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89
The objective method: Probabilistic combinatorial optimization and local weak convergence
, 2003
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Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees
, 2007
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Network Externalities and the Deployment of Security Features
 and Protocols in the Internet. ACM SIGMETRICS 08
"... Getting new security features and protocols to be widely adopted and deployed in the Internet has been a continuing challenge. There are several reasons for this, in particular economic reasons arising from the presence of network externalities. Indeed, like the Internet itself, the technologies to ..."
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Cited by 42 (10 self)
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Getting new security features and protocols to be widely adopted and deployed in the Internet has been a continuing challenge. There are several reasons for this, in particular economic reasons arising from the presence of network externalities. Indeed, like the Internet itself, the technologies to secure it exhibit network effects: their value to individual users changes as other users decide to adopt them or not. In particular, the benefits felt by early adopters of security solutions might fall significantly below the cost of adoption, making it difficult for those solutions to gain attraction and get deployed at a large scale. Our goal in this paper is to model and quantify the impact of such externalities on the adoptability and deployment of security features and protocols in the Internet. We study a network of interconnected agents, which are subject to epidemic
Maximum Weight Independent Sets and Matchings in Sparse Random Graphs. Exact Results Using the Local Weak Convergence Method
, 2005
"... Let G(n, c/n) and Gr(n) be an nnode sparse random graph and a sparse random rregular graph, respectively, and let I(n, r) and I(n, c) be the sizes of the largest independent set in G(n, c/n) and Gr(n). The asymptotic value of I(n, c)/n as n →∞, can be computed using the KarpSipser algorithm when ..."
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Cited by 39 (11 self)
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Let G(n, c/n) and Gr(n) be an nnode sparse random graph and a sparse random rregular graph, respectively, and let I(n, r) and I(n, c) be the sizes of the largest independent set in G(n, c/n) and Gr(n). The asymptotic value of I(n, c)/n as n →∞, can be computed using the KarpSipser algorithm when c ≤ e. For random cubic graphs, r = 3, it is only known that.432 ≤ lim infn I(n,3)/n ≤ lim sup n I(n,3)/n ≤.4591 with high probability (w.h.p.) as n →∞, as shown in Frieze and Suen [Random Structures Algorithms 5 (1994), 649–664] and Bollabas [European J Combin 1 (1980), 311–316], respectively. In this paper we assume in addition that the nodes of the graph are equipped with nonnegative weights, independently generated according to some common distribution, and we consider instead the maximum weight of an independent set. Surprisingly, we discover that for certain weight distributions, the limit limn I(n, c)/n can be computed exactly even when c> e, and limn I(n, r)/n can be computed exactly for some r ≥ 1. For example, when the weights are exponentially distributed with parameter 1, limn I(n,2e)/n ≈.5517, and limn I(n,3)/n ≈.6077. Our results are established using the recently developed local weak convergence method further reduced to a certain local optimality property exhibited by the models we
TIGHTNESS FOR A FAMILY OF RECURSION EQUATIONS
"... Abstract. In this paper, we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on treelike structures. Examples include the maximal displacement of branching random walk in one dimension, and the cover time of symmetric ..."
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Cited by 33 (6 self)
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Abstract. In this paper, we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on treelike structures. Examples include the maximal displacement of branching random walk in one dimension, and the cover time of symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corresponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will facilitate their application in other settings. 1.
A Local Mean Field Analysis of Security Investments in Networks. arXiv:0803.3455
, 2008
"... Getting agents in the Internet, and in networks in general, to invest in and deploy security features and protocols is a challenge, in particular because of economic reasons arising from the presence of network externalities. Our goal in this paper is to model and investigate the impact of such exte ..."
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Cited by 32 (8 self)
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Getting agents in the Internet, and in networks in general, to invest in and deploy security features and protocols is a challenge, in particular because of economic reasons arising from the presence of network externalities. Our goal in this paper is to model and investigate the impact of such externalities on security investments in a network. Specifically, we study a network of interconnected agents subject to epidemic risks such as viruses and worms where agents can decide whether or not to invest some amount to deploy security solutions. We consider both cases when the security solutions are strong (they perfectly protect the agents deploying them) and when they are weak. We make three contributions in the paper. First, we introduce a general model which combines an epidemic propagation model
Counting Without Sampling: Asymptotics of the LogPartition Function for Certain Statistical Physics Models
, 2006
"... In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard ..."
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Cited by 29 (6 self)
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In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hardcore (independent set) model when the activity parameter λ is small, and also the Potts (qcoloring) model. We only consider the graphs with large girth. In particular, we prove that asymptotically the logarithm of the number of independent sets of any rregular graph with large girth when rescaled is approximately constant if r ≤ 5. For example, we show that every 4regular nnode graph with large girth has approximately (1.494 ···) nmany independent sets, for large n. Further, we prove that for every rregular graph with r ≥ 2, with n nodes and large girth, the number of proper q ≥ r +1 colorings is approximately [q(1 − 1 q) r 2] n, for large n. We also show that these results hold for random regular graphs with high probability (w.h.p.) as well. As a byproduct of our method we obtain simple algorithms for the problem of computing approximately the logarithm of the number of independent sets and proper colorings, in low degree
Counting without sampling. New algorithms for enumeration problems using statistical physics
 IN PROCEEDINGS OF SODA
, 2006
"... We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by developments in stati ..."
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Cited by 29 (8 self)
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We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by developments in statistical physics in connection with correlation decay properties of Gibbs measures and its implications to uniqueness of Gibbs measures on infinite trees, reconstruction problems and local weak convergence methods. On a negative side, our algorithms provide ǫapproximations only to the logarithms of the size of a feasible set (also known as free energy in statistical physics). But on the positive side, our approach provides deterministic as opposed to probabilistic guarantee on approximations. Moreover, for some regular graphs we obtain explicit values for the counting problem. For example, we show that every 4regular nnode graph with large girth has approximately (1.494...) n independent sets, and in every rregular graph with n nodes and large girth the number of q ≥ r + 1proper colorings is approximately [q(1 − 1 r q) 2] n, for large n. In statistical physics terminology, we compute explicitly the limit of the logpartition function. We extend our results to random regular graphs. Our explicit results would be hard to derive via the Markov chain method.
RANDOM RECURRENCE EQUATIONS AND RUIN IN A MARKOVDEPENDENT STOCHASTIC ECONOMIC ENVIRONMENT
, 2009
"... We develop sharp large deviation asymptotics for the probability of ruin in a Markovdependent stochastic economic environment and study the extremes for some related Markovian processes which arise in financial and insurance mathematics, related to perpetuities and the ARCH(1) and GARCH(1, 1) time ..."
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Cited by 18 (3 self)
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We develop sharp large deviation asymptotics for the probability of ruin in a Markovdependent stochastic economic environment and study the extremes for some related Markovian processes which arise in financial and insurance mathematics, related to perpetuities and the ARCH(1) and GARCH(1, 1) time series models. Our results build upon work of Goldie [Ann. Appl. Probab. 1 (1991) 126–166], who has developed tail asymptotics applicable for independent sequences of random variables subject to a random recurrence equation. In contrast, we adopt a general approach based on the theory of Harris recurrent Markov chains and the associated theory of nonnegative operators, and meanwhile develop certain recurrence properties for these operators under a nonstandard “Gärtner–Ellis” assumption on the driving process.