Superdiffusion in a class of networks with marginal long-range connections

A class of cubic networks composed of a regular one-dimensional lattice and a set of long-range links is introduced. Networks parametrized by a positive integer k are constructed by starting from a one-dimensional lattice and iteratively connecting each site of degree 2 with a $k$th neighboring site of degree 2. Specifying the way pairs of sites to be connected are selected, various random and regular networks are defined, all of which have a power-law edge-length distribution of the form $P_>(l)\sim l^{-s}$ with the marginal exponent s=1. In all these networks, lengths of shortest paths grow as a power of the distance and random walk is super-diffusive. Applying a renormalization group method, the corresponding shortest-path dimensions and random-walk dimensions are calculated exactly for k=1 networks and for k=2 regular networks; in other cases, they are estimated by numerical methods. Although, s=1 holds for all representatives of this class, the above quantities are found to depend on the details of the structure of networks controlled by k and other parameters.Comment: 10 pages, 9 figure

Detecting the Baryons in Matter Power Spectra

We examine power spectra from the Abell/ACO rich cluster survey and the 2dF Galaxy Redshift Survey (2dfGRS) for observational evidence of features produced by the baryons. A non-negligible baryon fraction produces relatively sharp oscillatory features at specific wavenumbers in the matter power spectrum. However, the mere existence of baryons will also produce a global suppression of the power spectrum. We look for both of these features using the false discovery rate (FDR) statistic. We show that the window effects on the Abell/ACO power spectrum are minimal, which has allowed for the discovery of discrete oscillatory features in the power spectrum. On the other hand, there are no statistically significant oscillatory features in the 2dFGRS power spectrum, which is expected from the survey's broad window function. After accounting for window effects, we apply a scale-independent bias to the 2dFGRS power spectrum, P_{Abell}(k) = b^2P_{2dF}(k) and b = 3.2. We find that the overall shapes of the Abell/ACO and the biased 2dFGRS power spectra are entirely consistent over the range 0.02 <= k <= 0.15hMpc^-1. We examine the range of Omega_{matter} and baryon fraction for which these surveys could detect significant suppression in power. The reported baryon fractions for both the Abell/ACO and 2dFGRS surveys are high enough to cause a detectable suppression in power (after accounting for errors, windows and k-space sampling). Using the same technique, we also examine, given the best fit baryon density obtained from BBN, whether it is possible to detect additional suppression due to dark matter-baryon interaction. We find that the limit on dark matter cross section/mass derived from these surveys are the same as those ruled out in a recent study by Chen, Hannestad and Scherrer.Comment: 11 pages of text, 6 figures. Submitted to Ap

Harmonic Labeling of Graphs

Which graphs admit an integer value harmonic function which is injective and surjective onto $\Z$? Such a function, which we call harmonic labeling, is constructed when the graph is the $\Z^2$ square grid. It is shown that for any finite graph $G$ containing at least one edge, there is no harmonic labeling of $G \times \Z$

The Inverse Shapley Value Problem

For $f$ a weighted voting scheme used by $n$ voters to choose between two candidates, the $n$ \emph{Shapley-Shubik Indices} (or {\em Shapley values}) of $f$ provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 \cite{SS54} and are widely studied in social choice theory as a measure of the "influence" of voters. The \emph{Inverse Shapley Value Problem} is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley-Shubik indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work. We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant \eps > 0 our algorithm runs in fixed poly$(n)$ time (the degree of the polynomial is independent of \eps) and has the following performance guarantee: given as input a vector of desired Shapley values, if any "reasonable" weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley values to within error \eps. If there is a "reasonable" voting scheme in which all voting weights are integers at most \poly(n) that approximately achieves the desired Shapley values, then our algorithm runs in time \poly(n) and outputs a weighted voting scheme that achieves the target vector of Shapley values to within error $\eps=n^{-1/8}.

