Structural network embedding is a crucial step in enabling effective downstream tasks for complex systems that aims to project a network into a lower-dimensional space while preserving similarities among nodes. We introduce a simple and efficient embedding technique based on approximate variants of equitable partitions. The approximation consists in introducing a user-tunable tolerance parameter relaxing the otherwise strict condition for exact equitable partitions that can be hardly found in real-world networks. We exploit a relationship between equitable partitions and equivalence relations for Markov chains and ordinary differential equations to develop a partition refinement algorithm for computing an approximate equitable partition in polynomial time. We compare our method against state-of-the-art embedding techniques on benchmark networks. We report comparable -- when not superior -- performance for visualization, classification, and regression tasks at a cost between one and three orders of magnitude smaller using a prototype implementation, enabling the embedding of large-scale networks which could not be efficiently handled by most of the competing techniques.

翻译：结构网络嵌入是实现复杂系统有效下游任务的关键步骤，其目标是将网络投影到低维空间，同时保持节点间的相似性。我们提出一种基于公平划分近似变体的简单高效嵌入技术。该近似方法通过引入用户可调的容差参数，放宽了严格精确公平划分的条件——这种严格条件在实际网络中几乎无法满足。我们利用公平划分与马尔可夫链及常微分方程等价关系之间的联系，开发了一种可在多项式时间内计算近似公平划分的划分细化算法。我们在基准网络上将本方法与前沿嵌入技术进行比较。通过原型实现，本方法在可视化、分类和回归任务中展现出相当（甚至更优）的性能，且计算成本降低一至三个数量级，从而能够对大规模网络进行嵌入处理——而大多数竞争技术无法高效处理此类网络。