We present a new representation learning framework, Intensity Profile Projection, for continuous-time dynamic network data. Given triples $(i,j,t)$, each representing a time-stamped ($t$) interaction between two entities ($i,j$), our procedure returns a continuous-time trajectory for each node, representing its behaviour over time. The framework consists of three stages: estimating pairwise intensity functions, e.g. via kernel smoothing; learning a projection which minimises a notion of intensity reconstruction error; and constructing evolving node representations via the learned projection. The trajectories satisfy two properties, known as structural and temporal coherence, which we see as fundamental for reliable inference. Moreoever, we develop estimation theory providing tight control on the error of any estimated trajectory, indicating that the representations could even be used in quite noise-sensitive follow-on analyses. The theory also elucidates the role of smoothing as a bias-variance trade-off, and shows how we can reduce the level of smoothing as the signal-to-noise ratio increases on account of the algorithm `borrowing strength' across the network.

翻译：我们提出了一种新的表示学习框架——强度分布投影，用于连续时间动态网络数据。给定三元组$(i,j,t)$（每个三元组表示两个实体$i,j$在时间点$t$的交互），该过程为每个节点返回一条连续时间轨迹，以刻画其随时间变化的行为。该框架包含三个阶段：通过核平滑等方法估计成对强度函数；学习一个最小化强度重构误差的投影映射；以及通过投影映射构建演化的节点表示。这些轨迹满足两个性质——结构一致性和时间一致性，我们将其视为可靠推断的基础。此外，我们发展了估计理论，可为任意估计轨迹的误差提供紧界控制，表明这些表示甚至可用于对噪声敏感的后续分析。该理论还阐明了平滑作为偏差-方差权衡的作用，并展示了算法如何通过跨网络"借力"提高信噪比，从而在信噪比增大时降低平滑程度。