We consider the problem of network anomaly detection given the data collected and processed over large distributed systems. Our algorithmic framework can be seen as an approximate, distributed version of the well-known Principal Component Analysis (PCA) method, which is concerned with continuously tracking the behavior of the data projected onto the residual subspace of the principal components within error bound guarantees. Our approach consists of a protocol for local processing at individual monitoring devices, and global decision-making and monitoring feedback at a coordinator. A key ingredient of our framework is an analytical method based on stochastic matrix perturbation theory for balancing the tradeoff between the accuracy of our approximate network anomaly detection, and the amount of data communication over the network.