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For the linear metabolic cost, we have proved that the set of minimizers contains a loop-free structure. Moreover, we enriched the energy functional such that it accounts also for robustness of the network, measured in terms of the Fiedler number of the graph with edge weights given by their conductivities. We examined fundamental properties of the modified functional, in particular, its convexity and differentiability. We provided analytical insights into the new model by considering two simple examples. Subsequently, we employed the projected subgradient method to find global minimizers of the modified functional numerically. We then presented two numerical examples, illustrating how the optimal graph's structure and energy expenditure depend on the required robustness of the network.&lt;\/p&gt;<\/jats:p>","DOI":"10.3934\/nhm.2024035","type":"journal-article","created":{"date-parts":[[2024,8,9]],"date-time":"2024-08-09T11:28:57Z","timestamp":1723202937000},"page":"771-799","source":"Crossref","is-referenced-by-count":1,"title":["Robust network formation with biological applications"],"prefix":"10.3934","volume":"19","author":[{"given":"Jan","family":"Haskovec","sequence":"first","affiliation":[{"name":"Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia"}]},{"given":"Vyb\u00edral","family":"Jan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Trojanova 12, 12000 Praha, Czech Republic"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2024035-1","doi-asserted-by":"publisher","unstructured":"R. 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