River landscapes and optimal channel networks.

We study tree structures termed optimal channel networks (OCNs) that minimize the total gravitational energy loss in the system, an exact property of steady-state landscape configurations that prove dynamically accessible and strikingly similar to natural forms. Here, we show that every OCN is a so-called natural river tree, in the sense that there exists a height function such that the flow directions are always directed along steepest descent. We also study the natural river trees in an arbitrary graph in terms of forbidden substructures, which we call k-path obstacles, and OCNs on a d-dimensional lattice, improving earlier results by determining the minimum energy up to a constant factor for every [Formula: see text] Results extend our capabilities in environmental statistical mechanics.

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.

Phase transitions in the neuropercolation model of neural populations with mixed local and non-local interactions.

We model the dynamical behavior of the neuropil, the densely interconnected neural tissue in the cortex, using neuropercolation approach. Neuropercolation generalizes phase transitions modeled by percolation theory of random graphs, motivated by properties of neurons and neural populations. The generalization includes (1) a noisy component in the percolation rule, (2) a novel depression function in addition to the usual arousal function, (3) non-local interactions among nodes arranged on a multi-dimensional lattice. This paper investigates the role of non-local (axonal) connections in generating and modulating phase transitions of collective activity in the neuropil. We derived a relationship between critical values of the noise level and non-locality parameter to control the onset of phase transitions. Finally, we propose a potential interpretation of ontogenetic development of the neuropil maintaining a dynamical state at the edge of criticality.