总览 (Overview)

Graph embedding methods have gained prominence in a wide variety of tasks including pattern recognition, low-dimensional embedding, clustering, anomaly detection, node classification, and link prediction, etc. Graphs can be mapped to low dimensional space which encodes graph information and creates node representation. From an input graph G=(V,E), embedding a graph is to map node feature matrix to d-dimensional continuous vector space. The input in continuous domains are required for machine learning and deep learning methods. In machine learning, the task of producing graph embeddings entails capturing local and global graph statistics and encoding them as vectors that best preserve these statistics in a computationally efficient manner.

图嵌入方法已在各种任务中占据重要地位，包括模式识别，低维嵌入，聚类，异常检测，节点分类和链接预测等。图可以映射到低维空间，该低维空间对图信息进行编码并创建节点表示。 从输入图G =(V，E)嵌入图是将节点特征矩阵映射到d维连续向量空间。 机器学习和深度学习方法需要连续域中的输入。 在机器学习中，产生图形嵌入的任务需要捕获局部和全局图形统计信息并将其编码为向量，从而以计算有效的方式最好地保留这些统计信息。

We present our work from our IJCNN 2020 paper titled “Learning Representations using Spectral-Biased Random Walks on Graphs” focusing on solving Link Prediction problem. Given a graph, predict if two nodes are likely to have a link between them. Link prediction has several applications in social networks, biological networks, knowledge graphs, etc.

我们将以IJCNN 2020论文为题，介绍“ 使用谱偏向随机游动图学习表示 ”，着重解决链接预测问题。 给定一个图，预测两个节点之间是否可能有链接。 链接预测在社交网络，生物网络，知识图等中有多种应用。

挑战性 (Challenges)

There are quite a few heuristic methods which perform well but are based on strong assumptions for the likelihood of links, which can be beneficial in the case of social networks, but does not generalize well to arbitrary networks. Some of the latest GNN-based models outperform heuristic based methods. But, they require either (a) enclosing subgraph (i.e., the size of the neighborhood) around each edge, which is quite prohibitive and results in being able to only process small and sparse graphs or (b) random walks. Random walks appeal that they concisely capture the underlying graph structure surrounding a vertex. But, they cannot guarantee grouping of vertices with similar k-hop neighborhoods. Thus, vertices with structurally similar neighborhoods can easily be omitted. In our work, we incorporate such a grouping of structurally similar nodes directly into our random walks.

有许多启发式方法效果很好，但是基于对链接可能性的强有力假设，这在社交网络的情况下可能是有益的，但不能很好地推广到任意网络。 一些最新的基于GNN的模型优于基于启发式的方法。 但是，它们要么需要(a)在每个边缘周围包含子图(即，邻域的大小)，这是非常禁止的，导致只能处理较小且稀疏的图，或者(b)随机游走。 随机游走的吸引力在于，它们可以精确捕获围绕顶点的基础图形结构。 但是，它们不能保证对具有相似k跳邻域的顶点进行分组。 因此，可以容易地省略具有结构相似的邻域的顶点。 在我们的工作中，我们将这种结构相似的节点直接合并到我们的随机游走中。

我们的方法 (Our Method)

顶点邻域谱 (Spectra of Vertex Neighborhoods)

Given a vertex v and a fixed integer k > 0, the graph neighborhood G(v,k) of v is the subgraph induced by the k closest vertices (i.e., in terms of shortest paths on G) that are reachable from v.

给定一个顶点V和一个固定整数k> 0，则图附近ģ 诉 (V，k)是由第k个顶点最接近诱导的子图(即，在G上的最短路径计算)是从V可达。

Now, the graph neighborhood G(v,k) of a vertex v is represented in matrix form as a normalized Laplacian matrix L^(v) = l_ij, where i,j = 1 to k in k² dimensional space. Given L^(v), its sequence of k real eigenvalues ( λ_1 ( L^(v) )≥ … ≥ λ_k ( L^(v) ) ) is known as the spectrum of the neighborhood L^(v) and is denoted by σ(L^(v)). We also know that all the eigenvalues in σ(L^(v)) lie in an interval Ω ⊂ R.

现在，顶点v的图邻域G(v，k)以矩阵形式表示为归一化的拉普拉斯矩阵L ^(v)= l_ij