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

2025-03-16

AI Summary

This study addresses two fundamental theoretical problems concerning multilevel synchronous dynamical systems (MSyDSs): (1) deciding whether two MSyDSs are functionally equivalent, and (2) characterizing the expressive boundaries imposed by layer count—specifically, determining when a *k*-layer system admits an equivalent reduction to *k*−1 or fewer layers. Employing combinatorial dynamical systems theory, computational complexity analysis, Boolean/threshold function modeling, and multilevel graph-theoretic techniques, we establish that MSyDS equivalence checking is NP-complete—its first rigorous proof. For restricted topologies—including sparse inter-layer connectivity and monotone update rules—we design polynomial-time decision algorithms. We develop a structural theory of MSyDS phase spaces and derive necessary and sufficient conditions for layer reducibility. Collectively, these results systematically characterize the intrinsic modeling advantages and fundamental limitations conferred by multilevel architecture over single-level counterparts.

Technology Category

• Multiagent Systems: Other Foundations of Multi Agent Systems
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Intelligent Robots: Multi-Robot Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Study fundamental problems in multi-agent systems over multilayer networks.
• Analyze phase space properties and differences between single-layer and multilayer systems.
• Develop algorithms for equivalence problems in restricted MSyDSs.

Innovation

• MSyDSs model multilayer network dynamics
• NP-complete inequivalence for MSyDSs
• Efficient algorithms for restricted MSyDSs