Creating a network-state homomorphism through optimization

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.