We consider the problem of predicting edges in a graph from node attributes in an e-commerce setting. Specifically, given nodes labelled with search query text, we want to predict links to related queries that share products. Experiments with a range of deep neural architectures show that simple feedforward networks with an attention mechanism perform best for learning embeddings. The simplicity of these models allows us to explain the performance of attention. We propose an analytically tractable model of query generation, AttEST, that views both products and the query text as vectors embedded in a latent space. We prove (and empirically validate) that the point-wise mutual information (PMI) matrix of the AttEST query text embeddings displays a low-rank behavior analogous to that observed in word embeddings. This low-rank property allows us to derive a loss function that maximizes the mutual information between related queries which is used to train an attention network to learn query embeddings. This AttEST network beats traditional memory-based LSTM architectures by over 20% on F-1 score. We justify this out-performance by showing that the weights from the attention mechanism correlate strongly with the weights of the best linear unbiased estimator (BLUE) for the product vectors, and conclude that attention plays an important role in variance reduction.

The classical Erdös-Gallai theorem kicked off the study of graph realizability by characterizing degree sequences. We extend this line of research by investigating realizability of directed acyclic graphs (DAGs) given both a local constraint via degree sequences and a global constraint via a sequence of reachability values (number of nodes reachable from a given node). We show that, without degree constraints, DAG reachability realization is solvable in linear time, whereas it is strongly NP-complete given upper bounds on in-degree or out-degree. After defining a suitable notion of bicriteria approximation based on consistency, we give two approximation algorithms achieving O(log n) -reachability consistency and O(log n) -degree consistency; the first, randomized, uses LP (Linear Program) rounding, while the second, deterministic, employs a k-set packing heuristic. We end with two conjectures that we hope motivate further study of realizability with reachability constraints.

We define and construct a novel pseudorandom tool, the Markovian hitter. Given an input sequence of n independent random bits, a Markovian hitter produces a sequence of pseudorandom samples in {0, 1}k, in an online fashion, that hits any subset W ⊂ {0, 1}k of size ε2k with probability ≈ 1 -- 2--(n--k)ε. This is comparable to the behavior of truly random samples or classical pseudorandom hitting sets. A Markovian hitter has an additional "Markovian" property of interest: each pseudorandom sample is a function of only the O(k) most recent bits of the input sequence (of random bits).Such Markovian properties are useful in distributed online settings. In particular, we apply Markovian hitters to obtain a new algorithm for the well-studied blind rendezvous problem for cognitive radios. This is the problem faced by two parties equipped with radios that can access channels in potentially different subsets, S1 and S2, of a universe of n channels. Their challenge is to discover each other (by tuning their radios to the same channel at the same time) as quickly as possible. In prior work [3] it was shown that deterministic schedules have a lower bound for rendezvous time of Ω(|S1| · |S2|). We beat this quadratic barrier by utilizing a public source of randomness in conjunction with a Markovian hitter to achieve rendezvous in expected time[EQUATION]We counterbalance this result by establishing two lower bounds on expected rendezvous time: an[EQUATION]bound for the setting with public randomness, and an Ω(|S1| · |S1|) bound in the setting with private randomness but no public randomness, which is a strengthening of the result for deterministic schedules.

There are two key properties of the above. First, given only c, Eve should not be able to learn what m is. Second, Dec(Enc(m)) = m, so that Bob can accurately determine the message that Alice wanted to send. Creating functions Enc and Dec which satisfy this is a fundamental problem of cryptography. But given the above encryption scheme, knowing what these functions are is enough for anyone to decrypt c. This would make it impossible for practical cryptography to be analyzed in the public, for any scheme which is openly discussed would become useless.

Real-world networks often consist of multiple layers, be they infrastructure such as airline networks or social such as collaboration networks. A common aspect to these networks is that there are multiple sub-networks that evolve in parallel on the same node set - these are referred to as multiplex networks. For example, in the case of airline networks, the cities (nodes) have been well-established for several decades if not centuries, but over time new airlines (sub-networks) emerge and each airline creates its own flight linkages between cities. Similarly multiple modalities of communications evolve in parallel between individuals (nodes) such as E-mail, SMS, and Online Social Networks, e.g., Facebook and Twitter. While in some multiplex networks, each layer evolves independently from other layers over time, in other multiple networks, the evolution of a layer is coupled with that of other layers - a process referred to as co-evolution. In this paper, we propose a novel generative model, BINBALL, for a class of multiplex networks whose structure may co-evolve (that is, depend on as well as influence) the structure of the individual networks. We validate our model using a multiplex data set for the European Air Transportation Network (EATN). We also investigate questions regarding the algorithmic complexity of finding short paths through multiplex networks as well as coverage of nodes using a minimum number of layers. We show that while certain problems in this space can be solved in polynomial time, others are NP hard. Among the latter, some problems are approximable, whereas others are not. Finally, we demonstrate that BINBALL is a good generative model since it is able to generate random networks whose degree as well as path length distributions closely match those of the EATN.

In this work, we introduce the Minimum Trilateration Problem, the problem of placing distance measuring guards in a polygon in order to locate points in the interior. We provide the first non-trivial bounds on trilaterating simple polygons, by showing that b 8N 9 c guards suffice for any non-degenerate polygon of N sides, and present an O(N logN) algorithm for the corresponding placement. We also show how this mapping can be efficiently inverted, in order to determine a point’s location given its distances to the guards which can see it.