Effective dimensional reduction of complex systems based on tensor networks (2411.13364v2)

Abstract: The exact treatment of Markovian models of complex systems requires knowledge of probability distributions exponentially large in the number of components $n$. Mean-field approximations provide an effective reduction in complexity of the models, requiring only a number of phase space variables polynomial in system size. However, this comes at the cost of losing accuracy close to critical points in the systems dynamics and an inability to capture correlations in the system. In this work, we introduce a tunable approximation scheme for Markovian spreading models on networks based on Matrix Product States (MPS). By controlling the bond dimensions of the MPS, we can investigate the effective dimensionality needed to accurately represent the exact $2^n$ dimensional steady-state distribution. We introduce the entanglement entropy as a measure of the compressibility of the system and find that it peaks just after the phase transition on the disordered side, in line with the intuition that more complex states are at the 'edge of chaos'. We compare the accuracy of the MPS with exact methods on different types of small random networks and with Markov Chain Monte Carlo methods for a simplified version of the railway network of the Netherlands with 55 nodes. The MPS provides a systematic way to tune the accuracy of the approximation by reducing the dimensionality of the systems state vector, leading to an improvement over second-order mean-field approximations for sufficiently large bond dimensions.

• The paper demonstrates that the MPS-based framework effectively reduces high-dimensional probability distributions by tuning bond dimensions.
• It leverages entanglement entropy to quantify system compressibility and highlights the 'edge of chaos' near phase transitions.
• The study optimizes tensor networks using DMRG and Fiedler ordering, outperforming mean-field models on random and real-world networks.

Effective Dimensional Reduction of Complex Systems Based on Tensor Networks

The paper "Effective dimensional reduction of complex systems based on tensor networks" by Wout Merbis et al. explores a sophisticated approach for analyzing complex systems through the lens of tensor network methodologies. The researchers focus on the application of Matrix Product States (MPS) to the network $\epsilon$-SIS (Susceptible-Infected-Susceptible) model. This work aims to address the daunting task of understanding high-dimensional probability distributions inherent in complex systems by leveraging tensor networks for efficient dimensional reduction.

Overview

The authors begin by highlighting the intrinsic complexity of Markovian models of complex systems. These systems are typically characterized by probability distributions exponentially large in the number of components. To manage this complexity, mean-field approximations are often employed, albeit at the cost of losing critical dynamics and correlations near critical points. In contrast, the proposed MPS approach offers a tunable approximation scheme, capable of adjusting the level of accuracy by controlling bond dimensions.

The paper introduces entanglement entropy as a quantitative measure of compressibility within the system, providing insights into when and where dimensionality reduction retains sufficient accuracy. The authors posit that entanglement entropy peaks just after a phase transition on the disordered side, a point known as the "edge of chaos”—a theoretically intriguing region where states are less compressible.

Methodology

The MPS representation of the network $\epsilon$-SIS model is explicated in detail. Each system's state is decomposed into tensor networks, where the bond dimensions—dictated by singular values—serve as indicators of state compressibility. By managing the bond dimension, the research explores the minimal dimensionality necessary to accurately approximate the steady-state distribution.

The paper also regards the effect of bond dimension on MPS accuracy and efficiency. The authors employ the Density Matrix Renormalization Group (DMRG) algorithm to optimize the MPS representation iteratively. A particularly noteworthy aspect of the paper is the application of Fiedler vector ordering to optimize node arrangement within the network, which enhances computational efficiency by reducing MPO bond dimensions.

Results

The research outcomes are demonstrated through simulations on various random network models such as Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks. Results indicate that the MPS method can outperform traditional mean-field approximations in accuracy especially above a certain bond dimension threshold. The paper further extends to examining a real-world-inspired network model based on the Dutch railway system, thereby confirming the MPS's capability to handle large-scale, complex networks efficiently.

Implications and Future Work

This paper contributes significantly to both theoretical and practical understanding of complex systems modeling. By demonstrating how entanglement entropy correlates with system compressibility, it provides a new lens to evaluate where mean-field approximations falter. On the practical side, the MPS method presents a scalable, efficient alternative for studying high-dimensional systems and rare event statistics.

For future research, several intriguing paths emerge. Further exploration could extend to more complex network models, integrating higher-order interactions, or applying more sophisticated tensor network structures such as projected entangled pair states (PEPS) for systems beyond one-dimensional configurations. Another potential vector for investigation is the application of this framework to time-evolving systems, leveraging techniques like time-evolving block decimation (TEBD) to capture dynamics beyond steady states.

In conclusion, this paper opens new doors for efficiently managing the complexity of large networks using tensor networks, marking a compelling advance in our ability to analyze and simulate complex, high-dimensional systems.