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We study the verification problem in distributed networks, stated as follows. Let H be a subgraph of a network G where each vertex of G knows which edges incident on it are in H. We would like to verify whether H has some properties, e.g., if it is a tree or if it is connected. We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication.In this paper we initiate a systematic study of distributed verification, and give almost tight lower bounds on the running time of distributed verification algorithms for many fundamental problems such as connectivity, spanning connected subgraph, and s − t cut verification. We then show applications of these results in deriving strong unconditional time lower bounds on the hardness of distributed approximation for many classical optimization problems including minimum spanning tree, shortest paths, and minimum cut. Many of these results are the first non-trivial lower bounds for both exact and approximate distributed computation and they resolve previous open questions. Moreover, our unconditional lower bound of approximating minimum spanning tree (MST) subsumes and improves upon the previous hardness of approximation bound of Elkin [STOC 2004] as well as the lower bound for (exact) MST computation of Peleg and Rubinovich [FOCS 1999]. Our result implies that there can be no distributed approximation algorithm for MST that is significantly faster than the current exact algorithm, for any approximation factor.Our lower bound proofs show an interesting connection between communication complexity and distributed computing which turns out to be useful in establishing the time complexity of exact and approximate distributed computation of many problems.

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Hundreds of thousands of photographs are uploaded to the internet every minute through various social networking and photo sharing platforms. While some images get millions of views, others are completely ignored. Even from the same users, different photographs receive different number of views. This begs the question: What makes a photograph popular? Can we predict the number of views a photograph will receive even before it is uploaded? These are some of the questions we address in this work. We investigate two key components of an image that affect its popularity, namely the image content and social context. Using a dataset of about 2.3 million images from Flickr, we demonstrate that we can reliably predict the normalized view count of images with a rank correlation of 0.81 using both image content and social cues. In this paper, we show the importance of image cues such as color, gradients, deep learning features and the set of objects present, as well as the importance of various social cues such as number of friends or number of photos uploaded that lead to high or low popularity of images.