This paper studies how to provision edge computing and network resources for complex microservice-based applications (MSAs) in face of uncertain and dynamic geo-distributed demands. The complex inter-dependencies between distributed microservice components make load balancing for MSAs extremely challenging, and the dynamic geo-distributed demands exacerbate load imbalance and consequently congestion and performance loss. In this paper, we develop an edge resource provisioning model that accurately captures the inter-dependencies between microservices and their impact on load balancing across both computation and communication resources. We also propose a robust formulation that employs explicit risk estimation and optimization to hedge against potential worst-case load fluctuations, with controlled robustness-resource trade-off. Utilizing a data-driven approach, we provide a solution that provides risk estimation with measurement data of past load geo-distributions. Simulations with real-world datasets have validated that our solution provides the important robustness crucially needed in MSAs, and performs superiorly compared to baselines that neglect either network or inter-dependency constraints.