Results 1  10
of
31
VERTEXREINFORCED RANDOM WALK ON Z EVENTUALLY GETS STUCK ON FIVE POINTS
, 2004
"... Vertexreinforced random walk (VRRW), defined by Pemantle in 1988, is a random process that takes values in the vertex set of a graph G, which is more likely to visit vertices it has visited before. Pemantle and Volkov considered the case when the underlying graph is the onedimensional integer latt ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
Vertexreinforced random walk (VRRW), defined by Pemantle in 1988, is a random process that takes values in the vertex set of a graph G, which is more likely to visit vertices it has visited before. Pemantle and Volkov considered the case when the underlying graph is the onedimensional integer lattice Z. They proved that the range is almost surely finite and that with positive probability the range contains exactly five points. They conjectured that this second event holds with probability 1. The proof of this conjecture is the main purpose of this paper.
The Power of Choice in Random Walks: An Empirical Study
 In MSWiM
, 2006
"... In recent years randomwalkbased algorithms have been proposed for a variety of networking tasks. These proposals include searching, routing, selfstabilization, and query processing in wireless networks, peertopeer networks and other distributed systems. This approach is gaining popularity becau ..."
Abstract

Cited by 22 (0 self)
 Add to MetaCart
(Show Context)
In recent years randomwalkbased algorithms have been proposed for a variety of networking tasks. These proposals include searching, routing, selfstabilization, and query processing in wireless networks, peertopeer networks and other distributed systems. This approach is gaining popularity because random walks present locality, simplicity, lowoverhead and inherent robustness to structural changes. In this work we propose and investigate an enhanced algorithm that we refer to as random walks with choice. In this algorithm, instead of selecting just one neighbor at each step, the walk moves to the next node after examining a small number of neighbors sampled at random. Our empirical results on random geometric graphs, the model best suited for wireless networks, suggest a significant improvement in important metrics such as the cover time and loadbalancing properties of random walks. We also systematically investigate random walks with choice on networks with a square grid topology. For this case, our simulations indicate that there is an unbounded improvement in cover time even with a choice of only two neighbors. We also observe a large reduction in the variance of the cover time, and a significant improvement in visit load balancing.
Random Processes with Reinforcement
"... This paper surveys recent work in the area of random processes with reinforcement. This include urn schemes, reinforced random walks, and stochastic approximations, as well as some recent continuoustime negative reinforcement models related to selfavoiding walks. Applications of these various mode ..."
Abstract

Cited by 22 (1 self)
 Add to MetaCart
This paper surveys recent work in the area of random processes with reinforcement. This include urn schemes, reinforced random walks, and stochastic approximations, as well as some recent continuoustime negative reinforcement models related to selfavoiding walks. Applications of these various models are cited and discussed, as well as the main analytic tools and directions of present research.
Linearly edgereinforced random walks
, 2006
"... We review results on linearly edgereinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the random walk on infinite graphs. On trees, one has a represe ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
We review results on linearly edgereinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the random walk on infinite graphs. On trees, one has a representation as a random walk in an independent random environment. We review recent results for the random walk on ladders: recurrence, a representation as a random walk in a random environment, and estimates for the position of the random walker.
Edgereinforced random walk on a ladder
 Ann. Probab
, 2005
"... We prove that the edgereinforced random walk on the ladder Z × {1,2} with initial weights a> 3/4 is recurrent. The proof uses a known representation of the edgereinforced random walk on a finite piece of the ladder as a random walk in a random environment. This environment is given by a margina ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
(Show Context)
We prove that the edgereinforced random walk on the ladder Z × {1,2} with initial weights a> 3/4 is recurrent. The proof uses a known representation of the edgereinforced random walk on a finite piece of the ladder as a random walk in a random environment. This environment is given by a marginal of a multicomponent Gibbsian process. A transfer operator technique and entropy estimates from statistical mechanics are used to analyse this Gibbsian process. Furthermore, we prove spatially exponentially fast decreasing bounds for normalized local times of the edgereinforced random walk on a finite piece of the ladder, uniformly in the size of the finite piece. 1
Volkov: Learning to signal: analysis of a microlevel reinforcement model
, 2008
"... We consider the following signaling game. Nature plays first from the set {1, 2}. Player 1 (the Sender) sees this and plays from the set {A, B}. Player 2 (the Receiver) sees only Player 1’s play and plays from the set {1, 2}. Both players win if Player 2’s play equals Nature’s play and lose otherwis ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
We consider the following signaling game. Nature plays first from the set {1, 2}. Player 1 (the Sender) sees this and plays from the set {A, B}. Player 2 (the Receiver) sees only Player 1’s play and plays from the set {1, 2}. Both players win if Player 2’s play equals Nature’s play and lose otherwise. Players are told whether they have won or lost, and the game is repeated. An urn scheme for learning coordination in this game is as follows. Each node of the desicion tree for Players 1 and 2 contains an urn with balls of two colors for the two possible decisions. Players make decisions by drawing from the appropriate urns. After a win, each ball that was drawn is reinforced by adding another of the same color to the urn. A number of equilibria are possible for this game other than the optimal ones. However, we show that the urn scheme achieves asymptotically optimal coordination.
On the transience of processes defined on GaltonWatson trees
 ANN. PROBAB
, 2006
"... We introduce a simple technique for proving the transience of certain processes defined on the random tree G generated by a supercritical branching process. We prove the transience for oncereinforced random walks on G, that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theo ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
We introduce a simple technique for proving the transience of certain processes defined on the random tree G generated by a supercritical branching process. We prove the transience for oncereinforced random walks on G, that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567– 592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on G. Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229–1241] and [Ann. Probab. 18 (1990) 931–958] as part of more general results. A similar technique is applied to a vertexreinforced jump process. A byproduct of our result is that this process is transient on the 3ary tree. Davis and Volkov [Probab. Theory Related Fields 128 (2004) 42–62] proved that a vertexreinforced jump process defined on the bary tree is transient if b ≥ 4 and recurrent if b = 1. The case b = 2 is still open.
Dynamics of VertexReinforced Random Walks
, 2008
"... We generalize a result from Volkov (2001,[21]) and prove that, on an arbitrary connected graph of bounded degree (G, ∼) and for any symmetric reinforcement matrix a = (ai,j) i,j∈V (G), the vertexreinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
We generalize a result from Volkov (2001,[21]) and prove that, on an arbitrary connected graph of bounded degree (G, ∼) and for any symmetric reinforcement matrix a = (ai,j) i,j∈V (G), the vertexreinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete dpartite subgraph with possible loops plus its outer boundary. We first show that, in general, any stable equilibrium of a linear symmetric replicator dynamics with positive payoffs on a graph G satisfies the property that its support is a complete dpartite subgraph of G with possible loops, for some d � 1. This result is used here for the study of VRRWs, but also applies to other contexts such as evolutionary models in population genetics and game theory. Next we generalize the result of Pemantle (1992,[12]) and Benaïm (1997,[1]) relating the asymptotic behaviour of the VRRW to replicator dynamics. This enables us to conclude that, given any neighbourhood of a strictly stable equilibrium with support S, the following event occurs with positive probability: the walk localizes on S ∪ ∂S (where ∂S is the outer boundary of S) and the density of occupation of the VRRW converges, with polynomial rate, to a strictly stable equilibrium in this neighbourhood. 1 General introduction Let (Ω, F,P) be a probability space. Let (G, ∼) be a locally finite connected symmetric graph, and let V (G) be its vertex set which we sometimes also denote