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On Some Fundamental Problems for Multi-Agent Systems Over Multilayer Networks

Daniel J. Rosenkrantz, Madhav V. Marathe, Zirou Qiu, S. S. Ravi, Richard E. Stearns

TL;DR

This paper investigates fundamental questions for multilayer synchronous dynamical systems (MSyDS) on multilayer networks, focusing on phase-space structure, the complexity of equivalence, and the expressive power across layer counts. It proves that MSyDS inequivalence is NP-complete even under strong restrictions, and demonstrates long and even exponential cycles in phase spaces that contrast with single-layer behavior. The authors also develop efficient equivalence-testing algorithms for special MSyDS classes (e.g., fixed layer counts, symmetric or bounded-threshold local functions) and show how the number of layers shapes expressive power, including conditions where higher-layer dynamics cannot be reduced to fewer layers. These results illuminate the computational barriers and tractable regimes for analyzing MSyDS, with implications for understanding diffusion and control in complex multilayer networks.

Abstract

Many researchers have considered multi-agent systems over single-layer networks as models for studying diffusion phenomena. Since real-world networks involve connections between agents with different semantics (e.g., family member, friend, colleague), the study of multi-agent systems over multilayer networks has assumed importance. Our focus is on one class of multi-agent system models over multilayer networks, namely multilayer synchronous dynamical systems (MSyDSs). We study several fundamental problems for this model. We establish properties of the phase spaces of MSyDSs and bring out interesting differences between single-layer and multilayer dynamical systems. We show that, in general, the problem of determining whether two given MSyDSs are inequivalent is NP-complete. This hardness result holds even when the only difference between the two systems is the local function at just one node in one layer. We also present efficient algorithms for the equivalence problem for restricted versions of MSyDSs (e.g., systems where each local function is a bounded-threshold function, systems where the number of layers is fixed and each local function is symmetric). In addition, we investigate the expressive power of MSyDSs based on the number of layers. In particular, we examine conditions under which a system with k >= 2 layers has an equivalent system with k-1 or fewer layers.

On Some Fundamental Problems for Multi-Agent Systems Over Multilayer Networks

TL;DR

This paper investigates fundamental questions for multilayer synchronous dynamical systems (MSyDS) on multilayer networks, focusing on phase-space structure, the complexity of equivalence, and the expressive power across layer counts. It proves that MSyDS inequivalence is NP-complete even under strong restrictions, and demonstrates long and even exponential cycles in phase spaces that contrast with single-layer behavior. The authors also develop efficient equivalence-testing algorithms for special MSyDS classes (e.g., fixed layer counts, symmetric or bounded-threshold local functions) and show how the number of layers shapes expressive power, including conditions where higher-layer dynamics cannot be reduced to fewer layers. These results illuminate the computational barriers and tractable regimes for analyzing MSyDS, with implications for understanding diffusion and control in complex multilayer networks.

Abstract

Many researchers have considered multi-agent systems over single-layer networks as models for studying diffusion phenomena. Since real-world networks involve connections between agents with different semantics (e.g., family member, friend, colleague), the study of multi-agent systems over multilayer networks has assumed importance. Our focus is on one class of multi-agent system models over multilayer networks, namely multilayer synchronous dynamical systems (MSyDSs). We study several fundamental problems for this model. We establish properties of the phase spaces of MSyDSs and bring out interesting differences between single-layer and multilayer dynamical systems. We show that, in general, the problem of determining whether two given MSyDSs are inequivalent is NP-complete. This hardness result holds even when the only difference between the two systems is the local function at just one node in one layer. We also present efficient algorithms for the equivalence problem for restricted versions of MSyDSs (e.g., systems where each local function is a bounded-threshold function, systems where the number of layers is fixed and each local function is symmetric). In addition, we investigate the expressive power of MSyDSs based on the number of layers. In particular, we examine conditions under which a system with k >= 2 layers has an equivalent system with k-1 or fewer layers.

Paper Structure

This paper contains 16 sections, 12 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Contributions
  4. Additional Remarks
  5. Related Work
  6. Preliminaries
  7. Multilayer Synchronous Dynamical Systems
  8. Problem formulations
  9. Phase Space Properties of Multilayer Systems
  10. Complexity of Equivalence
  11. Efficient Algorithms for Equivalence for Special Classes of MSyDSs
  12. Fixed Number of Layers
  13. Bounded Threshold
  14. Expressive Power of MSyDSs
  15. Additional Remarks and Future Research Directions
  16. ...and 1 more sections

Key Result

Theorem 3.1

For every $n \geq 1$, there is a MSyDS $\mathcal{S}_n$ with two layers and $n$ nodes, where every local function is a threshold function, and every master function is symmetric, with the following properties: the phase space of $\mathcal{S}_n$ contains exactly one cycle, the length of this cycle is

Figures (1)

  • Figure 1: An example of a small MSyDS, where state-1 nodes are highlighted in blue. The master function at each node is OR. The thresholds of the nodes (same for the two layers) are: $v_1$ : $2$, $v_2$ : $2$, $v_3$ : $3$, $v_4$ : $1$ and $v_5$ : $3$.

Theorems & Definitions (12)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Theorem 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Theorem 5.4
  • Corollary 5.5
  • Proposition 6.1
  • ...and 2 more