194, 0.2893, 0.6784, 0.4997, 0.9144, 0.2833]) >>> # Message propagation from hyperedge to vertex >>> X_ = hg.e2v(Y_, e2v_weight=e2v_weight, aggr="mean") >>> # Print the new vertex messages >>> X_ tensor([[0.2172, 0.4691], [0.0936, 0.0988], [1.0335, 1.2427], [0.6650, 0.7853], [1.1605, 1.7178]])
从顶点集到顶点集的消息传递
每一条超边连接一个顶点集，其为两个顶点集合的信息桥梁。 在超图中，超边连接的源顶点集和汇顶点集是相同的。
>>> # Print the vertex messages >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]]) >>> # Message propagation from vertex set to vertex set >>> X_ = hg.v2v(X, aggr="mean") >>> # Print the new vertex messages >>> X_ tensor([[0.2955, 0.6381], [0.5107, 0.5386], [0.4110, 0.5845], [0.4495, 0.5667], [0.4114, 0.5893]])
从顶点集到顶点集依赖边权的两阶段消息传递
>>> # Print the vertex messages >>> X tensor([[0.3958, 0.9219], [0.7588, 0.3811], [0.0262, 0.3594], [0.7933, 0.7811], [0.4643, 0.6329]]) >>> hg.v2e_weight tensor([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]) >>> # Generate random edge weights for the first stage >>> v2e_weight = torch.rand(len(hg.v2e_weight)) >>> v2e_weight tensor([0.5739, 0.2444, 0.2476, 0.1210, 0.6869, 0.6617, 0.5168, 0.9089, 0.8799, 0.6949, 0.4609, 0.1263]) >>> hg.e2v_weight tensor([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]) >>> # Generate random edge weights for the second stage >>> e2v_weight = torch.rand(len(hg.e2v_weight)) >>> e2v_weight tensor([0.6332, 0.4839, 0.7779, 0.9180, 0.0768, 0.9693, 0.2956, 0.7251, 0.5438, 0.7403, 0.3211, 0.5044]) >>> # Message propagation from vertex set to vertex set >>> X_ = hg.v2v(X, v2e_weight=v2e_weight, e2v_weight=e2v_weight, aggr="mean") >>> # Print the new vertex messages >>> X_ tensor([[ 0.3082, 0.5642], [ 0.4297, 0.4918], [ 7.9027, 10.4666], [ 3.9316, 4.8732], [ 3.3256, 4.5806]])