Encoding dynamics for multiscale community detection: Markov time sweeping for the Map equation

The detection of community structure in networks is intimately related to finding a concise description of the network in terms of its modules. This notion has been recently exploited by the Map equation formalism (M. Rosvall and C.T. Bergstrom, PNAS, 105(4), pp.1118--1123, 2008) through an information-theoretic description of the process of coding inter- and intra-community transitions of a random walker in the network at stationarity. However, a thorough study of the relationship between the full Markov dynamics and the coding mechanism is still lacking. We show here that the original Map coding scheme, which is both block-averaged and one-step, neglects the internal structure of the communities and introduces an upper scale, the `field-of-view' limit, in the communities it can detect. As a consequence, Map is well tuned to detect clique-like communities but can lead to undesirable overpartitioning when communities are far from clique-like. We show that a signature of this behavior is a large compression gap: the Map description length is far from its ideal limit. To address this issue, we propose a simple dynamic approach that introduces time explicitly into the Map coding through the analysis of the weighted adjacency matrix of the time-dependent multistep transition matrix of the Markov process. The resulting Markov time sweeping induces a dynamical zooming across scales that can reveal (potentially multiscale) community structure above the field-of-view limit, with the relevant partitions indicated by a small compression gap.Comment: 10 pages, 6 figure

On generalized processor sharing and objective functions: analytical framework

Today, telecommunication networks host a wide range of heterogeneous services. Some demand strict delay minima, while others only need a best-effort kind of service. To achieve service differentiation, network traffic is partitioned in several classes which is then transmitted according to a flexible and fair scheduling mechanism. Telecommunication networks can, for instance, use an implementation of Generalized Processor Sharing (GPS) in its internal nodes to supply an adequate Quality of Service to each class. GPS is flexible and fair, but also notoriously hard to study analytically. As a result, one has to resort to simulation or approximation techniques to optimize GPS for some given objective function. In this paper, we set up an analytical framework for two-class discrete-time probabilistic GPS which allows to optimize the scheduling for a generic objective function in terms of the mean unfinished work of both classes without the need for exact results or estimations/approximations for these performance characteristics. This framework is based on results of strict priority scheduling, which can be regarded as a special case of GPS, and some specific unfinished-work properties in two-class GPS. We also apply our framework on a popular type of objective functions, i.e., convex combinations of functions of the mean unfinished work. Lastly, we incorporate the framework in an algorithm to yield a faster and less computation-intensive result for the optimum of an objective function

Isotopic evidence for biogenic molecular hydrogen production in the Atlantic Ocean

Oceans are a net source of molecular hydrogen (H2) to the atmosphere. The production of marine H2 is assumed to be mainly biological by N2 fixation, but photochemical pathways are also discussed. We present measurements of mole fraction and isotopic composition of dissolved and atmospheric H2 from the southern and northern Atlantic between 2008 and 2010. In total almost 400 samples were taken during five cruises along a transect between Punta Arenas (Chile) and Bremerhaven (Germany), as well as at the coast of Mauretania. The isotopic source signatures of dissolved H2 extracted from surface water are highly deuterium-depleted and correlate negatively with temperature, showing δD values of (−629 ± 54) ‰ for water temperatures at (27 ± 3) °C and (−249 ± 88) ‰ below (19 ± 1) °C. The results for warmer water masses are consistent with biological production of H2. This is the first time that marine H2 excess has been directly attributed to biological production by isotope measurements. However, the isotope values obtained in the colder water masses indicate that beside possible biological production a significant different source should be considered. The atmospheric measurements show distinct differences between both hemispheres as well as between seasons. Results from the global chemistry transport model TM5 reproduce the measured H2 mole fractions and isotopic composition well. The climatological global oceanic emissions from the GEMS database are in line with our data and previously published flux calculations. The good agreement between measurements and model results demonstrates that both the magnitude and the isotopic signature of the main components of the marine H2 cycle are in general adequately represented in current atmospheric models despite a proposed source different from biological production or a substantial underestimation of nitrogen fixation by several authors

