{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T10:53:52Z","timestamp":1775040832812,"version":"3.50.1"},"reference-count":53,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,13]],"date-time":"2021-09-13T00:00:00Z","timestamp":1631491200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2018\/21934-5"],"award-info":[{"award-number":["2018\/21934-5"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2018\/04654-9"],"award-info":[{"award-number":["2018\/04654-9"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["303855\/2019-3"],"award-info":[{"award-number":["303855\/2019-3"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002322","name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","doi-asserted-by":"publisher","award":["001"],"award-info":[{"award-number":["001"]}],"id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Graphs\/networks have become a powerful analytical approach for data modeling. Besides, with the advances in sensor technology, dynamic time-evolving data have become more common. In this context, one point of interest is a better understanding of the information flow within and between networks. Thus, we aim to infer Granger causality (G-causality) between networks\u2019 time series. In this case, the straightforward application of the well-established vector autoregressive model is not feasible. Consequently, we require a theoretical framework for modeling time-varying graphs. One possibility would be to consider a mathematical graph model with time-varying parameters (assumed to be random variables) that generates the network. Suppose we identify G-causality between the graph models\u2019 parameters. In that case, we could use it to define a G-causality between graphs. Here, we show that even if the model is unknown, the spectral radius is a reasonable estimate of some random graph model parameters. We illustrate our proposal\u2019s application to study the relationship between brain hemispheres of controls and children diagnosed with Autism Spectrum Disorder (ASD). We show that the G-causality intensity from the brain\u2019s right to the left hemisphere is different between ASD and controls.<\/jats:p>","DOI":"10.3390\/e23091204","type":"journal-article","created":{"date-parts":[[2021,9,13]],"date-time":"2021-09-13T21:37:12Z","timestamp":1631569032000},"page":"1204","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Granger Causality among Graphs and Application to Functional Brain Connectivity in Autism Spectrum Disorder"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4523-2062","authenticated-orcid":false,"given":"Ad\u00e8le Helena","family":"Ribeiro","sequence":"first","affiliation":[{"name":"Data Science Institute, Columbia University, New York, NY 10027, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maciel Calebe","family":"Vidal","sequence":"additional","affiliation":[{"name":"Insper Institute of Education and Research, S\u00e3o Paulo 04546-042, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jo\u00e3o Ricardo","family":"Sato","sequence":"additional","affiliation":[{"name":"Center of Mathematics, Computing and Cognition, Universidade Federal do ABC, Santo Andr\u00e9 09210-580, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7756-7051","authenticated-orcid":false,"given":"Andr\u00e9","family":"Fujita","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Statistics, University of S\u00e3o Paulo, S\u00e3o Paulo 05508-090, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1038\/nrm2503","article-title":"Modelling and analysis of gene regulatory networks","volume":"9","author":"Karlebach","year":"2008","journal-title":"Nat. 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