Generating complex connectivity structures for large-scale neural models

Martin Huelse: Generating complex connectivity structures for large-scale neural models. In: V. Kurkova, R. Neruda, and J. Koutnik (Eds.): ICANN 2008, Part II, LNCS 5164, pp. 849?858, 2008. Sponsorship: EPSRC

Modelling random fuzzy renewal reward processes

Shen, Q., Zhao, R., Tang, W. (2008). Modelling random fuzzy renewal reward processes. IEEE Transactions on Fuzzy Systems, 16 (5),1379-1385

On the Performance of Polynomial-time CLIQUE Approximation Algorithms on Very Large Graphs

The performance of a randomized version of the subgraph-exclusion algorithm (called Ramsey) for CLIQUE by Boppana and Halldorsson is studied on very large graphs. We compare the performance of this algorithm with the performance of two common heuristic algorithms, the greedy heuristic and a version of simulated annealing. These algorithms are tested on graphs with up to 10,000 vertices on a workstation and graphs as large as 70,000 vertices on a Connection Machine. Our implementations establish the ability to run clique approximation algorithms on very large graphs. We test our implementations on a variety of different graphs. Our conclusions indicate that on randomly generated graphs minor changes to the distribution can cause dramatic changes in the performance of the heuristic algorithms. The Ramsey algorithm, while not as good as the others for the most common distributions, seems more robust and provides a more even overall performance. In general, and especially on deterministically generated graphs, a combination of simulated annealing with either the Ramsey algorithm or the greedy heuristic seems to perform best. This combined algorithm works particularly well on large Keller and Hamming graphs and has a competitive overall performance on the DIMACS benchmark graphs.

Approximately Uniform Random Sampling in Sensor Networks

Recent work in sensor databases has focused extensively on distributed query problems, notably distributed computation of aggregates. Existing methods for computing aggregates broadcast queries to all sensors and use in-network aggregation of responses to minimize messaging costs. In this work, we focus on uniform random sampling across nodes, which can serve both as an alternative building block for aggregation and as an integral component of many other useful randomized algorithms. Prior to our work, the best existing proposals for uniform random sampling of sensors involve contacting all nodes in the network. We propose a practical method which is only approximately uniform, but contacts a number of sensors proportional to the diameter of the network instead of its size. The approximation achieved is tunably close to exact uniform sampling, and only relies on well-known existing primitives, namely geographic routing, distributed computation of Voronoi regions and von Neumann's rejection method. Ultimately, our sampling algorithm has the same worst-case asymptotic cost as routing a point-to-point message, and thus it is asymptotically optimal among request/reply-based sampling methods. We provide experimental results demonstrating the effectiveness of our algorithm on both synthetic and real sensor topologies.

Fast Shortest Path Distance Estimation in Large Networks

We study the problem of preprocessing a large graph so that point-to-point shortest-path queries can be answered very fast. Computing shortest paths is a well studied problem, but exact algorithms do not scale to huge graphs encountered on the web, social networks, and other applications. In this paper we focus on approximate methods for distance estimation, in particular using landmark-based distance indexing. This approach involves selecting a subset of nodes as landmarks and computing (offline) the distances from each node in the graph to those landmarks. At runtime, when the distance between a pair of nodes is needed, we can estimate it quickly by combining the precomputed distances of the two nodes to the landmarks. We prove that selecting the optimal set of landmarks is an NP-hard problem, and thus heuristic solutions need to be employed. Given a budget of memory for the index, which translates directly into a budget of landmarks, different landmark selection strategies can yield dramatically different results in terms of accuracy. A number of simple methods that scale well to large graphs are therefore developed and experimentally compared. The simplest methods choose central nodes of the graph, while the more elaborate ones select central nodes that are also far away from one another. The efficiency of the suggested techniques is tested experimentally using five different real world graphs with millions of edges; for a given accuracy, they require as much as 250 times less space than the current approach in the literature which considers selecting landmarks at random. Finally, we study applications of our method in two problems arising naturally in large-scale networks, namely, social search and community detection.

