# A Hierarchical Generative Embedding Model for Influence Maximization in Attributed Social Networks
## 6 Results and Analysis
### Overall Performance of HGE-GA Algorithm  
We adopt the metric of expected spread to evaluate the overall performance of all influence maximization algorithms. To compare the proposed HGE-GA algorithm with the baselines, we change the size of the seed node set \(K\) to be 1, 20, 30, 40, and 50. The experimental results are reported in Figure 6(a)-e. It can be observed from Figure 6(a)-e that:  
(1) The expected spread increases for all algorithms when enlarging the size of the seed node set. The proposed HGE-GA algorithm outperforms all the baselines significantly for \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Method** & \multicolumn{2}{c}{**Cora**} & \multicolumn{2}{c}{**DBLP**} & \multicolumn{2}{c}{**BlogCatalog**} & \multicolumn{2}{c}{**Flickr**} & \multicolumn{2}{c}{**Pubmed**} \\ \hline  & **AUC** & **AP** & **AUC** & **AP** & **AUC** & **AP** & **AUC** & **AP** & **AUC** & **AP** \\ \hline
**AANE** & 0.834 & 0.812 & 0.761 & 0.789 & 0.771 & 0.776 & 0.678 & 0.639 & 0.842 & 0.837 \\
**GraphSAGE** & 0.869 & 0.892 & 0.803 & 0.811 & 0.807 & 0.819 & 0.802 & 0.817 & 0.812 & 0.825 \\
**M-NMF** & 0.867 & 0.861 & 0.841 & 0.837 & 0.716 & 0.710 & 0.728 & 0.736 & 0.816 & 0.825 \\
**GNE** & 0.944 & 0.941 & 0.935 & 0.944 & 0.805 & 0.802 & 0.828 & 0.837 & 0.945 & 0.942 \\
**SpaceNE** & 0.927 & 0.914 & 0.927 & 0.933 & 0.815 & 0.827 & 0.908 & 0.917 & 0.936 & 0.958 \\
**HGE** & **0.979** & **0.974** & **0.986** & **0.989** & **0.836** & **0.845** & **0.929** & **0.913** & **0.977** & **0.972** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Link prediction performance with embedding size L = 64. Figure 6: The experimental results of network visualization in two-dimensional space. all seed sizes on the five datasets. For instance, on the Pubmed dataset, when the seed set size \(K=50\), the value of the expected spread of the proposed HGE-GA algorithm is 1445, compared with 721 for PMIA, 819 for IMM, 1052 for TSH-GA, and 1384 for HGE-1N-GA;  
(2) On all five datasets, all of the attribute-related algorithms, TSH-GA, HGE-1N-GA and HGE-GA, outperform the non-attribute-related algorithms, PMIA and IMM, in terms of the performance of the expected spread. This is because the PMIA and IMM algorithms do not have the step of predicting the influence probability of node pairs, with influence probability being set uniformly. Moreover, this demonstrates that homophily is a cause of similar behaviors and is useful for predicting future behaviors of users. Consequently, the expected spread can be increased obviously when node attributes are taken into account in modeling;  
(3) HGE-1N-GA and HGE-GA outperform TSH-GA on the five datasets for the metric of expected spread. The reason is that HGE-1N-GA and HGE-GA consider the general network structure, hierarchical network structure as well as node attributes to learn users' latent feature representations for predicting the influence probability of node pairs, but TSH-GA uses hand-crafted rules to extract the features of users' interactions, network structure and node homophily, heavily depending on the domain expert's knowledge and without considering hierarchical network structure;  
(4) The expected spread of HGE-GA is superior than HGE-1N-GA. The reason is obvious: HGE-1N-GA uses the one-normal form to compute the similarity of two nodes, while HGE-GA leverages the two-normal form to compute similarity, which can capture the relationships between nodes better. The running times of all algorithms are reported in Figure 7. This figure only presents the experimental results on the DBLP dataset due to space constraints. Please note that the DBLP dataset is the biggest dataset in the five datasets considering both the number of nodes and the number of edges. Similar trends of performance of the algorithms are found for the other four datasets. From Figure 7b,f, it can be observed that, for the DBLP dataset, compared with the baseline algorithms, the proposed algorithm, HGE-GA, obtains a better performance of the expected spread by spending more time.