Distances in random graphs with finite mean and infinite variance degrees

In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent $\tau\in (2,3)$. The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph with $N$ nodes is investigated when $N\to \infty$. When $\tau\in (2,3)$, this graph distance grows like $2\frac{\log\log N}{|\log(\tau-2)|}$. In different papers, the cases $\tau>3$ and $\tau\in (1,2)$ have been studied. We also study the fluctuations around these asymptotic means, and describe their distributions. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.Comment: 52 pages, 4 figure

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 $F$ is regularly varying with exponent $\tau\in (1,2)$. Thus, 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, in a graph with $N$ nodes is investigated when $N\to \infty$. The paper is part of a sequel of three papers. The other two papers study the case where $\tau \in (2,3)$, and $\tau \in (3,\infty),$ respectively. The main result of this paper is that the graph distance converges for $\tau\in (1,2)$ to a limit random variable with probability mass exclusively on the points 2 and 3. We also consider the case where we condition the degrees to be at most $N^{\alpha}$ for some $\alpha>0.$ For $\tau^{-1}<\alpha<(\tau-1)^{-1}$, the hopcount converges to 3 in probability, while for $\alpha>(\tau-1)^{-1}$, the hopcount converges to the same limit as for the unconditioned degrees. Our results give convincing asymptotics for the hopcount when the mean degree is infinite, using extreme value theory.Comment: 20 pages, 2 figure

On Lossless Compression of 1-bit Audio Signals

In this paper we consider the problem of lossless compression of 1-bit audio signals. We study the properties of some existing proposed solutions. We also discuss possible improvements. Other methods have been considered, and the results are reported

The Abelian sandpile: a mathematical introduction

We give a simple rigourous treatment of the classical results of the abelian sandpile model. Although we treat results which are well-known in the physics literature, in many cases we did not find complete proofs in the literature. The paper tries to fill the gap between the mathematics and the physics literature on this subject, and also presents some new proofs. It can also serve as an introduction to the model

A phase transition for the diameter of the configuration model

In this paper, we study the configuration model (CM) with i.i.d. degrees. We establish a phase transition for the diameter when the power-law exponent τ of the degrees satisfies τ ∈ (2, 3). Indeed, we show that for τ&gt; 2 and when vertices with degree 1 or 2 are present with positive probability, the diameter of the random graph is, with high probability, bounded from below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are 3 or more, we show that, for τ ∈ (2, 3), the diameter of the graph is, with high probability, bounded from above by a constant times the log log of the size of the graph

Random graphs with arbitrary i.i.d. degrees

In this paper we study distances and connectivity properties of random graphs with an arbitrary i.i.d. degree sequence. When the tail of the degree distribution is regularly varying with exponent 1 − τ there are three distinct cases: (i) τ&gt; 3, where the degrees have finite variance, (ii) τ ∈ (2, 3), where the degrees have infinite variance, but finite mean, and (iii) τ ∈ (1, 2), where the degrees have infinite mean. These random graphs can serve as models for complex networks where degree power laws are observed. The distances between pairs of nodes in the three cases mentioned above have been studied in three previous publications, and we survey the results obtained there. Apart from the critical cases τ = 1, τ = 2 and τ = 3, this completes the scaling picture. We explain the results heuristically and describe related work and open problems. We also compare the behavior in this model to Internet data, where a degree power law with exponent τ ≈ 2.2 is observed. Furthermore, in this paper we derive results concerning the connected components and the diameter. We give a criterion when there exists a unique largest connected component of size proportional to the size of the graph, and study sizes of the other connected components. Also, we show that for τ ∈ (2, 3), which is most often observed in real networks, the diameter in this model grows much faster than the typical distance between two arbitrary nodes