Results 1  10
of
88
Exact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics
, 1996
"... For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has ..."
Abstract

Cited by 406 (13 self)
 Add to MetaCart
For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the al...
Uniform spanning forests
 Ann. Probab
, 2001
"... Abstract. We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle (1991) proved that the free and wired spanning forests coincid ..."
Abstract

Cited by 61 (22 self)
 Add to MetaCart
Abstract. We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle (1991) proved that the free and wired spanning forests coincide in Z d and that they give a single tree iff d � 4. In the present work, we extend Pemantle’s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks, potential theory, invariant percolation, and amenability. The uniform spanning forest model is related to random cluster models in statistical physics, but, because of the preceding connections, its analysis can be carried further. Among our results are the following: • The FSF and WSF in a graph G coincide iff all harmonic Dirichlet functions on G are constant. • The tail σfields of the WSF and the FSF are trivial on any graph. • On any Cayley graph that is not a finite extension of Z, all component trees of the WSF have one end; this is new in Z d for d � 5. • On any tree, as well as on any graph with spectral radius less than 1, a.s. all components of the WSF are recurrent. • The basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space H d is analyzed. • A Cayley graph is amenable iff for all ɛ> 0, the union of the WSF and Bernoulli percolation with parameter ɛ is connected. • Harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees. We also present numerous open problems and conjectures.
Rayleigh processes, real trees, and root growth with regrafting
, 2004
"... Abstract. The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous’s Brownian continuum random tree, the random treelike object naturally associated with ..."
Abstract

Cited by 48 (11 self)
 Add to MetaCart
Abstract. The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous’s Brownian continuum random tree, the random treelike object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N → ∞ of both a critical GaltonWatson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous–Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N → ∞ of the Aldous–Broder chain. A key technical ingredient in this work is the use of a pointed Gromov–
Conformally invariant processes in the plane
 Mathematical Surveys and Monographs, 114. American Mathematical Society
, 2005
"... ..."
Scaling limits for minimal and random spanning trees in two dimensions
, 1998
"... A general formulation is presented for continuum scaling limits of stochastic spanning trees. Tightness of the distribution, as δ → 0, is established for the following twodimensional examples: the uniformly random spanning tree on δZ 2, the minimal spanning tree on δZ 2 (with random edge lengths), a ..."
Abstract

Cited by 32 (6 self)
 Add to MetaCart
A general formulation is presented for continuum scaling limits of stochastic spanning trees. Tightness of the distribution, as δ → 0, is established for the following twodimensional examples: the uniformly random spanning tree on δZ 2, the minimal spanning tree on δZ 2 (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in R 2 with density δ −2. A continuum limit is expressed through a consistent collection of trees (made of curves) which includes a spanning tree for every finite set of points in the plane. Sample trees are proven to have the following properties, with probability one with respect to any of the limiting measures: i) there is a single route to infinity (as was known for δ> 0), ii) the tree branches are given by curves which are regular in the sense of Hölder continuity, iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds one, iv) there is a random dense subset of R², of dimension strictly between one and two, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints, v) branching occurs at countably many points in R 2, and vi) the branching numbers are uniformly bounded. The results include tightness for the loop erased random walk (LERW) in two dimensions. The proofs proceed through the derivation of scaleinvariant power bounds on the probabilities of repeated crossings of annuli.
THE DIMENSION OF THE SLE CURVES
, 2008
"... Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ≥0. We prove that, with probability one, the Hausdorff dimension of γ is equal to Min(2,1 + κ/8). ..."
Abstract

Cited by 27 (0 self)
 Add to MetaCart
Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ≥0. We prove that, with probability one, the Hausdorff dimension of γ is equal to Min(2,1 + κ/8).
Limiting the Spread of Misinformation in Social Networks
"... In this work, we study the notion of competing campaigns in a social network. By modeling the spread of influence in the presence of competing campaigns, we provide necessary tools for applications such as emergency response where the goal is to limit the spread of misinformation. We study the probl ..."
Abstract

Cited by 26 (2 self)
 Add to MetaCart
In this work, we study the notion of competing campaigns in a social network. By modeling the spread of influence in the presence of competing campaigns, we provide necessary tools for applications such as emergency response where the goal is to limit the spread of misinformation. We study the problem of influence limitation where a “bad ” campaign starts propagating from a certain node in the network and use the notion of limiting campaigns to counteract the effect of misinformation. The problem can be summarized as identifying a subset of individuals that need to be convinced to adopt the competing (or “good”) campaign so as to minimize the number of people that adopt the “bad ” campaign at the end of both propagation processes. We show that this optimization problem is NPhard and provide approximation guarantees for a greedy solution for various definitions of this problem by proving that they are submodular. Although the greedy algorithm is a polynomial time algorithm, for today’s large scale social networks even this solution is computationally very expensive. Therefore, we study the performance of the degree centrality heuristic as well as other heuristics that have implications on our specific problem. The experiments on a number of closeknit regional networks obtained from the Facebook social network show that in most cases inexpensive heuristics do in fact compare well with the greedy approach.