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This assumption is not true of real interdependent infrastructures that have emergency procedures to buffer against cascades. In this work, we consider a node-to-link failure propagation mechanism and establish \u201cweak\u201d interdependence across layers via a tolerance parameter \u03b1 which quantifies the likelihood that a node survives when one of its interdependent neighbors fails. Analytical and numerical results show that weak interdependence produces a striking phenomenon: layers at different positions within the multilayer system experience distinct percolation transitions. Especially, layers with high super degree values percolate in an abrupt manner, while those with low super degree values exhibit both continuous and discontinuous transitions. This novel phenomenon we call mixed percolation transitions has significant implications for network robustness. Previous results that do not consider cascade tolerance and layer super degree may be under- or over-estimating the vulnerability of real systems. Moreover, our model reveals how nodal protection activities influence failure dynamics in interdependent, multilayer systems.<\/jats:p>","DOI":"10.1038\/s41598-018-20019-7","type":"journal-article","created":{"date-parts":[[2018,1,26]],"date-time":"2018-01-26T12:25:31Z","timestamp":1516969531000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":63,"title":["The \u201cweak\u201d interdependence of infrastructure systems produces mixed percolation transitions in multilayer networks"],"prefix":"10.1038","volume":"8","author":[{"given":"Run-Ran","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel A.","family":"Eisenberg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas P.","family":"Seager","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ying-Cheng","family":"Lai","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2018,2,1]]},"reference":[{"key":"20019_CR1","doi-asserted-by":"publisher","first-page":"378","DOI":"10.1038\/35019019","volume":"406","author":"R Albert","year":"2000","unstructured":"Albert, R., Jeong, H. & Barab\u00e1si, A.-L. 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