Ranking and clustering of nodes in networks with smart teleportation

Random teleportation is a necessary evil for ranking and clustering directed networks based on random walks. Teleportation enables ergodic solutions, but the solutions must necessarily depend on the exact implementation and parametrization of the teleportation. For example, in the commonly used PageRank algorithm, the teleportation rate must trade off a heavily biased solution with a uniform solution. Here we show that teleportation to links rather than nodes enables a much smoother trade-off and effectively more robust results. We also show that, by not recording the teleportation steps of the random walker, we can further reduce the effect of teleportation with dramatic effects on clustering.Comment: 10 pages, 7 figure

The Number of Convex Permutominoes

Permutominoes are polyominoes defined by suitable pairs of permutations. In this paper we provide a formula to count the number of convex permutominoes of given perimeter. To this aim we define the transform of a generic pair of permutations, we characterize the transform of any pair defining a convex permutomino, and we solve the counting problem in the transformed space

Efficiently Clustering Very Large Attributed Graphs

Attributed graphs model real networks by enriching their nodes with attributes accounting for properties. Several techniques have been proposed for partitioning these graphs into clusters that are homogeneous with respect to both semantic attributes and to the structure of the graph. However, time and space complexities of state of the art algorithms limit their scalability to medium-sized graphs. We propose SToC (for Semantic-Topological Clustering), a fast and scalable algorithm for partitioning large attributed graphs. The approach is robust, being compatible both with categorical and with quantitative attributes, and it is tailorable, allowing the user to weight the semantic and topological components. Further, the approach does not require the user to guess in advance the number of clusters. SToC relies on well known approximation techniques such as bottom-k sketches, traditional graph-theoretic concepts, and a new perspective on the composition of heterogeneous distance measures. Experimental results demonstrate its ability to efficiently compute high-quality partitions of large scale attributed graphs.Comment: This work has been published in ASONAM 2017. This version includes an appendix with validation of our attribute model and distance function, omitted in the converence version for lack of space. Please refer to the published versio

Local Ranking Problem on the BrowseGraph

The "Local Ranking Problem" (LRP) is related to the computation of a centrality-like rank on a local graph, where the scores of the nodes could significantly differ from the ones computed on the global graph. Previous work has studied LRP on the hyperlink graph but never on the BrowseGraph, namely a graph where nodes are webpages and edges are browsing transitions. Recently, this graph has received more and more attention in many different tasks such as ranking, prediction and recommendation. However, a web-server has only the browsing traffic performed on its pages (local BrowseGraph) and, as a consequence, the local computation can lead to estimation errors, which hinders the increasing number of applications in the state of the art. Also, although the divergence between the local and global ranks has been measured, the possibility of estimating such divergence using only local knowledge has been mainly overlooked. These aspects are of great interest for online service providers who want to: (i) gauge their ability to correctly assess the importance of their resources only based on their local knowledge, and (ii) take into account real user browsing fluxes that better capture the actual user interest than the static hyperlink network. We study the LRP problem on a BrowseGraph from a large news provider, considering as subgraphs the aggregations of browsing traces of users coming from different domains. We show that the distance between rankings can be accurately predicted based only on structural information of the local graph, being able to achieve an average rank correlation as high as 0.8

Random Surfing Without Teleportation

In the standard Random Surfer Model, the teleportation matrix is necessary to ensure that the final PageRank vector is well-defined. The introduction of this matrix, however, results in serious problems and imposes fundamental limitations to the quality of the ranking vectors. In this work, building on the recently proposed NCDawareRank framework, we exploit the decomposition of the underlying space into blocks, and we derive easy to check necessary and sufficient conditions for random surfing without teleportation.Comment: 13 pages. Published in the Volume: "Algorithms, Probability, Networks and Games, Springer-Verlag, 2015". (The updated version corrects small typos/errors

Criteria for Elves and Sprites on Schumann Resonance Observations

Ground flashes with positive polarity associated with both sprites and elves excite the Earth\u27s Schumann resonances to amplitudes several times greater than the background resonances. Theoretical predictions for dielectric breakdown in the mesosphere are tested using ELF methods to evaluate vertical charge moments of positive ground flashes. Comparisons of the measured time constants for lightning charge transfer with the electrostatic relaxation time at altitudes of nighttime sprite initiation (50–70 km) generally validate the electrostatic assumption in predictions made initially by Wilson [1925]. The measured charge moments (Q dS = 200–2000 C-km) are large in comparison with ordinary negative lightning but are generally insufficient to account for conventional air breakdown at sprite altitudes. The measured charge moments, however, are sufficient to account for electron runaway breakdown, and the long avalanche length in this mechanism also accounts for the exclusive association of sprites with ground flashes of positive polarity. The association of elves with large peak currents (50–200 kA) measured by the National Lightning Detection Network in a band pass beyond the Schumann resonance range is consistent with an electromagnetic pulse mechanism for these events

Fast matrix computations for pair-wise and column-wise commute times and Katz scores

