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Recently, Hermitian adjacency matrices have been proposed to investigate directed graph structures. Previous studies have demonstrated that these matrices can extract valuable information for clustering. In this paper, we propose the complex non-backtracking matrix that integrates the properties of the Hermitian adjacency matrix and the non-backtracking matrix. The proposed matrix has similar properties with the non-backtracking matrix of undirected graphs. We reveal relationships between the complex non-backtracking matrix and the Hermitian adjacency matrix. Also, we provide intriguing insights that this matrix representation holds cluster information, particularly for sparse directed graphs.<\/jats:p>","DOI":"10.1093\/comnet\/cnaf012","type":"journal-article","created":{"date-parts":[[2025,6,29]],"date-time":"2025-06-29T05:55:03Z","timestamp":1751176503000},"source":"Crossref","is-referenced-by-count":0,"title":["Complex non-backtracking matrix for directed graphs"],"prefix":"10.1093","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-1849-5486","authenticated-orcid":false,"given":"Keishi","family":"Sando","sequence":"first","affiliation":[{"name":"Department of Statistical Science, The Graduate University for Advanced Studies , Kanagawa,","place":["Japan"]}]},{"given":"Hideitsu","family":"Hino","sequence":"additional","affiliation":[{"name":"The Institute of Statistical Mathematics , 10-3 Midoricho , Tokyo,","place":["Japan"]}]}],"member":"286","published-online":{"date-parts":[[2025,6,29]]},"reference":[{"key":"2025072206313214800_cnaf012-B1","doi-asserted-by":"crossref","first-page":"219","DOI":"10.2969\/jmsj\/01830219","article-title":"On discrete subgroups of the two by two projective linear group over p-adic fields","volume":"18","author":"Ihara","year":"1966","journal-title":"J Math Soc Jpn"},{"key":"2025072206313214800_cnaf012-B2","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1007\/BFb0075662","volume-title":"Curvature and Topology of Riemannian Manifolds","author":"Sunada","year":"1986"},{"year":"1989","author":"Hashimoto","first-page":"211","key":"2025072206313214800_cnaf012-B3"},{"key":"2025072206313214800_cnaf012-B4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/16-AOP1142","article-title":"Nonbacktracking spectrum of random graphs: community detection and nonregular Ramanujan graphs","volume":"46","author":"Bordenave","year":"2018","journal-title":"Ann Probab"},{"key":"2025072206313214800_cnaf012-B5","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1137\/17M1112297","article-title":"The deformed graph Laplacian and its applications to network centrality analysis","volume":"39","author":"Grindrod","year":"2018","journal-title":"SIAM J Matrix Anal Appl"},{"key":"2025072206313214800_cnaf012-B6","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01908075","article-title":"Comparing partitions","volume":"2","author":"Hubert","year":"1985","journal-title":"J. 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