In the backbone of large-scale networks, traffic flows experience abrupt unusual changes which can result in congestion, and limit the extent to which end-user quality of service requirements are met. Diagnosing such traffic volume anomalies is a crucial task towards engineering the traffic in the network. This is challenging however, since the available data are the superposition of unobservable origin-to-destination (OD) flows per link. Leveraging the low intrinsic-dimensionality of OD flows and the sparse nature of anomalies, a convex program is formulated to unveil anomalies across flows and time. A centralized solver is developed using the proximal gradient algorithm, which offers provable iteration complexity guarantees. An equivalent nonconvex but separable criterion enables in-network processing of link-load measurements, when optimized via the alternating-direction method of multipliers. The novel distributed iterations entail reduced-complexity local tasks, and affordable message passing between neighboring nodes. Interestingly, under mild conditions the distributed algorithm approaches its centralized counterpart. Numerical tests with synthetic and real network data corroborate the effectiveness of the novel scheme.