Jadbabaie, Ali
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Publication Game Theoretic Analysis of a Strategic Model of Competitive Contagion and Product Adoption in Social Networks(2012-12-01) Fazeli, Arastoo; Jadbabaie, AliIn this paper we propose and study a strategic model of marketing and product adoption in social networks. Two firms compete for the spread of their products in a social network. Considering their fixed budgets, they initially determine the payoff of their products and the number of their initial seeds in a network. Afterwards, neighboring agents play a local coordination game over a fixed network which determines the dynamics of the spreading. Assuming myopic best response dynamics, agents choose a product based on the payoff received by actions of their neighbors. This local update dynamics results in a game-theoretic diffusion process in the network. Utilizing earlier results in the literature, we find a lower and an upper bound on the proportion of product adoptions. We derive an explicit characterization of these bounds based on the payoff of products offered by firms, the initial number of adoptions and the underlying structure of the network. We then consider a case in which after switching to the new product, agents might later switch back to the old product with some fixed rate. We show that depending on the rate of switching back to the old product, the new product might always die out in the network eventually. Finally, we consider a game between two firms aiming to optimize their products adoptions while considering their fixed budgets. We describe the Nash equilibrium of this game and show how the optimal payoffs offered by firms and the initial number of seeds depend on the relative budgets of firms.Publication Distributed coverage verification in sensor networks without location information(2008-12-09) Tahbaz-Salehi, Alireza; Jadbabaie, AliIn this paper, we present a distributed algorithm for detecting coverage holes in a sensor network with no location information. We demonstrate how, in the absence of localization devices, simplicial complexes and tools from computational homology can be used in providing valuable information on the properties of the cover. In particular, we capture the combinatorial relationships among the sensors by the means of the Rips complex, which is the generalization of the proximity graph of the network to higher dimensions. Our approach is based on computation of a certain generator of the first homology of the Rips complex of the network. We formulate the problem of localizing coverage holes as an optimization problem to compute the sparsest generator of the first homology classes. We also demonstrate how subgradient methods can be used in solving this optimization problem in a distributed manner. Finally, non-trivial simulations are provided that illustrate the performance of our algorithm.Publication Optimal Control of Spatially Distributed Systems(2008-08-01) Motee, Nader; Jadbabaie, AliIn this paper, we study the structural properties of optimal control of spatially distributed systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant-dependent coupling functions over arbitrary graphs. We study the structural properties of optimal control problems with infinite-horizon linear quadratic criteria, by analyzing the spatial structure of the solution to the corresponding operator Lyapunov and Riccati equations. The key idea of the paper is the introduction of a special class of operators called spatially decaying (SD). These operators are a generalization of translation invariant operators used in the study of spatially invariant systems. We prove that given a control system with a state-space representation consisting of SD operators, the solution of operator Lyapunov and Riccati equations are SD. Furthermore, we show that the kernel of the optimal state feedback for each subsystem decays in the spatial domain, with the type of decay (e.g., exponential, polynomial or logarithmic) depending on the type of coupling between subsystems.Publication Elastic Multi-Particle Systems for Bounded-Curvature Path Planning(2008-06-11) Ahmadzadeh, Ali; Jadbabaie, Ali; Pappas, George J; Kumar, VijayThis paper investigates a path planning algorithm for Dubins vehicles. Our approach is based on approximation of the trajectories of vehicles using sequence of waypoints and treating each way point as a moving particle in the space. We define interaction forces between the particles such that the resulting multi-particle system will be stable, moreover, the trajectories generated by the waypoints in the equilibria of the multi-particle system will satisfy all of the hard constraint such as bounded-curvature constraint and obstacle avoidance.Publication Density Functions for Navigation Function Based Systems(2006-12-15) Loizou, Savvas G; Jadbabaie, AliIn this paper, we present a scheme for constructing density functions for systems that are almost globally asymptotically stable (i.e., systems for which all trajectories converge to an equilibrium except for a set of measure zero) based on navigation functions. Although recently-proven converse theorems guarantee the existence of density functions for such systems, results are only existential and the construction of a density function for almost globally asymptotically stable systems remains a challenging task. We show that for a specific class of dynamical systems that are defined based on a navigation function, a density function can be easily derived from the system's underlying navigation functionPublication Distributed Quadratic