Scaling behavior of the contact process in networks with long-range connections

We present simulation results for the contact process on regular, cubic networks that are composed of a one-dimensional lattice and a set of long edges with unbounded length. Networks with different sets of long edges are considered, that are characterized by different shortest-path dimensions and random-walk dimensions. We provide numerical evidence that an absorbing phase transition occurs at some finite value of the infection rate and the corresponding dynamical critical exponents depend on the underlying network. Furthermore, the time-dependent quantities exhibit log-periodic oscillations in agreement with the discrete scale invariance of the networks. In case of spreading from an initial active seed, the critical exponents are found to depend on the location of the initial seed and break the hyper-scaling law of the directed percolation universality class due to the inhomogeneity of the networks. However, if the cluster spreading quantities are averaged over initial sites the hyper-scaling law is restored.Comment: 9 pages, 10 figure

A new source detection algorithm using FDR

The False Discovery Rate (FDR) method has recently been described by Miller et al (2001), along with several examples of astrophysical applications. FDR is a new statistical procedure due to Benjamini and Hochberg (1995) for controlling the fraction of false positives when performing multiple hypothesis testing. The importance of this method to source detection algorithms is immediately clear. To explore the possibilities offered we have developed a new task for performing source detection in radio-telescope images, Sfind 2.0, which implements FDR. We compare Sfind 2.0 with two other source detection and measurement tasks, Imsad and SExtractor, and comment on several issues arising from the nature of the correlation between nearby pixels and the necessary assumption of the null hypothesis. The strong suggestion is made that implementing FDR as a threshold defining method in other existing source-detection tasks is easy and worthwhile. We show that the constraint on the fraction of false detections as specified by FDR holds true even for highly correlated and realistic images. For the detection of true sources, which are complex combinations of source-pixels, this constraint appears to be somewhat less strict. It is still reliable enough, however, for a priori estimates of the fraction of false source detections to be robust and realistic.Comment: 17 pages, 7 figures, accepted for publication by A

On a random walk with memory and its relation to Markovian processes

We study a one-dimensional random walk with memory in which the step lengths to the left and to the right evolve at each step in order to reduce the wandering of the walker. The feedback is quite efficient and lead to a non-diffusive walk. The time evolution of the displacement is given by an equivalent Markovian dynamical process. The probability density for the position of the walker is the same at any time as for a random walk with shrinking steps, although the two-time correlation functions are quite different.Comment: 10 pages, 4 figure

P-values for high-dimensional regression

Assigning significance in high-dimensional regression is challenging. Most computationally efficient selection algorithms cannot guard against inclusion of noise variables. Asymptotically valid p-values are not available. An exception is a recent proposal by Wasserman and Roeder (2008) which splits the data into two parts. The number of variables is then reduced to a manageable size using the first split, while classical variable selection techniques can be applied to the remaining variables, using the data from the second split. This yields asymptotic error control under minimal conditions. It involves, however, a one-time random split of the data. Results are sensitive to this arbitrary choice: it amounts to a `p-value lottery' and makes it difficult to reproduce results. Here, we show that inference across multiple random splits can be aggregated, while keeping asymptotic control over the inclusion of noise variables. We show that the resulting p-values can be used for control of both family-wise error (FWER) and false discovery rate (FDR). In addition, the proposed aggregation is shown to improve power while reducing the number of falsely selected variables substantially.Comment: 25 pages, 4 figure

From brain to earth and climate systems: Small-world interaction networks or not?

We consider recent reports on small-world topologies of interaction networks derived from the dynamics of spatially extended systems that are investigated in diverse scientific fields such as neurosciences, geophysics, or meteorology. With numerical simulations that mimic typical experimental situations we have identified an important constraint when characterizing such networks: indications of a small-world topology can be expected solely due to the spatial sampling of the system along with commonly used time series analysis based approaches to network characterization

ExplainIt! -- A declarative root-cause analysis engine for time series data (extended version)

We present ExplainIt!, a declarative, unsupervised root-cause analysis engine that uses time series monitoring data from large complex systems such as data centres. ExplainIt! empowers operators to succinctly specify a large number of causal hypotheses to search for causes of interesting events. ExplainIt! then ranks these hypotheses, reducing the number of causal dependencies from hundreds of thousands to a handful for human understanding. We show how a declarative language, such as SQL, can be effective in declaratively enumerating hypotheses that probe the structure of an unknown probabilistic graphical causal model of the underlying system. Our thesis is that databases are in a unique position to enable users to rapidly explore the possible causal mechanisms in data collected from diverse sources. We empirically demonstrate how ExplainIt! had helped us resolve over 30 performance issues in a commercial product since late 2014, of which we discuss a few cases in detail.Comment: SIGMOD Industry Track 201