Causal and homogeneous networks

Growing networks have a causal structure. We show that the causality strongly influences the scaling and geometrical properties of the network. In particular the average distance between nodes is smaller for causal networks than for corresponding homogeneous networks. We explain the origin of this effect and illustrate it using as an example a solvable model of random trees. We also discuss the issue of stability of the scale-free node degree distribution. We show that a surplus of links may lead to the emergence of a singular node with the degree proportional to the total number of links. This effect is closely related to the backgammon condensation known from the balls-in-boxes model.Comment: short review submitted to AIP proceedings, CNET2004 conference; changes in the discussion of the distance distribution for growing trees, Fig. 6-right change

In search of lost introns

Many fundamental questions concerning the emergence and subsequent evolution of eukaryotic exon-intron organization are still unsettled. Genome-scale comparative studies, which can shed light on crucial aspects of eukaryotic evolution, require adequate computational tools. We describe novel computational methods for studying spliceosomal intron evolution. Our goal is to give a reliable characterization of the dynamics of intron evolution. Our algorithmic innovations address the identification of orthologous introns, and the likelihood-based analysis of intron data. We discuss a compression method for the evaluation of the likelihood function, which is noteworthy for phylogenetic likelihood problems in general. We prove that after $O(nL)$ preprocessing time, subsequent evaluations take $O(nL/\log L)$ time almost surely in the Yule-Harding random model of $n$-taxon phylogenies, where $L$ is the input sequence length. We illustrate the practicality of our methods by compiling and analyzing a data set involving 18 eukaryotes, more than in any other study to date. The study yields the surprising result that ancestral eukaryotes were fairly intron-rich. For example, the bilaterian ancestor is estimated to have had more than 90% as many introns as vertebrates do now