Siddhartha Banerjee

Greedy sensor selection: Leveraging submodularity

We consider the problem of sensor selection in resource constrained sensor networks. The fusion center selects a subset of k sensors from an available pool of m sensors according to the maximum a posteriori or the maximum likelihood rule. We cast the sensor selection problem as the maximization of a submodular function over uniform matroids, and demonstrate that a greedy sensor selection algorithm achieves performance within (1 − 1 over e ) of the optimal solution. The greedy algorithm has a complexity of O(n3mk), where n is the dimension of the measurement space. The complexity of the algorithm is further reduced to O(n2mk) by exploiting certain structural features of the problem. An application to the sensor selection in linear dynamical systems where the fusion center employs Kalman filtering for state estimation is considered. Simulation results demonstrate the superior performance of the greedy sensor selection algorithm over competing techniques based on convex relaxation.

FAST-PPR: scaling personalized pagerank estimation for large graphs

We propose a new algorithm, FAST-PPR, for computing personalized PageRank: given start node s and target node t in a directed graph, and given a threshold δ, it computes the Personalized PageRank π_s(t) from s to t, guaranteeing that the relative error is small as long πs(t) > δ. Existing algorithms for this problem have a running-time of Ω(1/δ in comparison, FAST-PPR has a provable average running-time guarantee of O(√d/δ) (where d is the average in-degree of the graph). This is a significant improvement, since δ is often O(1/n) (where n is the number of nodes) for applications. We also complement the algorithm with an Ω(1/√δ) lower bound for PageRank estimation, showing that the dependence on δ cannot be improved. We perform a detailed empirical study on numerous massive graphs, showing that FAST-PPR dramatically outperforms existing algorithms. For example, on the 2010 Twitter graph with 1.5 billion edges, for target nodes sampled by popularity, FAST-PPR has a 20 factor speedup over the state of the art. Furthermore, an enhanced version of FAST-PPR has a 160 factor speedup on the Twitter graph, and is at least 20 times faster on all our candidate graphs.

Pricing in Ride-Sharing Platforms: A Queueing-Theoretic Approach

We study optimal pricing strategies for ride-sharing platforms, using a queueing-theoretic economic model. Analysis of pricing in such settings is complex: On one hand these platforms are two-sided - this requires economic models that capture the incentives of both drivers and passengers. On the other hand, these platforms support very high temporal-resolution for data collection and pricing - this requires stochastic models that capture the dynamics of drivers and passengers in the system. We focus our attention on the value of dynamic pricing: where prices can react to instantaneous imbalances between available supply and incoming demand. We find two main results: We first show that profit under any dynamic pricing strategy cannot exceed profit under the optimal static pricing policy (i.e., one which is agnostic of stochastic fluctuations in the system load). This result belies the prevalence of dynamic pricing in practice. Our second result explains the apparent paradox: we show that dynamic pricing is much more robust to fluctuations in system parameters compared to static pricing. Moreover, these results hold even if the monopolist maximizes welfare or throughput. Thus dynamic pricing does not necessarily yield higher performance than static pricing - however, it lets platforms realize the benefits of optimal static pricing, even with imperfect knowledge of system parameters.

Greedy learning of Markov network structure

Markov Random Fields (MRFs), a.k.a. Graphical Models, serve as popular models for networks in the social and biological sciences, as well as communications and signal processing. A central problem is one of structure learning or model selection: given samples from the MRF, determine the graph structure of the underlying distribution. When the MRF is not Gaussian (e.g. the Ising model) and contains cycles, structure learning is known to be NP hard even with infinite samples. Existing approaches typically focus either on specific parametric classes of models, or on the sub-class of graphs with bounded degree; the complexity of many of these methods grows quickly in the degree bound. We develop a simple new ‘greedy’ algorithm for learning the structure of graphical models of discrete random variables. It learns the Markov neighborhood of a node by sequentially adding to it the node that produces the highest reduction in conditional entropy. We provide a general sufficient condition for exact structure recovery (under conditions on the degree/girth/correlation decay), and study its sample and computational complexity. We then consider its implications for the Ising model, for which we establish a self-contained condition for exact structure recovery.

