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85
A survey of maxtype recursive distributional equations
 Annals of Applied Probability 15 (2005
, 2005
"... In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent cop ..."
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Cited by 63 (6 self)
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In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(·) is essentially a “maximum ” or “minimum” function. We draw attention to the theoretical question of endogeny: inthe associated recursive tree process X i,aretheX i measurable functions of the innovations process (ξ i)? 1. Introduction. Write
Processes on unimodular random networks
 In preparation
, 2005
"... Abstract. We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasitransitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amen ..."
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Cited by 52 (4 self)
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Abstract. We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasitransitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasitransitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications
Asymptotic enumeration of spanning trees
 Combin. Probab. Comput
, 2005
"... Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that ..."
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Cited by 29 (6 self)
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Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call “tree entropy”, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasitransitive amenable graphs, extending a result of Burton and Pemantle (1993). §1. Introduction. Methods of enumeration of spanning trees in a finite graph G and relations to various areas of mathematics and physics have been investigated for more than 150 years. The number of spanning trees is often called the complexity of the graph, denoted here by τ(G). The best known formula for the complexity, proved in every basic text on graph
Maximum Weight Independent Sets and Matchings in Sparse Random Graphs. Exact Results Using the Local Weak . . .
, 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 28 (7 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
On exchangeable random variables and the statistics of large graphs and hypergraphs
, 2008
"... ..."
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 23 (9 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
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 21 (6 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.
Gaussian limits for random measures in geometric probability
 Ann. Appl. Probab
, 2005
"... We establish Gaussian limits for measures induced by binomial and Poisson point processes in ddimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general central limit theorems are applied to measures induced by random gr ..."
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Cited by 19 (6 self)
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We establish Gaussian limits for measures induced by binomial and Poisson point processes in ddimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general central limit theorems are applied to measures induced by random graphs (nearest neighbor, Voronoi, and sphere of influence graph), random sequential packing models (ballistic deposition and spatial birth growth models), and statistics of germgrain models. 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 17 (7 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