Statistical inverse problems on graphs with application to flow volume estimation in computer networks

Estimation of flow volumes in computer networks involves the use of data that are either highly aggregated or fairly noisy. We address several conceptual and practical aspects of the use of such data for flow volume estimation in this work. The results presented are often of general statistical interest in addition to their application in computer networks context. First, we study the problem of identifiability of joint distribution of flow volumes in a computer network from aggregate (lower dimensional) measurements collected on its edges. Conceptually, this is a canonical example of a statistical inverse problem. In a significant departure from previous approaches we investigate settings where flow-volumes exhibit dependence. We introduce a number of models that capture spatial, temporal and inter-modal (i.e. between packets and bytes) dependence between flow-volumes. We provide sufficient, sometimes necessary, conditions for the identifiability of the flow volumes distribution (up to mean) under these models. Next we use these results and models to perform computer network tomography using joint modeling for packet curl byte volumes. We highlight various technical challenges, propose different estimating procedures and investigate their properties. Finally, we examine the problem of optimal design in the context of filtering multiple random walks. Specifically we define the steady state E-optimal design criterion and show that the underlying optimization problem is convex. The developed methodology is applied to tracking network flow volumes using sampled data, where the design variable corresponds to controlling the sampling rate.