Epidemic Spreading With External Agents

We study epidemic spreading processes in large networks, when the spread is assisted by a small number of external agents: infection sources with bounded spreading power, but whose movement is unrestricted vis-à-vis the underlying network topology. For networks, which are spatially constrained, we show that the spread of infection can be significantly speeded up even by a few such external agents infecting randomly. Moreover, for general networks, we derive upper bounds on the order of the spreading time achieved by certain simple (random/greedy) external-spreading policies. Conversely, for certain common classes of networks such as line graphs, grids, and random geometric graphs, we also derive lower bounds on the order of the spreading time over all (potentially network-state aware and adversarial) external-spreading policies; these adversarial lower bounds match (up to logarithmic factors) the spreading time achieved by an external agent with a random spreading policy. This demonstrates that random, state-oblivious infection-spreading by an external agent is in fact order-wise optimal for spreading in such spatially constrained networks.

Random mobility and the spread of infection

We study infection spreading on large static networks when the spread is assisted by a small number of additional virtually mobile agents. For networks which are “spatially constrained”, we show that the spread of infection can be significantly sped up even by a few virtually mobile agents acting randomly. More specifically, for general networks with bounded virulence (e.g., a single or finite number of random virtually mobile agents), we derive upper bounds on the order of the time taken (as a function of network size) for infection to spread. Conversely, for certain common classes of networks such as linear graphs, grids and random geometric graphs, we also derive lower bounds on the order of the spreading time over all (potentially network-state aware and adversarial) virtual mobility strategies. We show that up to a logarithmic factor, these lower bounds for adversarial virtual mobility match the upper bounds on spreading via an agent with random virtual mobility. This demonstrates that random, state-oblivious virtual mobility is in fact order-wise optimal for dissemination in such spatially constrained networks.

On distributed scheduling with heterogeneously delayed network-state information

We study the problem of distributed scheduling in wireless networks, where each node makes individual scheduling decisions based on heterogeneously delayed network state information (NSI). This leads to inconsistency in the views of the network across nodes, which, coupled with interference, makes it challenging to schedule for high throughputs.We characterize the network throughput region for this setup, and develop optimal scheduling policies to achieve the same. Our scheduling policies have a threshold-based structure and, moreover, require the nodes to use only the “smallest critical subset” of the available delayed NSI to make decisions. In addition, using Markov chain mixing techniques, we quantify the impact of delayed NSI on the throughput region. This not only highlights the value of extra NSI for scheduling, but also characterizes the loss in throughput incurred by lower complexity scheduling policies which use homogeneously delayed NSI.

Towards a queueing-based framework for in-network function computation

We seek to develop network algorithms for function computation in sensor networks. Specifically, we want dynamic joint aggregation, routing, and scheduling algorithms that have analytically provable performance benefits due to in-network computation as compared to simple data forwarding. To this end, we define a class of functions, the Fully-Multiplexible functions, which includes several functions such as parity, MAX, and kth-order statistics. For such functions we characterize the maximum achievable refresh rate of the network in terms of an underlying graph primitive, the min-mincut. In acyclic wireline networks we show that the maximum refresh rate is achievable by a simple algorithm that is dynamic, distributed, and only dependent on local information. In the case of wireless networks we provide a MaxWeight-like algorithm with dynamic flow-splitting, which is shown to be throughput-optimal.

Greedy Sensor Selection under Channel Uncertainty

Estimation in resource constrained sensor networks where the fusion center selects a fixed-size subset from a pool of available sensors observing the states of a linear dynamical system is considered. With some probability, the communication between a selected sensor and the fusion center may fail. It is shown that when the fusion center employs a Kalman filter and desires to minimize a function of the error covariance matrix, sensor selection under communication uncertainty can be cast as the maximization of a submodular function over uniform matroids. We propose a computationally efficient greedy sensor selection scheme achieving performance within (1 -1/ e ) of the optimal non-adaptive policy. Additionally, we propose an efficient adaptive greedy algorithm which achieves (1-1/e) of the optimal adaptive policy. Structural features of the problem are exploited to reduce the complexity of the greedy selection algorithms. We analyze the complexity and present simulation studies which demonstrate efficacy of the proposed techniques.

Bidirectional PageRank Estimation: From Average-Case to Worst-Case

We present a new algorithm for estimating the Personalized PageRank PPR between a source and target node on undirected graphs, with sublinear running-time guarantees over the worst-case choice of source and target nodes. Our work builds on a recent line of work on bidirectional estimators for PPR, which obtained sublinear running-time guarantees but in an average-case sense, for a uniformly random choice of target node. Crucially, we show how the reversibility of random walks on undirected networks can be exploited to convert average-case to worst-case guarantees. While past bidirectional methods combine forward random walks with reverse local pushes, our algorithm combines forward local pushes with reverse random walks. We also discuss how to modify our methods to estimate random-walk probabilities for any length distribution, thereby obtaining fast algorithms for estimating general graph diffusions, including the heat kernel, on undirected networks.