A preferential attachment model with random initial degrees

In this paper, a random graph process ${G(t)}_{t\geq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{t\geq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+\delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $\tau=\min\{\tau_{W}, \tau_{P}\}$, where $\tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{t\geq 1}$ and $\tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.Comment: In the published form of the paper, the proof of Proposition 2.1 is incomplete. This version contains the complete proo

On a recursive formula for the moments of phase noise

We present a recursive formula for the moments of phase noise in communication systems. The phase noise is modeled using continuous Brownian motion. The recursion is simple and valid for an arbitrary initial phase value. The moments obtained bp the recursion are used to calculate approximations to the probability density function of the phase noise, using orthogonal polynomial series expansions and a maximum entropy criterio

The M/G/1 processor sharing queue as the almost sure limit of feedback queues

In the paper a probabilistic coupling between the M/G/1 processor sharing queue and the M/M/1 feedback queue, with general feedback probabilities, is established. This coupling is then used to prove the almost sure convergence of sojourn times in the feedback model to sojourn times in the M/G/1 processor sharing queue. Using the theory of regenerative processes it follows that for stable queues the stationary distribution of the sojourn time in the feedback model converges in law to the corresponding distribution in the processor sharing model. The results do not depend on Poisson arrival times, but are also valid for general arrival processes

Universality for first passage percolation on sparse random graphs

\u3cp\u3eWe consider first passage percolation on the configuration model with n vertices, and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform X\u3csup\u3e2\u3c/sup\u3e logX-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path namely the hopcount. Writing Ln for the weight of the optimal path, we show that L\u3csub\u3en\u3c/sub\u3e- (log n)/α\u3csub\u3en\u3c/sub\u3e converges to a limiting random variable, for some sequence α\u3csub\u3en\u3c/sub\u3e. Furthermore, the hopcount satisfies a central limit theorem (CLT) with asymptotic mean and variance of order log n. The sequence α\u3csub\u3en\u3c/sub\u3e and the norming constants for the CLT are expressible in terms of the parameters of an associated continuous-time branching process that describes the growth of neighborhoods around a uniformly chosen vertex in the random graph. The limit of L\u3csub\u3en\u3c/sub\u3e-(log n)/ α\u3csub\u3en\u3c/sub\u3e equals the sum of the logarithm of the product of two independent martingale limits, and a Gumbel random variable. So far, for sparse random graph models, such results have only been shown for the special case where the edge weights have an exponential distribution, wherein the Markov property of this distribution plays a crucial role in the technical analysis of the problem. The proofs in the paper rely on a refined coupling between shortest path trees and continuous-time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination.\u3c/p\u3

Distances in random graphs with infinite mean degrees

We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function is regularly varying with exponent . In particular, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed. The minimal number of edges between two arbitrary nodes, also called the graph distance or the hopcount, is investigated when the size of the graph tends to infinity. The paper is part of a sequel of three papers. The other two papers study the case where , and , respectively. The main result of this paper is that the graph distance for converges in distribution to a random variable with probability mass exclusively on the points and . We also consider the case where we condition the degrees to be at most for some , where is the size of the graph. For fixed and such that , the hopcount converges to in probability, while for , the hopcount converges to the same limit as for the unconditioned degrees. The proofs use extreme value theory