We first explore methods for approximating the commute time and Katz score between a pair of nodes. These methods are based on the approach of matrices, moments, and quadrature developed in the numerical linear algebra community. They rely on the Lanczos process and provide upper and lower bounds on an estimate of the pair-wise scores. We also explore methods to approximate the commute times and Katz scores from a node to all other nodes in the graph. Here, our approach for the commute times is based on a variation of the conjugate gradient algorithm, and it provides an estimate of all the diagonals of the inverse of a matrix. Our technique for the Katz scores is based on exploiting an empirical localization property of the Katz matrix. We adopt algorithms used for personalized PageRank computing to these Katz scores and theoretically show that this approach is convergent. We evaluate these methods on 17 real world graphs ranging in size from 1000 to 1,000,000 nodes. Our results show that our pair-wise commute time method and column-wise Katz algorithm both have attractive theoretical properties and empirical performance.Comment: 35 pages, journal version of http://dx.doi.org/10.1007/978-3-642-18009-5_13 which has been submitted for publication. Please see http://www.cs.purdue.edu/homes/dgleich/publications/2011/codes/fast-katz/ for supplemental code

Inversion of Multi-Station Schumann Resonance Background Records for Global Lightning Activity in Absolute Units

Every lightning flash contributes energy to the TEM mode of the natural global waveguide that contains the Earth’s Schumann resonances. The modest attenuation at ELF (0.1 dB/Mm) allows for the continuous monitoring of the global lightning with a small number of receiving stations worldwide. In this study, nine ELF receiving sites (in Antarctica (3 sites), Hungary, India, Japan, Poland, Spitsbergen and USA) are used to provide power spectra at 12-minute intervals in two absolutely calibrated magnetic fields and occasionally, one electric field, with up to five resonance modes each. The observables are the extracted modal parameters (peak intensity, peak frequency and Q-factor) for each spectrum. The unknown quantities are the geographical locations of three continental lightning ‘chimneys’ and their lightning source strengths in absolute units (C2 km2/sec). The unknowns are calculated from the observables by the iterative inversion of an evolving ‘sensitivity matrix’ whose elements are the partial derivatives of each observable for all receiving sites with respect to each unknown quantity. The propagation model includes the important day-night asymmetry of the natural waveguide. To overcome the problem of multiple minima (common in inversion problems of this kind), location information from the World Wide Lightning Location Network has been used to make initial guess solutions based on centroids of stroke locations in each chimney. Results for five consecutive days in 2009 (Jan 7-11) show UT variations with the African chimney dominating on four of five days, and America dominating on the fifth day. The amplitude variations in absolute source strength exceed that of the ‘Carnegie curve’ of the DC global circuit by roughly twofold. Day-to-day variations in chimney source strength are of the order of tens of percent. Examination of forward calculations performed with the global inversion solution often show good agreement with the observed diurnal variations at individual receiving sites, lending confidence to the 3-chimney model for global lightning

Aliasing of the Schumann resonance background signal by sprite-associated Q-bursts

The Earth's naturally occurring Schumann resonances (SR) are composed of a quasi-continuous background component and a larger-amplitude, short-duration transient component, otherwise called ‘Q-burst’ (Ogawa et al., 1967). Sprites in the mesosphere are also known to accompany the energetic positive ground flashes that launch the Q-bursts (Boccippio et al., 1995). Spectra of the background Schumann Resonances (SR) require a natural stabilization period of ~10–12 min for the three conspicuous modal parameters to be derived from Lorentzian fitting. Before the spectra are computed and the fitting process is initiated, the raw time series data need to be properly filtered for local cultural noise, narrow band interference as well as for large transients in the form of global Q-bursts. Mushtak and Williams (2009) describe an effective technique called Isolated Lorentzian (I-LOR), in which, the contributions from local cultural and various other noises are minimized to a great extent. An automated technique based on median filtering of time series data has been developed. These special lightning flashes are known to have greater contribution in the ELF range (below 1 kHz) compared to general negative CG strikes (Huang et al., 1999; Cummer et al., 2006). The global distributions of these Q-bursts have been studied by Huang et al. (1999) Rhode Island, USA by wave impedance methods from single station ELF measurements at Rhode Island, USA and from Japan Hobara et al. (2006). The present work aims to demonstrate the effect of Q- bursts on SR background spectra using GPS time-stamped observation of TLEs. It is observed that the Q-bursts selected for the present work do alias the background spectra over a 5-s period, though the amplitudes of these Q- bursts are far below the background threshold of 16 Core Standard Deviation (CSD) so that they do not strongly alias the background spectra of 10–12 min duration. The examination of one exceptional Q-burst shows that appreciable spectral aliasing can occur even when 12-min spectral integrations are considered. The statistical result shows that for a 12-min spectrum, events above 16 CSD are capable of producing significant frequency aliasing of the modal frequencies, although the intensity aliasing might have a negligible effect unless the events are exceptionally large (~200 CSD). The spectral CSD methodology may be used to extract the time of arrival of the Q-burst transients. This methodology may be combined with a hyperbolic ranging, thus becoming an effective tool to detect TLEs globally with a modest number of networked observational stations.Peer ReviewedPostprint (published version