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43
THE MARKOV CHAIN MONTE CARLO REVOLUTION
"... Abstract. The use of simulation for highdimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through microlocal analysis. 1. ..."
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Abstract. The use of simulation for highdimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through microlocal analysis. 1.
Distributed random access algorithm: Scheduling and congestion control
 IEEE TRANS. INFORM. THEORY
, 2009
"... This paper provides proofs of the rate stability, Harris recurrence, and εoptimality of CSMA algorithms where the backoff parameter of each node is based on its backlog. These algorithms require only local information and are easy to implement. The setup is a network of wireless nodes with a fixed ..."
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Cited by 40 (12 self)
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This paper provides proofs of the rate stability, Harris recurrence, and εoptimality of CSMA algorithms where the backoff parameter of each node is based on its backlog. These algorithms require only local information and are easy to implement. The setup is a network of wireless nodes with a fixed conflict graph that identifies pairs of nodes whose simultaneous transmissions conflict. The paper studies two algorithms. The first algorithm schedules transmissions to keep up with given arrival rates of packets. The second algorithm controls the arrivals in addition to the scheduling and attempts to maximize the sum of the utilities of the flows of packets at the different nodes. For the first algorithm, the paper proves rate stability for strictly feasible arrival rates and also Harris recurrence of the queues. For the second algorithm, the paper proves the ǫoptimality. Both algorithms operate with strictly local information in the case of decreasing step sizes, and operate with the additional information of the number of nodes in the network in the case of constant step size.
Randomized Scheduling Algorithm for Queueing Networks
, 2009
"... There has recently been considerable interest in design of lowcomplexity, myopic, distributed and stable scheduling policies for constrained queueing network models that arise in the context of emerging communication networks. Here, we consider two representative models. One, a model for the collec ..."
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Cited by 27 (7 self)
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There has recently been considerable interest in design of lowcomplexity, myopic, distributed and stable scheduling policies for constrained queueing network models that arise in the context of emerging communication networks. Here, we consider two representative models. One, a model for the collection of wireless nodes communicating through a shared medium, that represents randomly varying number of packets in the queues at the nodes of networks. Two, a buffered circuit switched network model for an optical core of future Internet, to capture the randomness in calls or flows present in the network. The maximum weight scheduling policy proposed by Tassiulas and Ephremide [32] leads to a myopic and stable policy for the packetlevel wireless network model. But computationally it is very expensive (NPhard) and centralized. It is not applicable to the buffered circuit switched network due to the requirement of nonpremption of the calls in the service. As the main contribution of this paper, we present a stable scheduling algorithm for both of these models. The algorithm is myopic, distributed and performs few logical operations at each node per unit time.
THE TOTAL sENERGY OF A MULTIAGENT SYSTEM
 SIAM J. CONTROL OPTIM, VOL. 49, NO. 4, PP. 1680–1706
, 2011
"... We introduce the total senergy of a multiagent system with timedependent links. This provides a new analytical perspective on bidirectional agreement dynamics, which we use to bound the convergence rates of dynamical systems for synchronization, flocking, opinion dynamics, and social epistemology ..."
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We introduce the total senergy of a multiagent system with timedependent links. This provides a new analytical perspective on bidirectional agreement dynamics, which we use to bound the convergence rates of dynamical systems for synchronization, flocking, opinion dynamics, and social epistemology.
Dynamics in congestion games
 In ACM SIGMETRICS/Performance
, 2010
"... Game theoretic modeling and equilibrium analysis of congestion games have provided insights in the performance of Internet congestion control, road transportation networks, etc. Despite the long history, very little is known about their transient (non equilibrium) performance. In this paper, we are ..."
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Cited by 15 (0 self)
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Game theoretic modeling and equilibrium analysis of congestion games have provided insights in the performance of Internet congestion control, road transportation networks, etc. Despite the long history, very little is known about their transient (non equilibrium) performance. In this paper, we are motivated to seek answers to questions such as how long does it take to reach equilibrium, when the system does operate near equilibrium in the presence of dynamics, e.g. nodes join or leave. In this pursuit, we provide three contributions in this paper. First, a novel probabilistic model to capture realistic behaviors of agents allowing for the possibility of arbitrariness in conjunction with rationality. Second, evaluation of (a) time to converge to equilibrium under this behavior model and (b) distance to Nash equilibrium. Finally, determination of tradeoff between the rate of dynamics and quality of performance (distance to equilibrium) which leads to an interesting uncertainty principle. The novel technical ingredients involve analysis of logarithmic Sobolov constant of Markov process with time varying state space and methodically this should be of broader interest in the context of dynamical systems.
Random quantum circuits are approximate 2designs
, 2008
"... Given a universal gate set on two qubits, it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately Haardistributed unitary. However, this requires exponential time. We show that random circuits of only polynomial length will approxima ..."
