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Online Learning Of Expanding Graphs

Samuel Rey, Bishwadeep Das, Elvin Isufi

TL;DR

This work tackles online topology inference for expanding graphs from streaming signals, addressing the difficulty of growing node sets and changing topology in delay-sensitive settings. It introduces a general online algorithm based on projected proximal gradient descent, equipped with a covariance update that uses zero-padding and a block-structured GSO to handle newly arriving nodes. The approach is specialized to Gaussian Markov Random Fields, with a dynamic cumulative regret analysis showing how tracking performance depends on the evolution of the optimal topology. Empirical validation on controlled and real-world data, including epidemic and financial networks, demonstrates fast adaptation and competitive accuracy with modest computational cost. The proposed framework enables real-time graph learning in applications where the network expands over time and quick decisions are essential.

Abstract

This paper addresses the problem of online network topology inference for expanding graphs from a stream of spatiotemporal signals. Online algorithms for dynamic graph learning are crucial in delay-sensitive applications or when changes in topology occur rapidly. While existing works focus on inferring the connectivity within a fixed set of nodes, in practice, the graph can grow as new nodes join the network. This poses additional challenges like modeling temporal dynamics involving signals and graphs of different sizes. This growth also increases the computational complexity of the learning process, which may become prohibitive. To the best of our knowledge, this is the first work to tackle this setting. We propose a general online algorithm based on projected proximal gradient descent that accounts for the increasing graph size at each iteration. Recursively updating the sample covariance matrix is a key aspect of our approach. We introduce a strategy that enables different types of updates for nodes that just joined the network and for previously existing nodes. To provide further insights into the proposed method, we specialize it in Gaussian Markov random field settings, where we analyze the computational complexity and characterize the dynamic cumulative regret. Finally, we demonstrate the effectiveness of the proposed approach using both controlled experiments and real-world datasets from epidemic and financial networks.

Online Learning Of Expanding Graphs

TL;DR

This work tackles online topology inference for expanding graphs from streaming signals, addressing the difficulty of growing node sets and changing topology in delay-sensitive settings. It introduces a general online algorithm based on projected proximal gradient descent, equipped with a covariance update that uses zero-padding and a block-structured GSO to handle newly arriving nodes. The approach is specialized to Gaussian Markov Random Fields, with a dynamic cumulative regret analysis showing how tracking performance depends on the evolution of the optimal topology. Empirical validation on controlled and real-world data, including epidemic and financial networks, demonstrates fast adaptation and competitive accuracy with modest computational cost. The proposed framework enables real-time graph learning in applications where the network expands over time and quick decisions are essential.

Abstract

This paper addresses the problem of online network topology inference for expanding graphs from a stream of spatiotemporal signals. Online algorithms for dynamic graph learning are crucial in delay-sensitive applications or when changes in topology occur rapidly. While existing works focus on inferring the connectivity within a fixed set of nodes, in practice, the graph can grow as new nodes join the network. This poses additional challenges like modeling temporal dynamics involving signals and graphs of different sizes. This growth also increases the computational complexity of the learning process, which may become prohibitive. To the best of our knowledge, this is the first work to tackle this setting. We propose a general online algorithm based on projected proximal gradient descent that accounts for the increasing graph size at each iteration. Recursively updating the sample covariance matrix is a key aspect of our approach. We introduce a strategy that enables different types of updates for nodes that just joined the network and for previously existing nodes. To provide further insights into the proposed method, we specialize it in Gaussian Markov random field settings, where we analyze the computational complexity and characterize the dynamic cumulative regret. Finally, we demonstrate the effectiveness of the proposed approach using both controlled experiments and real-world datasets from epidemic and financial networks.
Paper Structure (12 sections, 2 theorems, 36 equations, 3 figures, 1 algorithm)

This paper contains 12 sections, 2 theorems, 36 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Fundamentals of network topology inference
  3. Learning expanding graphs
  4. Online learning of expanding graphs
  5. Expanding covariance update
  6. Online algorithm
  7. Online expanding Gaussian graphical model
  8. Dynamic cumulative regret analysis
  9. Numerical experiments
  10. Experiments using controlled data
  11. Experiments using real-world data
  12. Conclusion

Key Result

theorem 1

Let $\{ \hbS_t\}_{t=1}^T$ be the sequence of estimates procured by Algorithm a:nti_expanding with $\nabla f_t(\bbS)$ and $\Pi_{\ccalS_{N_t}}(\bbS)$ as given in e:gradient_ggl and e:projection_ggl, respectively. Let $\{ \bbS_t^* \}_{t=1}^T$ be the sequence of minimizers of e:offline_ggl. Setting the with $\underline{\bbS^*}^{(N_t)}_{t-1}$ being the $N_t \times N_t$ zero-padded version of $\bbS^*_{

Figures (3)

  • Figure 1: Evolution of $\ccalG_t$ and $\bbS_t$ in the expanding graph setting. Green edges and blocks of the GSO denote the connectivity of previous nodes in the network while purple indicates the connectivity of incoming nodes. Starting at $t-1$, the initial GSO $\bbS_{t-1}$ is an $N_{t-1}\times N_{t-1}$ matrix. At time $t$, a new signal $\bbx_t \in \reals^{N_t}$ (yellow) arrives and a new node joins the network, enlarging the size of the graph. Similarly, two nodes arrive on time $t+1$ and so on.
  • Figure 2: Evaluating the performance of Algorithm 1 over controlled data. (a) and (b) respectively depict the instantaneous error and the average cumulative regret between the online and the offline solution. We consider new groups of nodes arriving with low or high frequency, denoted as "L" and "H", and perform 1, 10, or 50 iterations of Alfgorithm 1 per time instant. (c) illustrates the instantaneous error of various methods relative to the true graph $\bbS_t$.
  • Figure 3: Numerical evaluation using real-world data. a) and c) respectively display the median of the standardized incidence rates of COVID-19 and the closing value prices of stocks from S&P 500. b) and d) plot the error and cumulative regret of the estimated graph when using data from the COVID-19 or the Financial dataset..

Theorems & Definitions (2)

  • theorem 1
  • lemma 1