We investigate the implications of different forms of multi-group connectivity. Four multi-group connectivity modalities are considered: co-memberships, edge bundles, bridges, and liaison hierarchies. We propose generative models to generate these four modalities. Our models are variants of planted partition or stochastic block models conditioned under certain topological constraints. We report findings of a comparative analysis in which we evaluate these structures, controlling for their edge densities and sizes, on mean rates of information propagation, convergence times to consensus, and steady state deviations from the consensus value in the presence of noise as network size increases.

We study the problem of quickest detection of anomalies in an environment under extreme uncertainties in sensor measurements. The robotic roadmap corresponding to the environment can be represented as a graph with an arbitrary topology. We analyze the Ensemble CUSUM Algorithm for this surveillance problem. We quantify the delay in detection of anomalies using the Ensemble CUSUM Algorithm and also frame an optimization problem to minimize this detection delay. We then provide an upper bound on the optimal detection delay and frame a convex optimization problem to minimize this upper bound. We also propose an efficient policy that achieves this upper bound and can be computed by solving a semidefinite program. We illustrate the efficacy of the Ensemble CUSUM Algorithm using numerical simulations. We observe that the efficient policy outperforms policies based on other well-known Markov chains.

We introduce a novel dynamic vehicle routing problem termed the Radially Escaping Targets (RET) problem in which mobile targets appear uniformly randomly on a disk according to a stochastic process and move radially outward to escape the disk in a minimum amount of time. A single vehicle is assigned the task of intercepting the targets before they escape. We first obtain two fundamental upper bounds on the fraction of targets intercepted by the vehicle in the steady state-termed the capture fraction-for the RET problem. We then propose three policies to maximize the capture fraction for the RET problem, and identify parameter regimes in which they are suitable. All three policies are within a constant factor of the optimal in specific parameter regimes. For the asymptotic regime of low arrival rate, this factor is equal to one. For the asymptotic regime of high arrival rate, the factor is equal to 2.52 when the disk radius is greater than or equal to one. For moderate speed regimes, this factor is dependent on the target speed. We verify performance of the policies with numerical simulations.

This article provides analysis and optimization results for the mean first passage time, also known as the Kemeny constant, of a Markov chain. First, we generalize the notion of the Kemeny constant to environments with heterogeneous travel and service times, denote this generalization as the weighted Kemeny constant, and we characterize its properties. Second, for reversible Markov chains, we show that the minimization of the Kemeny constant and its weighted counterpart can be formulated as convex optimization problems and, moreover, as semidefinite programs. Third, we apply these results to the design of stochastic surveillance strategies for quickest detection of anomalies in network environments. We numerically illustrate the proposed design: compared with other well-known Markov chains, the performance of our Kemeny-based strategies are always better and in many cases substantially so.

We study a dynamic vehicle routing problem for moving targets. Our setup is as follows: (i) targets appear uniformly distributed on a unit disk via a temporal Poisson process and move radially outward with a constant speed v, (ii) a single vehicle with speed greater than that of the targets aims to intercepts them before they escape the disk. With the goal of maximizing the fraction of captured targets in the steady state, we propose three policies: Capturable Nearest Neighbor (CNN), Sector Wise (SW) and Stay Near Boundary (SNB) policy. We derive lower bounds on the fraction of targets captured by the CNN, SW and SNB policies. The CNN policy is shown to be optimal for arrival rate λ below a critical value. For arrival rates above another critical value, the fraction of targets captured by the SW policy is more than that of the CNN policy. Finally, the SNB policy is within a constant factor of optimal in the limiting regime of ν → 0+ and λ → +∞. We present numerical simulations to illustrate our results.

This paper analyzes a discrete-time algorithm to synchronize a group of agents moving back and forth on a ring. Each agent or "bead" changes direction upon encountering another bead moving in the opposite direction. Communication is sporadic: two beads are able to exchange information only when they come sufficiently close. This allows agents to update their state including their velocity and desired sweeping arc on the boundary. Our analysis makes use of consensus algorithms tools and guarantees that for a given set of initial conditions, synchrony is asymptotically reached.

We propose stochastic surveillance strategies for quickest detection of anomalies in discrete network environments. Our surveillance strategy is determined by optimizing the mean first passage time also known as the Kemeny constant of a Markov chain. We generalize the notion of the Kemeny constant to environments with heterogeneous travel and service times, denote this generalization as the weighted Kemeny constant, and characterize its properties. For reversible Markov chains, we show that both the Kemeny constant and its heterogeneous counterpart can be formulated as convex optimization problems and, moreover, can be expressed as semidefinite programs (SDPs). We numerically illustrate the proposed design: compared with other well-known Markov chains, the performance of our Kemeny-based strategies are always better and in many cases substantially so.

The problem of combustion instabilities arising from the thermoacoustic interactions in a ducted premixed flame model is considered. Contrary to the conventionally used low-order models to describe such systems, a high dimensional model governed by a set of the acoustic equations coupled to the equations for flame dynamics is employed. The flame front is discretized into finite flame elements to assign internal degrees of freedom to the front and track its evolution. Model reduction schemes, namely proper orthogonal decomposition (POD) and balanced truncation, are used to reduce the size of this model. Compared to POD modes, balanced modes show superior input-output characteristics, in agreement with the full model. POD modes on the other hand capture the transient growth in the model while the balanced modes do not. Performance of POD modes is highly dependent on the snapshots used for their computation. A linear quadratic Gaussian (LQG) framework using the reduced-order model based on 16 balanced modes is formulated to control thermoacoustic instabilities in the linear model. The controllers thus obtained are then used on the nonlinear model. They successfully curtail limit cycle oscillations in the nonlinear plant and also avoid subcritical transition to instability.