This letter extends the concept of graph-frequency to graph signals that evolve with time. Our goal is to generalize and, in fact, unify the familiar concepts from time-and graphfrequency analysis. To this end, we study a joint temporal and graph Fourier transform (JFT) and demonstrate its attractive properties. We build on our results to create filters which act on the joint (temporal and graph) frequency domain, and show how these can be used to perform interference cancellation. The proposed algorithms are distributed, have linear complexity, and can approximate any desired joint filtering objective.

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Networks are critical for the security of many computer systems. However, their complex and asynchronous nature often renders it difficult to formally reason about network behavior. Accordingly, it is challenging to provide correctness guarantees, especially during network updates.This paper studies how to update networks while maintaining a most basic safety property, Waypoint Enforcement (WPE): each packet is required to traverse a certain checkpoint (for instance, a firewall). Waypoint enforcement is particularly relevant in today's increasingly virtualized and software-defined networks, where new in-network functionality is introduced flexibly.We show that WPE can easily be violated during network updates, even though both the old and the new policy ensure WPE. We then present an algorithm WayUp that guarantees WPE at any time, while completing updates quickly. We also find that in contrast to other transient consistency properties, WPE cannot always be implemented in a waitfree manner, and that WPE may even conflict with LoopFreedom (LF). Finally, we present an optimal policy update algorithm OptRounds, which requires a minimum number of communication rounds while ensuring both WPE and LF, whenever this is possible.

Many network operations, ranging from attack investigation and mitigation to traffic management, require answering network-wide flow queries in seconds. Although flow records are collected at each router, using available traffic capture utilities, querying the resulting datasets from hundreds of routers across sites and over time, remains a significant challenge due to the sheer traffic volume and distributed nature of flow records. In this paper, we investigate how to improve the response time for a priori unknown network-wide queries. We present Flowyager, a system that is built on top of existing traffic capture utilities. Flowyager generates and analyzes tree data structures, that we call Flowtrees, which are succinct summaries of the raw flow data available by capture utilities. Flowtrees are self-adjusted data structures that drastically reduce space and transfer requirements, by 75% to 95%, compared to raw flow records. Flowyager manages the storage and transfers of Flowtrees, supports Flowtree operators, and provides a structured query language for answering flow queries across sites and time periods. By deploying a Flowyager prototype at both a large Internet Exchange Point and a Tier-1 Internet Service Provider, we showcase its capabilities for networks with hundreds of router interfaces. Our results show that the query response time can be reduced by an order of magnitude when compared with alternative data analytics platforms. Thus, Flowyager enables interactive network-wide queries and offers unprecedented drill-down capabilities to, e.g., identify DDoS culprits, pinpoint the involved sites, and determine the length of the attack. 1 NetFlow is a Cisco trademark, so other vendors market the NetFlow support with other names, e.g., Juniper Networks use the trademark Jflow or cflowd. 2 NetFlow and IPFIX capabilities are available in router series, e.g., Cisco IOS-XR, IOS and Catalyst router [