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Learning the Topology and Behavior of Discrete Dynamical Systems

Zirou Qiu, Abhijin Adiga, Madhav V. Marathe, S. S. Ravi, Daniel J. Rosenkrantz, Richard E. Stearns, Anil Vullikanti

TL;DR

Learning both topology and interaction rules of unknown networked discrete dynamical systems from observed dynamics under the PAC framework is analyzed. The work proves general intractability for the full unknown topology problem, and provides efficient PAC learners for specific graph classes (e.g., matchings and directed graphs with bounded indegree) as well as a partially observed topology setting. It introduces a threshold-compatibility graph approach for matchings and derives per-vertex consistency checks for directed graphs, along with concrete sample-complexity bounds. The expressive power of the hypothesis class is characterized via the Natarajan dimension, yielding tight quadratic bounds and informing sample complexity; collectively, the results establish a rigorous foundation for learning both behavior and topology in discrete dynamical systems with contagion-like dynamics.

Abstract

Discrete dynamical systems are commonly used to model the spread of contagions on real-world networks. Under the PAC framework, existing research has studied the problem of learning the behavior of a system, assuming that the underlying network is known. In this work, we focus on a more challenging setting: to learn both the behavior and the underlying topology of a black-box system. We show that, in general, this learning problem is computationally intractable. On the positive side, we present efficient learning methods under the PAC model when the underlying graph of the dynamical system belongs to some classes. Further, we examine a relaxed setting where the topology of an unknown system is partially observed. For this case, we develop an efficient PAC learner to infer the system and establish the sample complexity. Lastly, we present a formal analysis of the expressive power of the hypothesis class of dynamical systems where both the topology and behavior are unknown, using the well-known formalism of the Natarajan dimension. Our results provide a theoretical foundation for learning both the behavior and topology of discrete dynamical systems.

Learning the Topology and Behavior of Discrete Dynamical Systems

TL;DR

Learning both topology and interaction rules of unknown networked discrete dynamical systems from observed dynamics under the PAC framework is analyzed. The work proves general intractability for the full unknown topology problem, and provides efficient PAC learners for specific graph classes (e.g., matchings and directed graphs with bounded indegree) as well as a partially observed topology setting. It introduces a threshold-compatibility graph approach for matchings and derives per-vertex consistency checks for directed graphs, along with concrete sample-complexity bounds. The expressive power of the hypothesis class is characterized via the Natarajan dimension, yielding tight quadratic bounds and informing sample complexity; collectively, the results establish a rigorous foundation for learning both behavior and topology in discrete dynamical systems with contagion-like dynamics.

Abstract

Discrete dynamical systems are commonly used to model the spread of contagions on real-world networks. Under the PAC framework, existing research has studied the problem of learning the behavior of a system, assuming that the underlying network is known. In this work, we focus on a more challenging setting: to learn both the behavior and the underlying topology of a black-box system. We show that, in general, this learning problem is computationally intractable. On the positive side, we present efficient learning methods under the PAC model when the underlying graph of the dynamical system belongs to some classes. Further, we examine a relaxed setting where the topology of an unknown system is partially observed. For this case, we develop an efficient PAC learner to infer the system and establish the sample complexity. Lastly, we present a formal analysis of the expressive power of the hypothesis class of dynamical systems where both the topology and behavior are unknown, using the well-known formalism of the Natarajan dimension. Our results provide a theoretical foundation for learning both the behavior and topology of discrete dynamical systems.
Paper Structure (19 sections, 14 theorems, 8 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 19 sections, 14 theorems, 8 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete Dynamical Systems
  4. The Learning Problem
  5. PAC Learnability and Sample Complexity
  6. Intractability of PAC learning
  7. PAC Learnability for Matchings
  8. PAC Learnability for Directed Graphs
  9. Partially Observed Networks
  10. Missing At Most One Edge Per Vertex
  11. Tight Bounds on the Natarajan Dimension
  12. Conclusion and Future Work
  13. Additional Material for Section \ref{['sec:general_complexity']}
  14. Notation Used in the Paper
  15. Classes of SyDSs Considered
  16. ...and 4 more sections

Key Result

Lemma 3.1

The $\Gamma$-Consistency problem is NP-complete for the following classes of SyDSs: $(i)$ (Undir, Thresh)-SyDSs and $(ii)$ (Tree, Thresh$\,=2$)-SyDSs.

Figures (1)

  • Figure 1: An example of a single transition of a SyDS with threshold interaction functions. The threshold values of vertices are highlighted in red. State-1 vertices are in blue.

Theorems & Definitions (14)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Theorem 3.8
  • Theorem 4.1
  • Corollary 4.1.1
  • ...and 4 more