Stability for large forbidden subgraphs

We extend the classical stability theorem of Erdos and Simonovits for forbidden graphs of logarithmic order.Comment: Some polishing. Updated reference

Dynamical formation of correlations in a Bose-Einstein condensate

We consider the evolution of $N$ bosons interacting with a repulsive short range pair potential in three dimensions. The potential is scaled according to the Gross-Pitaevskii scaling, i.e. it is given by $N^2V(N(x_i-x_j))$. We monitor the behavior of the solution to the $N$-particle Schr\"odinger equation in a spatial window where two particles are close to each other. We prove that within this window a short scale interparticle structure emerges dynamically. The local correlation between the particles is given by the two-body zero energy scattering mode. This is the characteristic structure that was expected to form within a very short initial time layer and to persist for all later times, on the basis of the validity of the Gross-Pitaevskii equation for the evolution of the Bose-Einstein condensate. The zero energy scattering mode emerges after an initial time layer where all higher energy modes disperse out of the spatial window. We can prove the persistence of this structure up to sufficiently small times before three-particle correlations could develop.Comment: 36 pages, latex fil

Clique percolation

Derenyi, Palla and Vicsek introduced the following dependent percolation model, in the context of finding communities in networks. Starting with a random graph $G$ generated by some rule, form an auxiliary graph $G'$ whose vertices are the $k$-cliques of $G$, in which two vertices are joined if the corresponding cliques share $k-1$ vertices. They considered in particular the case where $G=G(n,p)$, and found heuristically the threshold for a giant component to appear in $G'$. Here we give a rigorous proof of this result, as well as many extensions. The model turns out to be very interesting due to the essential global dependence present in $G'$.Comment: 33 pages, 1 figur

CABeRNET: a Cytoscape app for augmented Boolean models of gene regulatory NETworks

Background. Dynamical models of gene regulatory networks (GRNs) are highly effective in describing complex biological phenomena and processes, such as cell differentiation and cancer development. Yet, the topological and functional characterization of real GRNs is often still partial and an exhaustive picture of their functioning is missing. Motivation. We here introduce CABeRNET, a Cytoscape app for the generation, simulation and analysis of Boolean models of GRNs, specifically focused on their augmentation when a only partial topological and functional characterization of the network is available. By generating large ensembles of networks in which user-defined entities and relations are added to the original core, CABeRNET allows to formulate hypotheses on the missing portions of real networks, as well to investigate their generic properties, in the spirit of complexity science. Results. CABeRNET offers a series of innovative simulation and modeling functions and tools, including (but not being limited to) the dynamical characterization of the gene activation patterns ruling cell types and differentiation fates, and sophisticated robustness assessments, as in the case of gene knockouts. The integration within the widely used Cytoscape framework for the visualization and analysis of biological networks, makes CABeRNET a new essential instrument for both the bioinformatician and the computational biologist, as well as a computational support for the experimentalist. An example application concerning the analysis of an augmented T-helper cell GRN is provided.Comment: 18 pages, 3 figure

Anti-Ramsey numbers of doubly edge-critical graphs

Given a graph H and a positive integer n, Anti-Ramsey number AR(n, H) is the maximum number of colors in an edge-coloring of Kn that contains no polychromatic copy of H. The anti-Ramsey numbers were introduced in the 1970s by Erdős, Simonovits, and Sós, who among other things, determined this function for cliques. In general, few exact values of AR(n, H) are known. Let us call a graph Hdoubly edge-critical if χ(H−e)≥p+ 1 for each edge e∈E(H) and there exist two edges e1, e2 of H for which χ(H−e1−e2)=p. Here, we obtain the exact value of AR(n, H) for any doubly edge-critical H when n⩾n0(H) is sufficiently large. A main ingredient of our proof is the stability theorem of Erdős and Simonovits for the Turán problem. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 210–218, 200