  Dynamic graph embedding has gained great attention recently due to its
capability of learning low dimensional graph representations for complex
temporal graphs with high accuracy. However, recent advances mostly focus on
learning node embeddings as deterministic "vectors" for static graphs yet
disregarding the key graph temporal dynamics and the evolving uncertainties
associated with node embedding in the latent space. In this work, we propose an
efficient stochastic dynamic graph embedding method (DynG2G) that applies an
inductive feed-forward encoder trained with node triplet-based contrastive
loss. Every node per timestamp is encoded as a time-dependent probabilistic
multivariate Gaussian distribution in the latent space, hence we can quantify
the node embedding uncertainty on-the-fly. We adopted eight different
benchmarks that represent diversity in size (from 96 nodes to 87,626 and from
13,398 edges to 4,870,863) and diversity in dynamics. We demonstrate via
extensive experiments on these eight dynamic graph benchmarks that DynG2G
achieves new state-of-the-art performance in capturing the underlying temporal
node embeddings. We also demonstrate that DynG2G can predict the evolving node
embedding uncertainty, which plays a crucial role in quantifying the intrinsic
dimensionality of the dynamical system over time. We obtain a universal
relation of the optimal embedding dimension, $L_o$, versus the effective
dimensionality of uncertainty, $D_u$, and we infer that $L_o=D_u$ for all
cases. This implies that the uncertainty quantification approach we employ in
the DynG2G correctly captures the intrinsic dimensionality of the dynamics of
such evolving graphs despite the diverse nature and composition of the graphs
at each timestamp. Moreover, this $L_0 - D_u$ correlation provides a clear path
to select adaptively the optimum embedding size at each timestamp by setting $L
\ge D_u$.
