Abstract
Generating general-purpose vector representations of networks allows us to analyze them without the need for extensive feature-engineering. Recent works have shown that the hyperbolic space can naturally represent the structure of networks, and that embedding networks into hyperbolic space is extremely efficient, especially in low dimensions. However, the existing hyperbolic embedding methods apply to static networks and cannot capture the dynamic evolution of the nodes and edges of a temporal network. In this paper, we present an unsupervised framework that uses temporal random walks to obtain training samples with both temporal and structural information to learn hyperbolic embeddings from continuous-time dynamic networks. We also show how the framework extends to attributed and heterogeneous information networks. Through experiments on five publicly available real-world temporal datasets, we show the efficacy of our model in embedding temporal networks in low-dimensional hyperbolic space compared to several other unsupervised baselines. We show that our model obtains state-of-the-art performance in low dimensions, outperforming all baselines, and has competitive performance in higher dimensions, outperforming the baselines in three of the five datasets. Our results show that embedding temporal networks in hyperbolic space is extremely effective when necessitating low dimensions.
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Appendices
Appendix A: Detailed experimental results
In Table 6 below, we show the detailed results of the experiments shown in Fig. 2. Recall that these are the temporal link prediction experiments in which Euclidean baselines threshold on inner products. As before, all results are averaged over 10 runs. We show the standard deviation and the statistical significance of the results. For each setting (i.e., dataset and dimension), the statistical significance between the performance of our model (Temp-Hyper) and every other baseline was calculated using a t-test. We denoted statistical significance (i.e., \(p < 0.05\)) differences between a baseline and our model with a *.
Appendix B: Higher dimensions
As discussed in the paper, the advantage of our proposed hyperbolic representation learning method for temporal networks is that it is more efficient than Euclidean methods with respect to dimensionality, in that the representations learned by our method in lower dimensions are comparable to those learned in higher dimensions and significantly outperform the low-dimensional representations learned by Euclidean methods. The purpose of this section is to explore whether the performance boost of our method in lower dimensions is severely diminished in higher dimensions. Tables 7, and 8 below show the results of our temporal link prediction experiments on 128 dimensions for our method and all baselines. Specifically, Table 7 shows the results for the first experiment where we rely solely on the learned representations (to showcase the advantage of our hyperbolic approach over Euclidean baselines), and Table 8 shows the results for the second experiment where use edge feature-engineering and logistic regression.
As Table 7 shows, in the first experiment our model still significantly (\(p < 0.05\)) outperforms all Euclidean baselines with relatively large margins. Moreover, our model is one of the more stable ones with relatively low standard deviations across datasets.
Table 8 highlights the limitations of our model in higher dimensions. Though still the best model for some of the datasets and comparable for the rest, the advantage of our model in lower dimensions does diminish in 128 dimensions. This points to our model not being ideal in larger dimensions. However, as seen here, in cases where low dimensionality is important, our model is the best candidate, and even in larger dimensions, our model’s performance is not far off from the top-performing models.
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Wang, L., Huang, C., Ma, W. et al. Hyperbolic node embedding for temporal networks. Data Min Knowl Disc 35, 1906–1940 (2021). https://doi.org/10.1007/s10618-021-00774-4
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DOI: https://doi.org/10.1007/s10618-021-00774-4