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Given a universal gate set on two qubits, it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately Haardistributed unitary. However, this requires exponential time. We show that random circuits of only polynomial length will approximate the first and second moments of the Haar distribution, thus forming approximate 1 and 2designs. Previous constructions required longer circuits and worked only for specific gate sets. As a corollary of our main result, we also improve previous bounds on the convergence rate of random walks on the Clifford group. 1
Geometric Ergodicity and the Spectral Gap of NonReversible Markov Chains
, 2009
"... We argue that the spectral theory of nonreversible Markov chains may often be more effectively cast within the framework of the naturally associated weightedL ∞ space LV ∞, instead of the usual Hilbert space L2 = L2(π), where π is the invariant measure of the chain. This observation is, in part, b ..."
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Cited by 13 (0 self)
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We argue that the spectral theory of nonreversible Markov chains may often be more effectively cast within the framework of the naturally associated weightedL ∞ space LV ∞, instead of the usual Hilbert space L2 = L2(π), where π is the invariant measure of the chain. This observation is, in part, based on the following results. A discretetime Markov chain with values in a general state space is geometrically ergodic if and only if its transition. If the chain is reversible, the same equivalence holds kernel admits a spectral gap in LV ∞ with L2 in place of LV ∞, but in the absence of reversibility it fails: There are (necessarily nonreversible, geometrically ergodic) chains that admit a spectral gap in LV ∞ but not in L2. Moreover, if a chain admits a spectral gap in L2, then for any h ∈ L2 there exists a Lyapunov function Vh ∈ L1 such that Vh dominates h and the chain admits a spectral gap in LVh which the chain converges to equilibrium is also briefly discussed. ∞. The relationship between the size of the spectral gap in L V ∞ or L2, and the rate at
Zero” temperature stochastic 3D Ising model and Dimer covering fluctuation: a first step towards mean curvature motion
 Comm. Pure Appl. Math
"... Abstract. We consider the Glauber dynamics for the Ising model with “+ ” boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of “− ” spins disa ..."
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Cited by 11 (5 self)
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Abstract. We consider the Glauber dynamics for the Ising model with “+ ” boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of “− ” spins disappears within a time τ+ which is at most L 2 (log L) c and at least L 2 /(c log L), for some c> 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large timescales, the evolution of the interface between “+ ” and “− ” domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior τ+ ≃ const × L 2, conjectured on heuristic grounds [13, 7]. In dimension d = 2, τ+ can be shown to be of order L 2 without logarithmic corrections: the upper bound was proven in [8] and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [3].
Interacting particle systems as stochastic social dynamics, Bernoulli (to appear
, 2013
"... The style of mathematical models known to probabilists as Interacting Particle Systems and exemplified by the Voter, Exclusion and Contact processes have found use in many academic disciplines. In many such disciplines the underlying conceptual picture is of a social network, where individuals meet ..."
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Cited by 9 (3 self)
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The style of mathematical models known to probabilists as Interacting Particle Systems and exemplified by the Voter, Exclusion and Contact processes have found use in many academic disciplines. In many such disciplines the underlying conceptual picture is of a social network, where individuals meet pairwise and update their “state ” (opinion, activity etc) in a way depending on the two previous states. This picture motivates a precise general setup we call Finite Markov Information Exchange (FMIE) processes. We briefly describe a few less familiar models (Averaging, Compulsive Gambler, Deference, Fashionista) suggested by the social network picture, as well as a few familiar ones.
MEDIUM ACCESS USING QUEUES
 SUBMITTED TO THE ANNALS OF APPLIED PROBABILITY
"... Consider a wireless network of n nodes represented by a graph G = (V, E) where an edge (i, j) ∈ E models the fact that transmissions of i and j interfere with each other, i.e. simultaneous transmissions of i and j become unsuccessful. Hence it is required that at each time instance a set of nonint ..."
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Cited by 9 (2 self)
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Consider a wireless network of n nodes represented by a graph G = (V, E) where an edge (i, j) ∈ E models the fact that transmissions of i and j interfere with each other, i.e. simultaneous transmissions of i and j become unsuccessful. Hence it is required that at each time instance a set of noninterfering nodes (corresponding to an independent set in G) access the wireless medium. To utilize wireless resources efficiently, it is required to arbitrate the access of medium among interfering nodes properly. Moreover, to be of practical use, such a mechanism is required to be fully distributed as well as simple. As the main result of this paper, we provide such a medium access algorithm. It is randomized, fully distributed and simple: each node attempts to access medium at each time with probability that is a function of its local information. We establish its efficiency by showing that the corresponding network Markov chain is positive recurrent as long as the demand imposed on the network can be supported by the wireless network (using any algorithm). In that sense, the proposed algorithm is optimal in terms of utilizing wireless resources. The algorithm is oblivious to the network graph structure, in contrast with the socalled polynomial backoff algorithm that was established to be optimal only for a certain class of graphs. The key methodological innovations are (a) establishing the positive recurrence of coupled Markov chains, and (b) a comparison relation between stationary distributions of Markov chains building upon the classical Markov chain tree theorem.