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Topology Identification and Inference over Graphs

Gonzalo Mateos, Yanning Shen, Georgios B. Giannakis, Ananthram Swami

TL;DR

The chapter addresses the problem of identifying latent network topology and inferring processes evolving over graphs from nodal observations. It surveys a spectrum of approaches, from correlation-based networks and covariance selection to graph learning from smooth signals, and extends to directed, nonlinear, and dynamic graphs through SEMs, VARMs, kernel-based SVARMs, and MKL formulations. It introduces scalable algorithms with convergence guarantees (e.g., dual proximal gradient, ADMM) and tensor-based methods for dynamic and multilayer graphs, including joint topology-signal inference under partial observations. The framework provides a versatile toolkit applicable across domains such as brain, transportation, finance, and social networks, enabling robust topology identification, predictive modeling, and discriminative representations.

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.

Topology Identification and Inference over Graphs

TL;DR

The chapter addresses the problem of identifying latent network topology and inferring processes evolving over graphs from nodal observations. It surveys a spectrum of approaches, from correlation-based networks and covariance selection to graph learning from smooth signals, and extends to directed, nonlinear, and dynamic graphs through SEMs, VARMs, kernel-based SVARMs, and MKL formulations. It introduces scalable algorithms with convergence guarantees (e.g., dual proximal gradient, ADMM) and tensor-based methods for dynamic and multilayer graphs, including joint topology-signal inference under partial observations. The framework provides a versatile toolkit applicable across domains such as brain, transportation, finance, and social networks, enabling robust topology identification, predictive modeling, and discriminative representations.

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.

Paper Structure

This paper contains 13 sections, 40 equations, 2 figures.

Table of Contents

  1. Topology Identification and Inference over Graphs
  2. Introduction
  3. A tale of signal correlations and smoothness
  4. Correlation networks
  5. From covariance selection to graphical lasso
  6. Graph learning from observations of smooth signals
  7. Discriminative graph learning for classification
  8. Directional, nonlinear, and dynamic interactions
  9. Directed graphs and causality
  10. Nonlinear models for topology identification
  11. Dynamic networks and multilayer graphs
  12. Joint inference of graph topologies and signals
  13. Concluding summary

Figures (2)

  • Figure 1: (a) Significantly different connections between low and high valence with $p\leq 0.002$. Mean of the eigenvector magnitudes corresponding to (b) low, (c) mid, and (d) high frequencies. These results depict distinct discernible patterns between low and high valence, where most of these differences are aligned with the literature; see also saboksayr21icassp.
  • Figure 2: Performance evaluation on GDP prediction: a) cNMSE for GDP estimates; and b) Greece GDP along with JISG estimate; see also ioannidis2019semi.