{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T22:12:26Z","timestamp":1773267146811,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,8,23]],"date-time":"2020-08-23T00:00:00Z","timestamp":1598140800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["1DP1DA048968-01"],"award-info":[{"award-number":["1DP1DA048968-01"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We propose a new metric to characterize the complexity of weighted complex networks. Weighted complex networks represent a highly organized interactive process, for example, co-varying returns between stocks (financial networks) and coordination between brain regions (brain connectivity networks). Although network entropy methods have been developed for binary networks, the measurement of non-randomness and complexity for large weighted networks remains challenging. We develop a new analytical framework to measure the complexity of a weighted network via graph embedding and point pattern analysis techniques in order to address this unmet need. We first perform graph embedding to project all nodes of the weighted adjacency matrix to a low dimensional vector space. Next, we analyze the point distribution pattern in the projected space, and measure its deviation from the complete spatial randomness. We evaluate our method via extensive simulation studies and find that our method can sensitively detect the difference of complexity and is robust to noise. Last, we apply the approach to a functional magnetic resonance imaging study and compare the complexity metrics of functional brain connectivity networks from 124 patients with schizophrenia and 103 healthy controls. The results show that the brain circuitry is more organized in healthy controls than schizophrenic patients for male subjects while the difference is minimal in female subjects. These findings are well aligned with the established sex difference in schizophrenia.<\/jats:p>","DOI":"10.3390\/e22090925","type":"journal-article","created":{"date-parts":[[2020,8,23]],"date-time":"2020-08-23T21:28:06Z","timestamp":1598218086000},"page":"925","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Characterizing the Complexity of Weighted Networks via Graph Embedding and Point Pattern Analysis"],"prefix":"10.3390","volume":"22","author":[{"given":"Shuo","family":"Chen","sequence":"first","affiliation":[{"name":"Division of Biostatistics and Bioinformatics, Department of Epidemiology and Public Health, School of Medicine, University of Maryland, Baltimore, MD 21201, USA"},{"name":"Maryland Psychiatric Research Center, Department of Psychiatry, School of Medicine, University of Maryland, Baltimore, MD 21228, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhen","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Accounting, College of Business and Economics, Towson University, Towson, MD 21252, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chen","family":"Mo","sequence":"additional","affiliation":[{"name":"Maryland Psychiatric Research Center, Department of Psychiatry, School of Medicine, University of Maryland, Baltimore, MD 21228, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiong","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Maryland, College Park, MD 20742, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Kochunov","sequence":"additional","affiliation":[{"name":"Maryland Psychiatric Research Center, Department of Psychiatry, School of Medicine, University of Maryland, Baltimore, MD 21228, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L. Elliot","family":"Hong","sequence":"additional","affiliation":[{"name":"Maryland Psychiatric Research Center, Department of Psychiatry, School of Medicine, University of Maryland, Baltimore, MD 21228, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Latora, V., Nicosia, V., and Russo, G. (2017). Complex Networks: Principles, Methods and Applications, Cambridge University Press.","DOI":"10.1017\/9781316216002"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1381","DOI":"10.1016\/j.physa.2007.10.055","article-title":"Complex network-based time series analysis","volume":"387","author":"Yang","year":"2008","journal-title":"Phys. A Stat. Mech. 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