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Burst-parallel serverless applications invoke thousands of short-lived distributed functions to complete complex jobs such as data analytics, video encoding, or compilation. While these tasks execute in seconds, starting and configuring the virtual network they rely on is a major bottleneck that can consume up to 84% of total startup time. In this paper we characterize the magnitude of this network cold start problem in three popular overlay networks, Docker Swarm, Weave, and Linux Overlay. We focus on end-to-end startup time that encompasses both the time to boot a group of containers as well as interconnecting them. Our primary observation is that existing overlay approaches for serverless networking scale poorly in short-lived serverless environments. Based on our findings we develop Particle, a network stack tailored for multi-node serverless overlay networks that optimizes network creation without sacrificing multi-tenancy, generality, or throughput. When integrated into a serverless burst-parallel video processing pipeline, Particle improves application runtime by 2.4--3X over existing overlays.