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Online Network Inference from Graph-Stationary Signals with Hidden Nodes

Andrei Buciulea, Madeline Navarro, Samuel Rey, Santiago Segarra, Antonio G. Marques

TL;DR

This work tackles online graph learning when some nodes are hidden by exploiting graph stationarity, which implies $C S = S C$. It introduces a convex online objective that jointly estimates the observed-subgraph $S_O$ and a hidden-nodes influence term $P$, with a proximal-gradient solver that can operate in real time on streaming data. The method handles incomplete observations by using an informative block structure and known-edge constraints, and provides tracking guarantees showing the online solution can follow the batch solution as the network evolves. Experiments on synthetic graphs and real financial data demonstrate improved accuracy over online baselines and robustness to hidden nodes, enabling real-time topology inference in partially observed networks.

Abstract

Graph learning is the fundamental task of estimating unknown graph connectivity from available data. Typical approaches assume that not only is all information available simultaneously but also that all nodes can be observed. However, in many real-world scenarios, data can neither be known completely nor obtained all at once. We present a novel method for online graph estimation that accounts for the presence of hidden nodes. We consider signals that are stationary on the underlying graph, which provides a model for the unknown connections to hidden nodes. We then formulate a convex optimization problem for graph learning from streaming, incomplete graph signals. We solve the proposed problem through an efficient proximal gradient algorithm that can run in real-time as data arrives sequentially. Additionally, we provide theoretical conditions under which our online algorithm is similar to batch-wise solutions. Through experimental results on synthetic and real-world data, we demonstrate the viability of our approach for online graph learning in the presence of missing observations.

Online Network Inference from Graph-Stationary Signals with Hidden Nodes

TL;DR

This work tackles online graph learning when some nodes are hidden by exploiting graph stationarity, which implies . It introduces a convex online objective that jointly estimates the observed-subgraph and a hidden-nodes influence term , with a proximal-gradient solver that can operate in real time on streaming data. The method handles incomplete observations by using an informative block structure and known-edge constraints, and provides tracking guarantees showing the online solution can follow the batch solution as the network evolves. Experiments on synthetic graphs and real financial data demonstrate improved accuracy over online baselines and robustness to hidden nodes, enabling real-time topology inference in partially observed networks.

Abstract

Graph learning is the fundamental task of estimating unknown graph connectivity from available data. Typical approaches assume that not only is all information available simultaneously but also that all nodes can be observed. However, in many real-world scenarios, data can neither be known completely nor obtained all at once. We present a novel method for online graph estimation that accounts for the presence of hidden nodes. We consider signals that are stationary on the underlying graph, which provides a model for the unknown connections to hidden nodes. We then formulate a convex optimization problem for graph learning from streaming, incomplete graph signals. We solve the proposed problem through an efficient proximal gradient algorithm that can run in real-time as data arrives sequentially. Additionally, we provide theoretical conditions under which our online algorithm is similar to batch-wise solutions. Through experimental results on synthetic and real-world data, we demonstrate the viability of our approach for online graph learning in the presence of missing observations.
Paper Structure (9 sections, 12 equations, 1 figure, 1 algorithm)

This paper contains 9 sections, 12 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Graph signal processing
  4. Incomplete and streaming graph data
  5. Graph learning with hidden nodes
  6. Online graph learning
  7. Proximal gradient algorithm
  8. Numerical experiments
  9. Conclusions

Figures (1)

  • Figure 1: Normalized graph estimation error ${\rm err}({\mathbf S}_{{ {\mathcal{O}}}})$ versus the number of samples while considering 3 approaches combined with (a) 3 values for the number of iterations $\{1,10,100\}$ performed at each time $t$ for OnST, OnST-H, (b) 2 values for the hidden nodes $H = \{2,5\}$. (c) ${\rm err}({\mathbf S}_{{ {\mathcal{O}}}})$ for OnST and OnST-H (left $y$-axis) and standardized average stock values (right $y$-axis) versus the number of samples.