Tensor Network Renormalization as an Ultra-calculus for Complex System Dynamics

Abstract

In this chapter, following the previous one, we briefly present the modern approach to real-space renormalization group (RG) theory based on tensor network formulations which was developed during the last two decades. The aim of this sequel is to suggest a novel framework based on tensor networks in order to find the fixed points of complex systems via coarse-graining. The main result of RG is that it provides a systematic way to study the collective dynamics of a large ensemble of elements that interact according to a complex underlying network topology. RG explicitly seeks the fixed points of the complex system in the space of interactions and unravels the universality class of the complex system as well as calculates a plethora of important observables. We hope that tensor networks can particularly pave the way for better understanding of the sustainable interdependent networks (Amini et al., Sustainable interdependent networks: from theory to application, 2018) through proposing efficient computational strategies and discovering insightful features of the network behaviors.