Rigorous confidence intervals for critical probabilities.

We use the method of Balister, Bollobás, and Walters [Random Struct. Algorithms 26, 392 (2005)] to give rigorous 99.9999% confidence intervals for the critical probabilities for site and bond percolation on the 11 Archimedean lattices. In our computer calculations, the emphasis is on simplicity and ease of verification, rather than obtaining the best possible results. Nevertheless, we obtain intervals of width at most 0.0005 in all cases.

The consequences of Zipf's law for syntax and symbolic reference.

Although many species possess rudimentary communication systems, humans seem to be unique with regard to making use of syntax and symbolic reference. Recent approaches to the evolution of language formalize why syntax is selectively advantageous compared with isolated signal communication systems, but do not explain how signals naturally combine. Even more recent work has shown that if a communication system maximizes communicative efficiency while minimizing the cost of communication, or if a communication system constrains ambiguity in a non-trivial way while a certain entropy is maximized, signal frequencies will be distributed according to Zipf's law. Here we show that such communication principles give rise not only to signals that have many traits in common with the linking words in real human languages, but also to a rudimentary sort of syntax and symbolic reference.

Shortest paths and load scaling in scale-free trees.

Szabó, Alava, and Kertész [Phys. Rev. E 66, 026101 (2002)] considered two questions about the scale-free random tree given by the m=1 case of the Barabási-Albert (BA) model (identical with a random tree model introduced by Szymanski in 1987): what is the distribution of the node to node distances, and what is the distribution of node loads, where the load on a node is the number of shortest paths passing through it? They gave heuristic answers to these questions using a "mean-field" approximation, replacing the random tree by a certain fixed tree with carefully chosen branching ratios. By making use of our earlier results on scale-free random graphs, we shall analyze the random tree rigorously, obtaining and proving very precise answers to these questions. We shall show that, after dividing by N (the number of nodes), the load distribution converges to an integer distribution X with Pr(X=c)=2/[(2c+1)(2c+3)], c=0,1,2,..., confirming the asymptotic power law with exponent -2 predicted by Szabó, Alava, and Kertész. For the distribution of node-node distances, we show asymptotic normality, and give a precise form for the (far from normal) large deviation law. We note that the mean-field methods used by Szabó, Alava, and Kertész give very good results for this model.

Achlioptas processes are not always self-averaging.

We consider a class of percolation models, called Achlioptas processes, discussed in Science 323, 1453 (2009) and Science 333, 322 (2011). For these, the evolution of the order parameter (the rescaled size of the largest connected component) has been the main focus of research in recent years. We show that, in striking contrast to "classical" models, self-averaging is not a universal feature of these new percolation models: there are natural Achlioptas processes whose order parameter has random fluctuations that do not disappear in the thermodynamic limit.

Explosive percolation is continuous.

"Explosive percolation" is said to occur in an evolving network when a macroscopic connected component emerges in a number of steps that is much smaller than the system size. Recent predictions based on simulations suggested that certain Achlioptas processes (much-studied local modifications of the classical mean-field growth model of Erdos and Rényi) exhibit this phenomenon, undergoing a phase transition that is discontinuous in the scaling limit. We show that, in fact, all Achlioptas processes have continuous phase transitions, although related models in which the number of nodes sampled may grow with the network size can indeed exhibit explosive percolation.