Mark correlations: relating physical properties to spatial distributions

Mark correlations provide a systematic approach to look at objects both distributed in space and bearing intrinsic information, for instance on physical properties. The interplay of the objects' properties (marks) with the spatial clustering is of vivid interest for many applications; are, e.g., galaxies with high luminosities more strongly clustered than dim ones? Do neighbored pores in a sandstone have similar sizes? How does the shape of impact craters on a planet depend on the geological surface properties? In this article, we give an introduction into the appropriate mathematical framework to deal with such questions, i.e. the theory of marked point processes. After having clarified the notion of segregation effects, we define universal test quantities applicable to realizations of a marked point processes. We show their power using concrete data sets in analyzing the luminosity-dependence of the galaxy clustering, the alignment of dark matter halos in gravitational $N$-body simulations, the morphology- and diameter-dependence of the Martian crater distribution and the size correlations of pores in sandstone. In order to understand our data in more detail, we discuss the Boolean depletion model, the random field model and the Cox random field model. The first model describes depletion effects in the distribution of Martian craters and pores in sandstone, whereas the last one accounts at least qualitatively for the observed luminosity-dependence of the galaxy clustering.Comment: 35 pages, 12 figures. to be published in Lecture Notes of Physics, second Wuppertal conference "Spatial statistics and statistical physics

Structures in magnetohydrodynamic turbulence: detection and scaling

We present a systematic analysis of statistical properties of turbulent current and vorticity structures at a given time using cluster analysis. The data stems from numerical simulations of decaying three-dimensional (3D) magnetohydrodynamic turbulence in the absence of an imposed uniform magnetic field; the magnetic Prandtl number is taken equal to unity, and we use a periodic box with grids of up to 1536^3 points, and with Taylor Reynolds numbers up to 1100. The initial conditions are either an X-point configuration embedded in 3D, the so-called Orszag-Tang vortex, or an Arn'old-Beltrami-Childress configuration with a fully helical velocity and magnetic field. In each case two snapshots are analyzed, separated by one turn-over time, starting just after the peak of dissipation. We show that the algorithm is able to select a large number of structures (in excess of 8,000) for each snapshot and that the statistical properties of these clusters are remarkably similar for the two snapshots as well as for the two flows under study in terms of scaling laws for the cluster characteristics, with the structures in the vorticity and in the current behaving in the same way. We also study the effect of Reynolds number on cluster statistics, and we finally analyze the properties of these clusters in terms of their velocity-magnetic field correlation. Self-organized criticality features have been identified in the dissipative range of scales. A different scaling arises in the inertial range, which cannot be identified for the moment with a known self-organized criticality class consistent with MHD. We suggest that this range can be governed by turbulence dynamics as opposed to criticality, and propose an interpretation of intermittency in terms of propagation of local instabilities.Comment: 17 pages, 9 figures, 5 table

Topological network alignment uncovers biological function and phylogeny

Sequence comparison and alignment has had an enormous impact on our understanding of evolution, biology, and disease. Comparison and alignment of biological networks will likely have a similar impact. Existing network alignments use information external to the networks, such as sequence, because no good algorithm for purely topological alignment has yet been devised. In this paper, we present a novel algorithm based solely on network topology, that can be used to align any two networks. We apply it to biological networks to produce by far the most complete topological alignments of biological networks to date. We demonstrate that both species phylogeny and detailed biological function of individual proteins can be extracted from our alignments. Topology-based alignments have the potential to provide a completely new, independent source of phylogenetic information. Our alignment of the protein-protein interaction networks of two very different species--yeast and human--indicate that even distant species share a surprising amount of network topology with each other, suggesting broad similarities in internal cellular wiring across all life on Earth.Comment: Algorithm explained in more details. Additional analysis adde

Voids in cosmological simulations over cosmic time

We study evolution of voids in cosmological simulations using a new method for tracing voids over cosmic time. The method is based on tracking watershed basins (contiguous regions around density minima) of well developed voids at low redshift, on a regular grid of density field. It enables us to construct a robust and continuous mapping between voids at different redshifts, from initial conditions to the present time. We discuss how the new approach eliminates strong spurious effects of numerical origin when voids evolution is traced by matching voids between successive snapshots (by analogy to halo merger trees). We apply the new method to a cosmological simulation of a standard LambdaCDM cosmological model and study evolution of basic properties of typical voids (with effective radii between 6Mpc/h and 20Mpc/h at redshift z=0) such as volumes, shapes, matter density distributions and relative alignments. The final voids at low redshifts appear to retain a significant part of the configuration acquired in initial conditions. Shapes of voids evolve in a collective way which barely modifies the overall distribution of the axial ratios. The evolution appears to have a weak impact on mutual alignments of voids implying that the present state is in large part set up by the primordial density field. We present evolution of dark matter density profiles computed on iso-density surfaces which comply with the actual shapes of voids. Unlike spherical density profiles, this approach enables us to demonstrate development of theoretically predicted bucket-like shape of the final density profiles indicating a wide flat core and a sharp transition to high-density void walls.Comment: 13 pages, 13 figures; accepted for publication in MNRA

