Topology Identification and Inference over Graphs

TL;DR

This paper reviews methods for identifying and inferring the structure of evolving processes over graphs, covering undirected and directed relations, dynamic and nonlinear dependencies, and advanced modeling techniques.

Contribution

It provides a comprehensive overview of graph topology inference methods, including novel frameworks for dynamic, causal, and nonlinear relations, with convergence guarantees and tensor-based approaches.

Findings

Survey of correlation and covariance selection methods

Introduction of kernel-based frameworks for causal relations

Discussion of tensor and high-order statistical methods

Abstract

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.