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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.<\/jats:p>","DOI":"10.1051\/ita\/2024014","type":"journal-article","created":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T08:52:48Z","timestamp":1732524768000},"page":"17","source":"Crossref","is-referenced-by-count":1,"title":["Creating a network-state homomorphism through optimization"],"prefix":"10.1051","volume":"58","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2817-3400","authenticated-orcid":false,"given":"Yilun","family":"Shang","sequence":"first","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2024,11,25]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"Hahn G. and Tardif C., Graph homomorphisms: structure and symmetry, in Graph Symmetry, edited by Hahn G., Sabidussi G.. 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