This paper introduces and formalizes the problem of Robust Task Networking Under Attrition (RTNUA), which extends connectivity maintenance in multi-robot systems to explicitly address proactive redundancy and attrition recovery. We then introduce Physics-Informed Robust Employment of Multi-Agent Networks (ΦIREMAN), a topological algorithm leveraging physics-inspired potential fields to solve this problem. In our evaluations, ΦIREMAN consistently outperforms baselines, and is able to maintain greater than 99.9% task uptime despite substantial attrition in simulations with up to 100 tasks and 500 drones, demonstrating both effectiveness and scalability.