# Structural Hierarchy-Enhanced Network Representation Learning
## 3 The Proposed SHE-NRL Model
In the top-down way, when node embeddings are produced by NRL in \(H^{h+1}\) (Phase 2), they are instantaneously employed to initialize the embeddings of nodes in \(H^{h}\) via average pooling (Phase 3). We utilize Figure 2 to better elaborate the interweaved process between Phase 2 and Phase 3 in an alternative view. In this paragraph, that said, we will describe the bottom-up process, followed by the top-down process. Given the structural hierarchy, consisting of three graphs at different levels, i.e., \(H^{0}\), \(H^{1}\), and \(H^{2}\), in the bottom-up process, we first perform a typical NRL method on \(H^{0}\) and obtain node embedding \(\mathbf{x}_{v}^{0}\) for every \(v\in H^{0}\). Then we apply max pooling on nodes, which belong to the same level-1 community, to initialize the embedding of each level-1 community node in \(H^{1}\). We again apply the same NRL method to \(H^{1}\) to obtain node embedding for every level-1 community node, and use max pooling to initialize each level-2 community node in \(H^{2}\). Then NRL method is applied again on \(H^{2}\). Next, we are doing the top-down process. We utilize average pooling in Equation (2) to initialize level-1 community node embedding from level-2 community node embedding. Then the NRL method is applied to \(H^{1}\), followed by performing average pooling to initialize node embeddings in \(H^{0}\). The last NRL is executed on \(H^{0}\) to generate the embeddings of nodes in the original graph. Phase 4: Generating Final Node Embeddings. Equipped with the derived node embedding matrix \(\mathbf{\chi}^{h}\) at every level-\(h\) graph, we can generate the final embeddings for all nodes in the original network \(G\) (i.e., \(H^{0}\)). To let final node embeddings contain both fine-grained and coarse-grained information, the concatenation operation is adopted. We concatenate the embedding vector of node \(v\) in \(H^{0}\) with all of its corresponding higher-level embedding vectors in \(H^{h}\), where \(h=1,2,\ldots,\tau\). Suppose the dimension of each level's node embedding is equal and denoted as \(b\). The dimension of final node embeddings will be \((\tau+1)b\). Note that it is apparent that there can be alternative approaches, such as the operators of average, Hadamard, and weighted-L2 [3], to fuse node embeddings at different levels of the hierarchy. We leave the design of better embedding aggregation for future investigation. Remark. Two of the important ideas in the proposed SHE-NRL model are the bottom-up and top-down correlations on embedding vectors. If we first generate node embeddings for every level from \(H^{0}\) to \(H^{\tau}\), then apply the top-down corrections, our model can encounter a critical issue--missing the interactions and connections between levels. Bottom-up and top-down correlations are to bring the fine-grained semantics encoded by lower-level node embeddings to higher-levels ones, and to deliver the coarse-grained semantics to lower-level nodes, respectively. Such a design follows the realistic intuition that, for an example in the university, a graduate student is depicted by her advisor, her college, and her school in order. We need to make the advisor recognize that student, and make the college see the advisor, and so on. By doing so, through the immediate correction of vectors in \(H^{h}\) level based on the \(H^{h+1}\) vectors, the node embeddings can better encode the semantics, and become more robust.