Programming over Arbitrary Graphs(2007-01-01) Motee, Nader; Jadbabaie, AliIn this paper, the locality features of infinitedimensional quadratic programming (QP) optimization problems are studied. Our approach is based on tools from operator theory and ideas from Multi Parametric Quadratic Programming (MPQP). The key idea is to use the spatially decaying operators (SD), which has been recently developed to study spatially distributed systems in [1], to capture couplings between optimization variables in the quadratic cost functional and linear constraints. As an application, it is shown that the problem of receding horizon control of spatially distributed systems with heterogeneous subsystems, input and state constraints, and arbitrary interconnection topologies can be modeled as an infinitedimensional QP problem. Furthermore, we prove that for a convex infinite-dimensional QP in which the couplings are through SD operators, optimal solution is piece-wise affine– represented as convolution sums. More importantly, we prove that the kernel of each convolution sum decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the optimization problem, thereby providing evidence that even centralized solutions to the infinite-dimensional QP has inherent spatial locality.Publication Consensus Over Martingale Graph Processes(2012-06-01) Fazeli, Arastoo; Jadbabaie, AliIn this paper, we consider a consensus seeking process based on repeated averaging in a randomly changing network. The underlying graph of such a network at each time is generated by a martingale random process. We prove that consensus is reached almost surely if and only if the expected graph of the network contains a directed spanning tree. We then provide an example of a consensus seeking process based on local averaging of opinions in a dynamic model of social network formation which is a martingale. At each time step, individual agents randomly choose some other agents to interact with according to some arbitrary probabilities. The interaction is one-sided and results in the agent averaging her opinion with those of her randomly chosen neighbors based on the weights she assigns to them. Once an agent chooses a neighbor, the weights are updated in such a way that the expected values of the weights are preserved. We show that agents reach consensus in this random dynamical network almost surely. Finally, we demonstrate that a Polya Urn process is a martingale process, and our prior results in [1] is a special case of the model proposed in this paper.Publication On Consensus in a Correlated Model of Network Formation Based on a Polya Urn Process(2011-12-01) Fazeli, Arastoo; Jadbabaie, AliIn this paper, we consider a consensus seeking process based on local averaging of opinions in a dynamic model of social network formation. At each time step, individual agents randomly choose another agent to interact with. The interaction is one-sided and results in the agent averaging her opinion with that of her randomly chosen neighbor. Once an agent chooses a neighbor, the probabilities of interactions are updated in such a way that prior interactions are reinforced and future interactions become more likely, resulting in a random consensus process in which networks are highly correlated with each other. Using results of Skyrm and Pemantle and utilizing the de Finetti representation theorem as well as properties of Polya urn processes, we show that this highly correlated process is equivalent to a mixture of i.i.d. processes whose parameters are drawn from a random limit distribution. Therefore, prior results on consensus on i.i.d. processes can be used to show consensus and to compute the statistics of the consensus value in terms of the initial conditions. We provide simple expressions for the mean and the variance of the asymptotic random consensus value in terms of the number of nodes. We also show that the variance converges to a factor of the empirical variance of the initial values that depends only on the size of the network and goes to zero as the size of the network grows.Publication A Necessary and Sufficient Condition for Consensus Over Random Networks(2008-04-01) Tahbaz-Salehi, Alireza; Jadbabaie, AliWe consider the consensus problem for stochastic discrete time linear dynamical systems. The underlying graph of such systems at a given time instance is derived from a random graph process, independent of other time instances. For such a framework, we present a necessary and sufficient condition for almost sure asymptotic consensus using simple ergodicity and probabilistic arguments. This easily verifiable condition uses the spectrum of the average weight matrix. Finally, we investigate a special case for which the linear dynamical system converges to a fixed vector with probability 1.Publication Distributed Geodesic Control Laws for Flocking of Nonholonomic Agents(2007-04-01) Moshtagh, Nima; Jadbabaie, AliWe study the problem of flocking and velocity alignment in a group of kinematic nonholonomic agents in 2 and 3 dimensions. By analyzing the velocity vectors of agents on a circle (for planar motion) or sphere (for 3-D motion), we develop a geodesic control law that minimizes a misalignment potential and results in velocity alignment and flocking. The proposed control laws are distributed and will provably result in flocking when the underlying proximity graph which represents the neighborhood relation among agents is connected. We further show that flocking is possible even when the topology of the proximity graph changes over time, so long as a weaker notion of joint connectivity is preserved.