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 introduce Physics-Informed Robust Employment of Multi-Agent Networks (ΦIREMAN), a topological algorithm leveraging physics-inspired potential fields to solve this problem. Through simulation across 25 problem configurations, ΦIREMAN consistently outperforms the DCCRS baseline, and on large-scale problems with up to 100 tasks and 500 drones, maintains >99.9% task uptime despite substantial attrition, demonstrating both effectiveness and scalability.