Hierarchical hyperbolicity of graphs of multicurves

We show that many graphs naturally associated to a connected, compact, orientable surface are hierarchically hyperbolic spaces in the sense of Behrstock, Hagen and Sisto. They also automatically have the coarse median property defined by Bowditch. Consequences for such graphs include a distance formula analogous to Masur and Minsky's distance formula for the mapping class group, an upper bound on the maximal dimension of quasiflats, and the existence of a quadratic isoperimetric inequality. The hierarchically hyperbolic structure also gives rise to a simple criterion for when such graphs are Gromov hyperbolic.Comment: 27 pages, 4 figures. Minor changes from previous version. Addition of appendix describing a hierarchically hyperbolic structure on the arc grap

Coarse grained description of the protein folding

We consider two- and three-dimensional lattice models of proteins which were characterized previously. We coarse grain their folding dynamics by reducing it to transitions between effective states. We consider two methods of selection of the effective states. The first method is based on the steepest descent mapping of states to underlying local energy minima and the other involves an additional projection to maximally compact conformations. Both methods generate connectivity patterns that allow to distinguish between the good and bad folders. Connectivity graphs corresponding to the folding funnel have few loops and are thus tree-like. The Arrhenius law for the median folding time of a 16-monomer sequence is established and the corresponding barrier is related to easily identifiable kinetic trap states.Comment: REVTeX, 9 pages, 15 EPS figures, to appear in Phys. Rev.

Large-scale rank and rigidity of the Weil-Petersson metric

We study the large-scale geometry of Weil–Petersson space, that is, Teichmüller space equipped with theWeil–Petersson metric. We show that this admits a natural coarse median structure of a specific rank. Given that this is equal to the maximal dimension of a quasi-isometrically embedded euclidean space,we recover a result of Eskin,Masur and Rafi which gives the coarse rank of the space. We go on to show that, apart from finitely many cases, the Weil–Petersson spaces are quasi-isometrically distinct, and quasi-isometrically rigid. In particular, any quasi-isometry between such spaces is a bounded distance from an isometry. By a theorem of Brock,Weil–Petersson space is equivariantly quasi-isometric to the pants graph, so our results apply equally well to that space

Kazhdan and Haagerup properties from the median viewpoint

We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affine isometric actions on $L^p$-spaces. It also allows us to answer an open problem on a dynamical characterization of property (T), generalizing results of Robertson-Steger.Comment: final versio

Protein Evolution within a Structural Space

Understanding of the evolutionary origins of protein structures represents a key component of the understanding of molecular evolution as a whole. Here we seek to elucidate how the features of an underlying protein structural "space" might impact protein structural evolution. We approach this question using lattice polymers as a completely characterized model of this space. We develop a measure of structural comparison of lattice structures that is analgous to the one used to understand structural similarities between real proteins. We use this measure of structural relatedness to create a graph of lattice structures and compare this graph (in which nodes are lattice structures and edges are defined using structural similarity) to the graph obtained for real protein structures. We find that the graph obtained from all compact lattice structures exhibits a distribution of structural neighbors per node consistent with a random graph. We also find that subgraphs of 3500 nodes chosen either at random or according to physical constraints also represent random graphs. We develop a divergent evolution model based on the lattice space which produces graphs that, within certain parameter regimes, recapitulate the scale-free behavior observed in similar graphs of real protein structures.Comment: 27 pages, 7 figure

Weak hyperbolicity of cube complexes and quasi-arboreal groups

We examine a graph $\Gamma$ encoding the intersection of hyperplane carriers in a CAT(0) cube complex $\widetilde X$. The main result is that $\Gamma$ is quasi-isometric to a tree. This implies that a group $G$ acting properly and cocompactly on $\widetilde X$ is weakly hyperbolic relative to the hyperplane stabilizers. Using disc diagram techniques and Wright's recent result on the aymptotic dimension of CAT(0) cube complexes, we give a generalization of a theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs of asymptotically finite-dimensional groups. More precisely, we prove asymptotic finite-dimensionality for finitely-generated groups acting on finite-dimensional cube complexes with 0-cube stabilizers of uniformly bounded asymptotic dimension. Finally, we apply contact graph techniques to prove a cubical version of the flat plane theorem stated in terms of complete bipartite subgraphs of $\Gamma$.Comment: Corrections in Sections 2 and 4. Simplification in Section

Graph Summarization

The continuous and rapid growth of highly interconnected datasets, which are both voluminous and complex, calls for the development of adequate processing and analytical techniques. One method for condensing and simplifying such datasets is graph summarization. It denotes a series of application-specific algorithms designed to transform graphs into more compact representations while preserving structural patterns, query answers, or specific property distributions. As this problem is common to several areas studying graph topologies, different approaches, such as clustering, compression, sampling, or influence detection, have been proposed, primarily based on statistical and optimization methods. The focus of our chapter is to pinpoint the main graph summarization methods, but especially to focus on the most recent approaches and novel research trends on this topic, not yet covered by previous surveys.Comment: To appear in the Encyclopedia of Big Data Technologie