# Spectral Properties of Complex Distributed Intelligence Systems Coupled with an Environment

## TL;DR

This paper introduces a quantum-inspired framework to study how AI systems interact with their environment, showing how network structure affects collective behavior and coherence.

## Contribution

A novel quantum-inspired graph signal processing framework is introduced to model and analyze collective behavior in distributed AI systems.

## Key findings

- Commutative alignment between network structure and external influence leads to coherent dynamics and full phase synchronization.
- Non-commutative interactions with antagonistic couplings cause spectral disorder and disrupt phase coherence.
- Spectral entropy effectively quantifies disorder and external influence in distributed AI systems.

## Abstract

The increasing integration of artificial intelligence agents (AIAs) based on large language models (LLMs) is transforming many spheres of society. These agents act as human assistants, forming Distributed Intelligent Systems (DISs) and engaging in opinion formation, consensus-building, and collective decision-making. However, complex DIS network topologies introduce significant uncertainty into these processes. We propose a quantum-inspired graph signal processing framework to model collective behavior in a DIS interacting with an external environment represented by an influence matrix (IM). System topology is captured using scale-free and Watts–Strogatz graphs. Two contrasting interaction regimes are considered. In the first case, the internal structure fully aligns with the external influence, as expressed by the commutativity between the adjacency matrix and the IM. Here, a renormalization-group-based scaling approach reveals minimal reservoir influence, characterized by full phase synchronization and coherent dynamics. In the second case, the IM includes heterogeneous negative (antagonistic) couplings that do not commute with the network, producing partial or complete spectral disorder. This disrupts phase coherence and may fragment opinions, except for the dominant collective (Perron) mode, which remains robust. Spectral entropy quantifies disorder and external influence. The proposed framework offers insights into designing LLM-participated DISs that can maintain coherence under environmental perturbations.