Finding community structure in very large networks

The discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community, but most methods proposed so far are unsuitable for very large networks because of their computational cost. Here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms: its running time on a network with n vertices and m edges is O(m d log n) where d is the depth of the dendrogram describing the community structure. Many real-world networks are sparse and hierarchical, with m ~ n and d ~ log n, in which case our algorithm runs in essentially linear time, O(n log^2 n). As an example of the application of this algorithm we use it to analyze a network of items for sale on the web-site of a large online retailer, items in the network being linked if they are frequently purchased by the same buyer. The network has more than 400,000 vertices and 2 million edges. We show that our algorithm can extract meaningful communities from this network, revealing large-scale patterns present in the purchasing habits of customers

Structural Examination of Au/Ge(001) by Surface X-Ray Diffraction and Scanning Tunneling Microscopy

The one-dimensional reconstruction of Au/Ge(001) was investigated by means of autocorrelation functions from surface x-ray diffraction (SXRD) and scanning tunneling microscopy (STM). Interatomic distances found in the SXRD-Patterson map are substantiated by results from STM. The Au coverage, recently determined to be 3/4 of a monolayer of gold, together with SXRD leads to three non-equivalent positions for Au within the c(8x2) unit cell. Combined with structural information from STM topography and line profiling, two building blocks are identified: Au-Ge hetero-dimers within the top wire architecture and Au homo-dimers within the trenches. The incorporation of both components is discussed using density functional theory and model based Patterson maps by substituting Germanium atoms of the reconstructed Ge(001) surface.Comment: 5 pages, 3 figure

A measure of centrality based on the spectrum of the Laplacian

We introduce a family of new centralities, the k-spectral centralities. k-Spectral centrality is a measurement of importance with respect to the deformation of the graph Laplacian associated with the graph. Due to this connection, k-spectral centralities have various interpretations in terms of spectrally determined information. We explore this centrality in the context of several examples. While for sparse unweighted networks 1-spectral centrality behaves similarly to other standard centralities, for dense weighted networks they show different properties. In summary, the k-spectral centralities provide a novel and useful measurement of relevance (for single network elements as well as whole subnetworks) distinct from other known measures.Comment: 12 pages, 6 figures, 2 table

Dynamic Computation of Network Statistics via Updating Schema

In this paper we derive an updating scheme for calculating some important network statistics such as degree, clustering coefficient, etc., aiming at reduce the amount of computation needed to track the evolving behavior of large networks; and more importantly, to provide efficient methods for potential use of modeling the evolution of networks. Using the updating scheme, the network statistics can be computed and updated easily and much faster than re-calculating each time for large evolving networks. The update formula can also be used to determine which edge/node will lead to the extremal change of network statistics, providing a way of predicting or designing evolution rule of networks.Comment: 17 pages, 6 figure

Optical study of superconducting Ga-rich layers in silicon

We performed phase-sensitive terahertz (0.12 - 1.2 THz) transmission measurements of Ga-enriched layers in silicon. Below the superconducting transition, T_{c} = 6.7 K, we find clear signatures of the formation of a superconducting condensate and of the opening of an energy gap in the optical spectra. The London penetration depth, \lambda(T), and the condensate density, n_{s} = \lambda^{2} 0)/\lambda^{2}(T), as functions of temperature demonstrate behavior, typical for conventional superconductors with \lambda(0) = 1.8 \mu m. The terahertz spectra can be well described within the framework of Eliashberg theory with strong electron-phonon coupling: the zero-temperature energy gap is 2\Delta(0) = 2.64 meV and 2\Delta(0)/k_{B}T_{c} = 4.6 \pm 0.1, consistent with the amorphous state of Ga. At temperatures just above T_{c}, the optical spectra demonstrate Drude behavior.Comment: 5 pages, 4 figure