Unsupervised Domain Adaptation for 3D Keypoint Estimation via View Consistency

In this paper, we introduce a novel unsupervised domain adaptation technique for the task of 3D keypoint prediction from a single depth scan or image. Our key idea is to utilize the fact that predictions from different views of the same or similar objects should be consistent with each other. Such view consistency can provide effective regularization for keypoint prediction on unlabeled instances. In addition, we introduce a geometric alignment term to regularize predictions in the target domain. The resulting loss function can be effectively optimized via alternating minimization. We demonstrate the effectiveness of our approach on real datasets and present experimental results showing that our approach is superior to state-of-the-art general-purpose domain adaptation techniques.Comment: ECCV 201

Biological network comparison using graphlet degree distribution

Analogous to biological sequence comparison, comparing cellular networks is an important problem that could provide insight into biological understanding and therapeutics. For technical reasons, comparing large networks is computationally infeasible, and thus heuristics such as the degree distribution have been sought. It is easy to demonstrate that two networks are different by simply showing a short list of properties in which they differ. It is much harder to show that two networks are similar, as it requires demonstrating their similarity in all of their exponentially many properties. Clearly, it is computationally prohibitive to analyze all network properties, but the larger the number of constraints we impose in determining network similarity, the more likely it is that the networks will truly be similar. We introduce a new systematic measure of a network's local structure that imposes a large number of similarity constraints on networks being compared. In particular, we generalize the degree distribution, which measures the number of nodes 'touching' k edges, into distributions measuring the number of nodes 'touching' k graphlets, where graphlets are small connected non-isomorphic subgraphs of a large network. Our new measure of network local structure consists of 73 graphlet degree distributions (GDDs) of graphlets with 2-5 nodes, but it is easily extendible to a greater number of constraints (i.e. graphlets). Furthermore, we show a way to combine the 73 GDDs into a network 'agreement' measure. Based on this new network agreement measure, we show that almost all of the 14 eukaryotic PPI networks, including human, are better modeled by geometric random graphs than by Erdos-Reny, random scale-free, or Barabasi-Albert scale-free networks.Comment: Proceedings of the 2006 European Conference on Computational Biology, ECCB'06, Eilat, Israel, January 21-24, 200

Visible Volume: a Robust Measure for Protein Structure Characterization

We propose a new characterization of protein structure based on the natural tetrahedral geometry of the β carbon and a new geometric measure of structural similarity, called visible volume. In our model, the side-chains are replaced by an ideal tetrahedron, the orientation of which is fixed with respect to the backbone and corresponds to the preferred rotamer directions. Visible volume is a measure of the non-occluded empty space surrounding each residue position after the side-chains have been removed. It is a robust, parameter-free, locally-computed quantity that accounts for many of the spatial constraints that are of relevance to the corresponding position in the native structure. When computing visible volume, we ignore the nature of both the residue observed at each site and the ones surrounding it. We focus instead on the space that, together, these residues could occupy. By doing so, we are able to quantify a new kind of invariance beyond the apparent variations in protein families, namely, the conservation of the physical space available at structurally equivalent positions for side-chain packing. Corresponding positions in native structures are likely to be of interest in protein structure prediction, protein design, and homology modeling. Visible volume is related to the degree of exposure of a residue position and to the actual rotamers in native proteins. In this article, we discuss the properties of this new measure, namely, its robustness with respect to both crystallographic uncertainties and naturally occurring variations in atomic coordinates, and the remarkable fact that it is essentially independent of the choice of the parameters used in calculating it. We also show how visible volume can be used to align protein structures, to identify structurally equivalent positions that are conserved in a family of proteins, and to single out positions in a protein that are likely to be of biological interest. These properties qualify visible volume as a powerful tool in a variety of applications, from the detailed analysis of protein structure to homology modeling, protein structural alignment, and the definition of better scoring functions for threading purposes.National Library of Medicine (LM05205-13

Identifying networks with common organizational principles

Many complex systems can be represented as networks, and the problem of network comparison is becoming increasingly relevant. There are many techniques for network comparison, from simply comparing network summary statistics to sophisticated but computationally costly alignment-based approaches. Yet it remains challenging to accurately cluster networks that are of a different size and density, but hypothesized to be structurally similar. In this paper, we address this problem by introducing a new network comparison methodology that is aimed at identifying common organizational principles in networks. The methodology is simple, intuitive and applicable in a wide variety of settings ranging from the functional classification of proteins to tracking the evolution of a world trade network.Comment: 26 pages, 7 figure