Topology identification and inference of processes evolving over graphs arise in timely applications involving brain, transportation, financial, power, as well as social and information networks. This chapter provides an overview of graph topology identification and statistical inference methods for multidimensional relational data. Approaches for undirected links connecting graph nodes are outlined, going all the way from correlation metrics to covariance selection, and revealing ties with smooth signal priors. To account for directional (possibly causal) relations among nodal variables and address the limitations of linear time-invariant models in handling dynamic as well as nonlinear dependencies, a principled framework is surveyed to capture these complexities through judiciously selected kernels from a prescribed dictionary. Generalizations are also described via structural equations and vector autoregressions that can exploit attributes such as low rank, sparsity, acyclicity, and smoothness to model dynamic processes over possibly time-evolving topologies. It is argued that this approach supports both batch and online learning algorithms with convergence rate guarantees, is amenable to tensor (that is, multi-way array) formulations as well as decompositions that are well-suited for multidimensional network data, and can seamlessly leverage high-order statistical information.

翻译：图上演化过程的拓扑识别与推断在涉及脑网络、交通网络、金融网络、电力网络以及社会与信息网络等时效性应用中具有重要意义。本章概述了面向多维关系数据的图拓扑识别与统计推断方法。首先概述了连接图节点的无向链接方法，涵盖从相关性度量到协方差选择的全过程，并揭示了其与平滑信号先验的联系。为考虑节点变量间的方向性（可能为因果性）关系，并解决线性时不变模型在处理动态及非线性依赖关系时的局限性，本文综述了一种基于原则的框架，该框架通过从预设字典中审慎选择核函数来捕捉这些复杂性。此外，还通过结构方程和向量自回归模型描述了方法的推广形式，这些模型可利用低秩性、稀疏性、无环性和平滑性等属性，对可能随时间演化的拓扑结构上的动态过程进行建模。本文指出，该方法既支持具有收敛速率保证的批处理和在线学习算法，又适用于张量（即多维数组）公式及其分解——这些形式特别适合处理多维网络数据，并能无缝利用高阶统计信息。