Anisotropic Interpolation on Graphs: The Combinatorial Dirichlet Problem

The combinatorial Dirichlet problem is formulated, and an algorithm for solving it is presented. This provides an effective method for interpolating missing data on weighted graphs of arbitrary connectivity. Image processing examples are shown, and the relation to anistropic diffusion is discussed.

The Graph Analysis Toolbox: Image Processing on Arbitrary Graphs

Office of Naval Research (N00014-01-1-0624)

A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Telomere disruption results in non-random formation of de novo dicentric chromosomes involving acrocentric human chromosomes.

Genome rearrangement often produces chromosomes with two centromeres (dicentrics) that are inherently unstable because of bridge formation and breakage during cell division. However, mammalian dicentrics, and particularly those in humans, can be quite stable, usually because one centromere is functionally silenced. Molecular mechanisms of centromere inactivation are poorly understood since there are few systems to experimentally create dicentric human chromosomes. Here, we describe a human cell culture model that enriches for de novo dicentrics. We demonstrate that transient disruption of human telomere structure non-randomly produces dicentric fusions involving acrocentric chromosomes. The induced dicentrics vary in structure near fusion breakpoints and like naturally-occurring dicentrics, exhibit various inter-centromeric distances. Many functional dicentrics persist for months after formation. Even those with distantly spaced centromeres remain functionally dicentric for 20 cell generations. Other dicentrics within the population reflect centromere inactivation. In some cases, centromere inactivation occurs by an apparently epigenetic mechanism. In other dicentrics, the size of the alpha-satellite DNA array associated with CENP-A is reduced compared to the same array before dicentric formation. Extra-chromosomal fragments that contained CENP-A often appear in the same cells as dicentrics. Some of these fragments are derived from the same alpha-satellite DNA array as inactivated centromeres. Our results indicate that dicentric human chromosomes undergo alternative fates after formation. Many retain two active centromeres and are stable through multiple cell divisions. Others undergo centromere inactivation. This event occurs within a broad temporal window and can involve deletion of chromatin that marks the locus as a site for CENP-A maintenance/replenishment.

Gene selection using iterative feature elimination random forests for survival outcomes.

Although many feature selection methods for classification have been developed, there is a need to identify genes in high-dimensional data with censored survival outcomes. Traditional methods for gene selection in classification problems have several drawbacks. First, the majority of the gene selection approaches for classification are single-gene based. Second, many of the gene selection procedures are not embedded within the algorithm itself. The technique of random forests has been found to perform well in high-dimensional data settings with survival outcomes. It also has an embedded feature to identify variables of importance. Therefore, it is an ideal candidate for gene selection in high-dimensional data with survival outcomes. In this paper, we develop a novel method based on the random forests to identify a set of prognostic genes. We compare our method with several machine learning methods and various node split criteria using several real data sets. Our method performed well in both simulations and real data analysis.Additionally, we have shown the advantages of our approach over single-gene-based approaches. Our method incorporates multivariate correlations in microarray data for survival outcomes. The described method allows us to better utilize the information available from microarray data with survival outcomes.

Regularity of invariant densities for 1D systems with random switching

© 2015 IOP Publishing Ltd & London Mathematical Society.This is a detailed analysis of invariant measures for one-dimensional dynamical systems with random switching. In particular, we prove the smoothness of the invariant densities away from critical points and describe the asymptotics of the invariant densities at critical points.

Stochastic switching in infinite dimensions with applications to random parabolic PDE

© 2015 Society for Industrial and Applied Mathematics.We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDEs forced by disparate Gaussian noise. Our general results also have applications to other types of stochastic hybrid systems, such as ODEs with randomly switching right-hand sides.

Exact colorations of graphs and digraphs

A coloration is an exact regular coloration if whenever two vertices are colored the same they have identically colored neighborhoods. For example, if one of the two vertices that are colored the same is connected to three yellow vertices, two white and red, then the other vertex is as well. Exact regular colorations have been discussed informally in the social network literature. However they have been part of the mathematical literature for some time, though in a different format. We explore this concept in terms of social networks and illustrate some important results taken from the mathematical literature. In addition we show how the concept can be extended to ecological and perfect colorations, and discuss how the CATREGE algorithm can be extended to find the maximal exact regular coloration of a graph.