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Concentration inequalities and martingale inequalities – a survey
 Internet Math
"... Abstract. We examine a number of generalized and extended versions of concentration inequalities and martingale inequalities. These inequalities are effective for analyzing processes with quite general conditions as illustrated in an example for an infinite Polya process and web graphs. 1. ..."
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Cited by 66 (2 self)
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Abstract. We examine a number of generalized and extended versions of concentration inequalities and martingale inequalities. These inequalities are effective for analyzing processes with quite general conditions as illustrated in an example for an infinite Polya process and web graphs. 1.
Random trees and general branching processes
 2005 Preprint math.PR/0503728
, 2007
"... ABSTRACT: We consider a tree that grows randomly in time. Each time a new vertex appears, it chooses exactly one of the existing vertices and attaches to it. The probability that the new vertex chooses vertex x is proportional to w(deg(x)), a weight function of the actual degree of x. The weight fun ..."
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ABSTRACT: We consider a tree that grows randomly in time. Each time a new vertex appears, it chooses exactly one of the existing vertices and attaches to it. The probability that the new vertex chooses vertex x is proportional to w(deg(x)), a weight function of the actual degree of x. The weight function w: N → R+ is the parameter of the model. In [4] and [11] the authors derive the asymptotic degree distribution for a model that is equivalent to the special case, when the weight function is linear. The proof therein strongly relies on the linear choice of w. Using wellestablished results from the theory of general branching processes we give the asymptotical degree distribution for a wide range of weight functions. Moreover, we provide the asymptotic distribution of the tree itself as seen from a randomly selected vertex. The latter approach gives greater insight to the limiting structure of the tree. Our proof is robust and we believe that the method may be used to answer several other questions related to the model. It relies on the fact that considering the evolution of the random tree in continuous time, the process may be viewed as a general branching process, this way classical results can be
Artificial Language Evolution on a Dynamical Interaction Network
, 2007
"... This dissertation studies the impact of a dynamical interaction network on the distributed learning of a common language. In recent years there has been much interest is developing algorithms for enabling populations of agents to converge upon a shared language, and in studying the role of the inter ..."
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This dissertation studies the impact of a dynamical interaction network on the distributed learning of a common language. In recent years there has been much interest is developing algorithms for enabling populations of agents to converge upon a shared language, and in studying the role of the interaction network in this process. The focus so far has been on fixed networks, with various topologies, and simple algorithms, which do not provide a general framework for associating tasks in the environment with language. We try to overcome both these limitations in this work. We derive a new algorithm for generating realistic complex networks, called Noisy Preferential Attachment (NPA). This is a modification of preferential attachment that unifies it with the quasispecies model of molecular evolution. The growing network can now be seen as a process in which the links in the network are undergoing selection, replication, and mutation. We also demonstrate that by varying the mutation rate over time, we can reproduce features of growing networks in the real world. We then model a population of language learning agents on an interaction topology evolving according to NPA and demonstrate that under certain conditions they can converge very rapidly. However, we also note that they always converge to a maximally simple language. This leads us to introduce a method of relating language to task based on an analogy between the agents ’ hypothesis space and an information
A generalized Pólya’s urn with graph based interactions. Arxiv preprint arXiv:1211.1247
, 2013
"... Abstract. Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls raised ..."
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Cited by 4 (2 self)
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Abstract. Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls raised by some fixed power α> 0. We characterize the limiting behavior of the proportion of balls in the bins. The proof uses a dynamical approach to relate the proportion of balls to a vector field. Our main result is that the limit set of the proportion of balls is contained in the equilibria set of the vector field. We also prove that if α < 1 then there is a single point v = v(G,α) with nonzero entries such that the proportion converges to v almost surely. A special case is when G is regular and α ≤ 1. We show e.g. that if G is nonbipartite then the proportion of balls in the bins converges to the uniform measure almost surely. Contents
On a preferential attachment and generalized Pólya’s urn model
 Ann. Appl. Probab
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Streaming Balanced Graph Partitioning Algorithms for Random Graphs
"... The has been a recent explosion in the size of stored data, partially due to advances in storage technology, and partially due to the growing popularity of cloudcomputing and the vast quantities of data generated, motivates the need for streaming algorithms that can compute approximate solutions wit ..."
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The has been a recent explosion in the size of stored data, partially due to advances in storage technology, and partially due to the growing popularity of cloudcomputing and the vast quantities of data generated, motivates the need for streaming algorithms that can compute approximate solutions without full random access to all of the data. We address the problem of computing a balanced kpartitioning of a graph with only one pass over the data. Based on experimental results in [11] we analyze two variants of a randomized greedy algorithm, one that prefers the arg max and one that is proportional, on random graphs with embedded balanced kcuts and theoretically bound the performance of each algorithms the arg max algorithm is able to asymptotically recover the embedded kcut, while, surprisingly, the proportional variant can not. 1
Social Influence and Evolution of Market Share Simla Ceyhan Management Science and Engineering
"... We propose a model for the evolution of market share in the presence of social influence. We study a simple market in which the individuals arrive sequentially and choose one of the products. Their decision of which product to choose is a stochastic function of the inherent quality of the product an ..."
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We propose a model for the evolution of market share in the presence of social influence. We study a simple market in which the individuals arrive sequentially and choose one of the products. Their decision of which product to choose is a stochastic function of the inherent quality of the product and its market share. Using techniques from stochastic approximation theory, we show that market shares converge to an equilibrium. We also derive the market shares at equilibrium in terms of the level of social influence and the inherent fitness of the products. In a special case, when the choice model is a multinomial logit model, we show that inequality in the market increases with social influence and with strong enough social influence, monopoly occurs. These results support the observations made by Salganik et. al. [27] in their experimental study of cultural markets.
Large deviations for the degree structure in preferential attachment schemes
 Ann. Appl. Probab
, 2011
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Strongly reinforced Pólya urns with graphbased competition
, 2015
"... We introduce a class of reinforcement models where, at each time step t, one first chooses a random subset At of colours (independently of the past) from n colours of balls, and then chooses a colour i from this subset with probability proportional to the number of balls of colour i in the urn raise ..."
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We introduce a class of reinforcement models where, at each time step t, one first chooses a random subset At of colours (independently of the past) from n colours of balls, and then chooses a colour i from this subset with probability proportional to the number of balls of colour i in the urn raised to the power α> 1. We consider stability of equilibria for such models and establish the existence of phase transitions in a number of examples, including when the colours are the edges of a graph, a context which is a toy model for the formation and reinforcement of neural connections. We conjecture that for any graph G and all α sufficiently large, the set of stable equilibria is supported on socalled whiskerforests, which are forests whose components have diameter between 1 and 3.
Noisy Preferential Attachment and Language Evolution
"... Abstract. We study the role of the agent interaction topology in distributed language learning. In particular, we utilize the replicatormutator framework of language evolution for the creation of an emergent agent interaction topology that leads to quick convergence. In our system, it is the links ..."
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Abstract. We study the role of the agent interaction topology in distributed language learning. In particular, we utilize the replicatormutator framework of language evolution for the creation of an emergent agent interaction topology that leads to quick convergence. In our system, it is the links between agents that are treated as the units of selection and replication, rather than the languages themselves. We use the Noisy Preferential Attachment algorithm, which is a special case of the replicatormutator process, for generating the topology. The advantage of the NPA algorithm is that, in the shortterm, it produces a scalefree interaction network, which is helpful for rapid exploration of the space of languages present in the population. A change of parameter settings then ensures convergence because it guarantees the emergence of a single dominant node which is chosen as teacher almost always. 1