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.

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 well-established 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

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.

Abstract—In this paper we consider a one-dimensional random geometric graph process with the inter-nodal gaps evolving according to an exponential AR(1) process. The transition probability matrix and stationary distribution are derived for the Markov chains concerning connectivity and the number of components. We analyze the algorithm for hitting time regarding disconnectivity. In addition to dynamical properties, we also study topological properties for static snapshots. We obtain the degree distributions as well as asymptotic precise bounds and strong law of large numbers for connectivity threshold distance and the largest nearest neighbor distance amongst others. Both exact results and limit theorems are provided in this paper. Keywords—random geometric graph, autoregressive process, degree, connectivity, Markovian, wireless network. I.