We provide a systematic study of the problem of finding the source of a computer virus in a network. We model virus spreading in a network with a variant of the popular SIR model and then construct an estimator for the virus source. This estimator is based upon a novel combinatorial quantity which we term rumor centrality. We establish that this is an ML estimator for a class of graphs. We find the following surprising threshold phenomenon: on trees which grow faster than a line, the estimator always has non-trivial detection probability, whereas on trees that grow like a line, the detection probability will go to 0 as the network grows. Simulations performed on synthetic networks such as the popular small-world and scale-free networks, and on real networks such as an internet AS network and the U.S. electric power grid network, show that the estimator either finds the source exactly or within a few hops in different network topologies. We compare rumor centrality to another common network centrality notion known as distance centrality. We prove that on trees, the rumor center and distance center are equivalent, but on general networks, they may differ. Indeed, simulations show that rumor centrality outperforms distance centrality in finding virus sources in networks which are not tree-like.

In the past year or so, an exciting progress has led to throughput optimal design of CSMA-based algorithms for wireless networks. However, such an algorithm suffers from very poor delay performance. A recent work suggests that it is impossible to design a CSMA-like simple algorithm that is throughput optimal and induces low delay for any wireless network. However, wireless networks arising in practice are formed by nodes placed, possibly arbitrarily, in some geographic area. In this paper, we propose a CSMA algorithm with per-node average-delay bounded by a constant, independent of the network size, when the network has geometry (precisely, polynomial growth structure) that is present in any practical wireless network. Two novel features of our algorithm, crucial for its performance, are (a) choice of access probabilities as an appropriate function of queue-sizes, and (b) use of local network topological structures. Essentially, our algorithm is a queue-based CSMA with a minor difference that at each time instance a very small fraction of frozen nodes do not execute CSMA. Somewhat surprisingly, appropriate selection of such frozen nodes, in a distributed manner, lead to the delay optimal performance.

In this work, we consider the popular tree-based search strategy within the framework of reinforcement learning, the Monte Carlo Tree Search (MCTS), in the context of infinite-horizon discounted cost Markov Decision Process (MDP) with deterministic transitions. While MCTS is believed to provide an approximate value function for a given state with enough simulations, cf. [5,6], the claimed proof of this property is incomplete. This is due to the fact that the variant of MCTS, the Upper Confidence Bound for Trees (UCT), analyzed in prior works utilizes "logarithmic" bonus term for balancing exploration and exploitation within the tree-based search, following the insights from stochastic multi-arm bandit (MAB) literature, cf. [1,3]. In effect, such an approach assumes that the regret of the underlying recursively dependent non-stationary MABs concentrates around their mean exponentially in the number of steps, which is unlikely to hold as pointed out in [2], even for stationary MABs.As the key contribution of this work, we establish polynomial concentration property of regret for a class of non-stationary multiarm bandits. This in turn establishes that the MCTS with appropriate polynomial rather than logarithmic bonus term in UCB has the claimed property of [5,6]. Interestingly enough, empirically successful approaches (cf. [10]) utilize a similar polynomial form of MCTS as suggested by our result. Using this as a building block, we argue that MCTS, combined with nearest neighbor supervised learning, acts as a "policy improvement" operator, i.e., it iteratively improves value function approximation for all states, due to combining with supervised learning, despite evaluating at only finitely many states. In effect, we establish that to learn an ε-approximation of the value function for deterministic MDPs with respect to ℓ ∞ norm, MCTS combined with nearest neighbor requires a sample size scaling as O ε −(d+4) , where d is the dimension of the state space. This is nearly optimal due to a minimax lower bound of Ω ε −(d+2) [8] suggesting the strength of the variant of MCTS we propose here and our resulting analysis. 1

The need to rank items based on user input arises in many practical applications such as elections, group decision making and recommendation systems. The primary challenge in such scenarios is to decide on a global ranking based on partial preferences provided by users. The standard approach to address this challenge is to ask users to provide explicit numerical ratings (cardinal information) of a subset of the items. The main appeal of such an approach is the ease of aggregation. However, the rating scale as well as the individual ratings are often arbitrary and may not be consistent from one user to another. A more natural alternative to numerical ratings requires users to compare pairs of items (ordinal information). On the one hand, such comparisons provide an "absolute" indicator of the user's preference. On the other hand, it is often hard to combine or aggregate these comparisons to obtain a consistent global ranking. In this work, we provide a tractable framework for utilizing comparison data as well as first-order marginal information (see Section 2) for the purpose of ranking. We treat the available information as partial samples from an unknown distribution over permutations. We then reduce ranking problems of interest to performing inference on this distribution. Specifically, we consider the problems of (a) finding an aggregate ranking of n items, (b) learning the mode of the distribution, and (c) identifying the top k items. For many of these problems, we provide efficient algorithms to infer the ranking directly from the data without the need to estimate the underlying distribution. In other cases, we use the Principle of Maximum Entropy to devise a concise parameterization of a distribution consistent with observations using only O(n 2 ) parameters, where n is the number of items in question. We propose a distributed, iterative algorithm for estimating the parameters of the distribution. We establish the correctness of the algorithm and identify its rate of convergence explicitly.

We consider a switched (queueing) network in which there are constraints on which queues may be served simultaneously; such networks have been used to effectively model input-queued switches and wireless networks. The scheduling policy for such a network specifies which queues to serve at any point in time, based on the current state or past history of the system. In the main result of this paper, we provide a new class of online scheduling policies that achieve optimal average queue-size scaling for a class of switched networks including input-queued switches. In particular, it establishes the validity of a conjecture (documented in [24]) about optimal queue-size scaling for input-queued switches.