Creating a network-state homomorphism through optimization

Department of Computer and Information Sciences, Northumbria University, Newcastle upon Tyne NE1 8ST, UK

^* Corresponding author: yilun.shang@northumbria.ac.uk,shylmath@hotmail.com

Received: 9 May 2022
Accepted: 15 October 2024

Abstract

In graph theory, a mapping between two graphs that generally preserves the structure is called a graph homomorphism, which has been a fundamental notion and extensively studied in combinatorial and algebraic areas. Real-valued states are often assigned to the nodes of graphs (also called networks) in theory and applications underpinning the emerging science of networks. In this paper, we present a simple way to create homomorphisms between a network and its state space. The distance-induced structure in the state space is of practical relevance. We characterize the optimal homomorphism with minimum cost in terms of a constrained optimization problem, and demonstrate the calculation with concrete examples.