Moments of vicious walkers and M\"obius graph expansions

A system of Brownian motions in one-dimension all started from the origin and conditioned never to collide with each other in a given finite time-interval $(0, T]$ is studied. The spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue-statistics of random Hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian orthogonal ensemble (GOE) statistics as the time $t$ is going on from 0 to $T$. In the present paper, we characterize this GUE-to-GOE transition by presenting the graphical expansion formula for the moments of positions of vicious walkers. In the GUE limit $t \to 0$, only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus. Following the time evolution of the vicious walkers, however, the graphs with twisted ribbons, called M\"obius graphs, increase their contribution to our expansion formula, and we have to deal with the topology of non-orientable surfaces. Application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the M\"obius expansion using the Stirling numbers of the first kind.Comment: REVTeX4, 11 pages, 1 figure. v.2: calculations of the Green function and references added. v.3: minor additions and corrections made for publication in Phys.Rev.

Complex Networks on Hyperbolic Surfaces

We explore a novel method to generate and characterize complex networks by means of their embedding on hyperbolic surfaces. Evolution through local elementary moves allows the exploration of the ensemble of networks which share common embeddings and consequently share similar hierarchical properties. This method provides a new perspective to classify network-complexity both on local and global scale. We demonstrate by means of several examples that there is a strong relation between the network properties and the embedding surface.Comment: 8 Pages 3 Figure

Shape from Shading through Shape Evolution

In this paper, we address the shape-from-shading problem by training deep networks with synthetic images. Unlike conventional approaches that combine deep learning and synthetic imagery, we propose an approach that does not need any external shape dataset to render synthetic images. Our approach consists of two synergistic processes: the evolution of complex shapes from simple primitives, and the training of a deep network for shape-from-shading. The evolution generates better shapes guided by the network training, while the training improves by using the evolved shapes. We show that our approach achieves state-of-the-art performance on a shape-from-shading benchmark

Disordered locality in loop quantum gravity states

We show that loop quantum gravity suffers from a potential problem with non-locality, coming from a mismatch between micro-locality, as defined by the combinatorial structures of their microscopic states, and macro-locality, defined by the metric which emerges from the low energy limit. As a result, the low energy limit may suffer from a disordered locality characterized by identifications of far away points. We argue that if such defects in locality are rare enough they will be difficult to detect.Comment: 11 pages, 4 figures, revision with extended discussion of result

A simple physical model for scaling in protein-protein interaction networks

It has recently been demonstrated that many biological networks exhibit a scale-free topology where the probability of observing a node with a certain number of edges (k) follows a power law: i.e. p(k) ~ k^-g. This observation has been reproduced by evolutionary models. Here we consider the network of protein-protein interactions and demonstrate that two published independent measurements of these interactions produce graphs that are only weakly correlated with one another despite their strikingly similar topology. We then propose a physical model based on the fundamental principle that (de)solvation is a major physical factor in protein-protein interactions. This model reproduces not only the scale-free nature of such graphs but also a number of higher-order correlations in these networks. A key support of the model is provided by the discovery of a significant correlation between number of interactions made by a protein and the fraction of hydrophobic residues on its surface. The model presented in this paper represents the first physical model for experimentally determined protein-protein interactions that comprehensively reproduces the topological features of interaction networks. These results have profound implications for understanding not only protein-protein interactions but also other types of scale-free networks.Comment: 50 pages, 17 figure

Microbial Similarity between Students in a Common Dormitory Environment Reveals the Forensic Potential of Individual Microbial Signatures.

The microbiota of the built environment is an amalgamation of both human and environmental sources. While human sources have been examined within single-family households or in public environments, it is unclear what effect a large number of cohabitating people have on the microbial communities of their shared environment. We sampled the public and private spaces of a college dormitory, disentangling individual microbial signatures and their impact on the microbiota of common spaces. We compared multiple methods for marker gene sequence clustering and found that minimum entropy decomposition (MED) was best able to distinguish between the microbial signatures of different individuals and was able to uncover more discriminative taxa across all taxonomic groups. Further, weighted UniFrac- and random forest-based graph analyses uncovered two distinct spheres of hand- or shoe-associated samples. Using graph-based clustering, we identified spheres of interaction and found that connection between these clusters was enriched for hands, implicating them as a primary means of transmission. In contrast, shoe-associated samples were found to be freely interacting, with individual shoes more connected to each other than to the floors they interact with. Individual interactions were highly dynamic, with groups of samples originating from individuals clustering freely with samples from other individuals, while all floor and shoe samples consistently clustered together.IMPORTANCE Humans leave behind a microbial trail, regardless of intention. This may allow for the identification of individuals based on the "microbial signatures" they shed in built environments. In a shared living environment, these trails intersect, and through interaction with common surfaces may become homogenized, potentially confounding our ability to link individuals to their associated microbiota. We sought to understand the factors that influence the mixing of individual signatures and how best to process sequencing data to best tease apart these signatures

On the spectral dimension of causal triangulations

We introduce an ensemble of infinite causal triangulations, called the uniform infinite causal triangulation, and show that it is equivalent to an ensemble of infinite trees, the uniform infinite planar tree. It is proved that in both cases the Hausdorff dimension almost surely equals 2. The infinite causal triangulations are shown to be almost surely recurrent or, equivalently, their spectral dimension is almost surely less than or equal to 2. We also establish that for certain reduced versions of the infinite causal triangulations the spectral dimension equals 2 both for the ensemble average and almost surely. The triangulation ensemble we consider is equivalent to the causal dynamical triangulation model of two-dimensional quantum gravity and therefore our results apply to that model.Comment: 22 pages, 6 figures; typos fixed, one extra figure, references update